[ { "description": "JEE Adv 2016 Paper 1", "index": 1, "subject": "phy", "type": "MCQ", "question": "In a historical experiment to determine Planck's constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength $(\\lambda)$ of incident light and the corresponding stopping potential $\\left(V_{0}\\right)$ are given below :\n\n\\begin{center}\n\n\\begin{tabular}{cc}\n\n\\hline\n\n$\\lambda(\\mu \\mathrm{m})$ & $V_{0}($ Volt $)$ \\\\\n\n\\hline\n\n0.3 & 2.0 \\\\\n\n0.4 & 1.0 \\\\\n\n0.5 & 0.4 \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}\n\nGiven that $c=3 \\times 10^{8} \\mathrm{~m} \\mathrm{~s}^{-1}$ and $e=1.6 \\times 10^{-19} \\mathrm{C}$, Planck's constant (in units of $J \\mathrm{~s}$ ) found from such an experiment is\n\n(A) $6.0 \\times 10^{-34}$\n\n(B) $6.4 \\times 10^{-34}$\n\n(C) $6.6 \\times 10^{-34}$\n\n(D) $6.8 \\times 10^{-34}$", "gold": "B" }, { "description": "JEE Adv 2016 Paper 1", "index": 2, "subject": "phy", "type": "MCQ", "question": "A uniform wooden stick of mass $1.6 \\mathrm{~kg}$ and length $l$ rests in an inclined manner on a smooth, vertical wall of height $h(20$. For an unstable nucleus having $N / P$ ratio less than 1 , the possible mode(s) of decay is(are)\n\n(A) $\\beta^{-}$-decay $(\\beta$ emission)\n\n(B) orbital or $K$-electron capture\n\n(C) neutron emission\n\n(D) $\\beta^{+}$-decay (positron emission)", "gold": "BD" }, { "description": "JEE Adv 2016 Paper 1", "index": 26, "subject": "chem", "type": "MCQ(multiple)", "question": "The crystalline form of borax has\n\n(A) tetranuclear $\\left[\\mathrm{B}_{4} \\mathrm{O}_{5}(\\mathrm{OH})_{4}\\right]^{2-}$ unit\n\n(B) all boron atoms in the same plane\n\n(C) equal number of $s p^{2}$ and $s p^{3}$ hybridized boron atoms\n\n(D) one terminal hydroxide per boron atom", "gold": "ACD" }, { "description": "JEE Adv 2016 Paper 1", "index": 27, "subject": "chem", "type": "MCQ(multiple)", "question": "The compound(s) with TWO lone pairs of electrons on the central atom is(are)\n\n(A) $\\mathrm{BrF}_{5}$\n\n(B) $\\mathrm{ClF}_{3}$\n\n(C) $\\mathrm{XeF}_{4}$\n\n(D) $\\mathrm{SF}_{4}$", "gold": "BC" }, { "description": "JEE Adv 2016 Paper 1", "index": 28, "subject": "chem", "type": "MCQ(multiple)", "question": "The reagent(s) that can selectively precipitate $\\mathrm{S}^{2-}$ from a mixture of $\\mathrm{S}^{2-}$ and $\\mathrm{SO}_{4}^{2-}$ in aqueous solution is(are)\n\n(A) $\\mathrm{CuCl}_{2}$\n\n(B) $\\mathrm{BaCl}_{2}$\n\n(C) $\\mathrm{Pb}\\left(\\mathrm{OOCCH}_{3}\\right)_{2}$\n\n(D) $\\mathrm{Na}_{2}\\left[\\mathrm{Fe}(\\mathrm{CN})_{5} \\mathrm{NO}\\right]$", "gold": "A" }, { "description": "JEE Adv 2016 Paper 1", "index": 32, "subject": "chem", "type": "Integer", "question": "The mole fraction of a solute in a solution is 0.1 . At $298 \\mathrm{~K}$, molarity of this solution is the same as its molality. Density of this solution at $298 \\mathrm{~K}$ is $2.0 \\mathrm{~g} \\mathrm{~cm}^{-3}$. What is the ratio of the molecular weights of the solute and solvent, $\\left(\\frac{M W_{\\text {solute }}}{M W_{\\text {solvent }}}\\right)$?", "gold": "9" }, { "description": "JEE Adv 2016 Paper 1", "index": 33, "subject": "chem", "type": "Integer", "question": "The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the diffusion coefficient of this gas increases $x$ times. What is the value of $x$?", "gold": "4" }, { "description": "JEE Adv 2016 Paper 1", "index": 34, "subject": "chem", "type": "Integer", "question": "In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitatively oxidize thiosulphate anions to produce $\\mathbf{X}$ moles of a sulphur containing product. What is the magnitude of $\\mathbf{X}$?", "gold": "6" }, { "description": "JEE Adv 2016 Paper 1", "index": 37, "subject": "math", "type": "MCQ", "question": "Let $-\\frac{\\pi}{6}<\\theta<-\\frac{\\pi}{12}$. Suppose $\\alpha_{1}$ and $\\beta_{1}$ are the roots of the equation $x^{2}-2 x \\sec \\theta+1=0$ and $\\alpha_{2}$ and $\\beta_{2}$ are the roots of the equation $x^{2}+2 x \\tan \\theta-1=0$. If $\\alpha_{1}>\\beta_{1}$ and $\\alpha_{2}>\\beta_{2}$, then $\\alpha_{1}+\\beta_{2}$ equals\n\n(A) $2(\\sec \\theta-\\tan \\theta)$\n\n(B) $2 \\sec \\theta$\n\n(C) $-2 \\tan \\theta$\n\n(D) 0", "gold": "C" }, { "description": "JEE Adv 2016 Paper 1", "index": 38, "subject": "math", "type": "MCQ", "question": "A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is\n\n(A) 380\n\n(B) 320\n\n(C) 260\n\n(D) 95", "gold": "A" }, { "description": "JEE Adv 2016 Paper 1", "index": 39, "subject": "math", "type": "MCQ", "question": "Let $S=\\left\\{x \\in(-\\pi, \\pi): x \\neq 0, \\pm \\frac{\\pi}{2}\\right\\}$. The sum of all distinct solutions of the equation $\\sqrt{3} \\sec x+\\operatorname{cosec} x+2(\\tan x-\\cot x)=0$ in the set $S$ is equal to\n\n(A) $-\\frac{7 \\pi}{9}$\n\n(B) $-\\frac{2 \\pi}{9}$\n\n(C) 0\n\n(D) $\\frac{5 \\pi}{9}$", "gold": "C" }, { "description": "JEE Adv 2016 Paper 1", "index": 40, "subject": "math", "type": "MCQ", "question": "A computer producing factory has only two plants $T_{1}$ and $T_{2}$. Plant $T_{1}$ produces $20 \\%$ and plant $T_{2}$ produces $80 \\%$ of the total computers produced. $7 \\%$ of computers produced in the factory turn out to be defective. It is known that\n\n$P$ (computer turns out to be defective given that it is produced in plant $T_{1}$ )\n\n$=10 P\\left(\\right.$ computer turns out to be defective given that it is produced in plant $\\left.T_{2}\\right)$,\n\nwhere $P(E)$ denotes the probability of an event $E$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_{2}$ is\n\n(A) $\\frac{36}{73}$\n\n(B) $\\frac{47}{79}$\n\n(C) $\\frac{78}{93}$\n\n(D) $\\frac{75}{83}$", "gold": "C" }, { "description": "JEE Adv 2016 Paper 1", "index": 41, "subject": "math", "type": "MCQ", "question": "The least value of $\\alpha \\in \\mathbb{R}$ for which $4 \\alpha x^{2}+\\frac{1}{x} \\geq 1$, for all $x>0$, is\n\n(A) $\\frac{1}{64}$\n\n(B) $\\frac{1}{32}$\n\n(C) $\\frac{1}{27}$\n\n(D) $\\frac{1}{25}$", "gold": "C" }, { "description": "JEE Adv 2016 Paper 1", "index": 42, "subject": "math", "type": "MCQ(multiple)", "question": "Consider a pyramid $O P Q R S$ located in the first octant $(x \\geq 0, y \\geq 0, z \\geq 0)$ with $O$ as origin, and $O P$ and $O R$ along the $x$-axis and the $y$-axis, respectively. The base $O P Q R$ of the pyramid is a square with $O P=3$. The point $S$ is directly above the mid-point $T$ of diagonal $O Q$ such that $T S=3$. Then\n\n(A) the acute angle between $O Q$ and $O S$ is $\\frac{\\pi}{3}$\n\n(B) the equation of the plane containing the triangle $O Q S$ is $x-y=0$\n\n(C) the length of the perpendicular from $P$ to the plane containing the triangle $O Q S$ is $\\frac{3}{\\sqrt{2}}$\n\n(D) the perpendicular distance from $O$ to the straight line containing $R S$ is $\\sqrt{\\frac{15}{2}}$", "gold": "BCD" }, { "description": "JEE Adv 2016 Paper 1", "index": 43, "subject": "math", "type": "MCQ(multiple)", "question": "Let $f:(0, \\infty) \\rightarrow \\mathbb{R}$ be a differentiable function such that $f^{\\prime}(x)=2-\\frac{f(x)}{x}$ for all $x \\in(0, \\infty)$ and $f(1) \\neq 1$. Then\n\n(A) $\\lim _{x \\rightarrow 0+} f^{\\prime}\\left(\\frac{1}{x}\\right)=1$\n\n(B) $\\lim _{x \\rightarrow 0+} x f\\left(\\frac{1}{x}\\right)=2$\n\n(C) $\\lim _{x \\rightarrow 0+} x^{2} f^{\\prime}(x)=0$\n\n(D) $|f(x)| \\leq 2$ for all $x \\in(0,2)$", "gold": "A" }, { "description": "JEE Adv 2016 Paper 1", "index": 44, "subject": "math", "type": "MCQ(multiple)", "question": "Let $P=\\left[\\begin{array}{ccc}3 & -1 & -2 \\\\ 2 & 0 & \\alpha \\\\ 3 & -5 & 0\\end{array}\\right]$, where $\\alpha \\in \\mathbb{R}$. Suppose $Q=\\left[q_{i j}\\right]$ is a matrix such that $P Q=k I$, where $k \\in \\mathbb{R}, k \\neq 0$ and $I$ is the identity matrix of order 3 . If $q_{23}=-\\frac{k}{8}$ and $\\operatorname{det}(Q)=\\frac{k^{2}}{2}$, then\n\n(A) $\\alpha=0, k=8$\n\n(B) $4 \\alpha-k+8=0$\n\n(C) $\\operatorname{det}(P \\operatorname{adj}(Q))=2^{9}$\n\n(D) $\\operatorname{det}(Q \\operatorname{adj}(P))=2^{13}$", "gold": "BC" }, { "description": "JEE Adv 2016 Paper 1", "index": 45, "subject": "math", "type": "MCQ(multiple)", "question": "In a triangle $X Y Z$, let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$, respectively, and $2 s=x+y+z$. If $\\frac{s-x}{4}=\\frac{s-y}{3}=\\frac{s-z}{2}$ and area of incircle of the triangle $X Y Z$ is $\\frac{8 \\pi}{3}$, then\n\n(A) area of the triangle $X Y Z$ is $6 \\sqrt{6}$\n\n(B) the radius of circumcircle of the triangle $X Y Z$ is $\\frac{35}{6} \\sqrt{6}$\n\n(C) $\\sin \\frac{X}{2} \\sin \\frac{Y}{2} \\sin \\frac{Z}{2}=\\frac{4}{35}$\n\n(D) $\\sin ^{2}\\left(\\frac{X+Y}{2}\\right)=\\frac{3}{5}$", "gold": "ACD" }, { "description": "JEE Adv 2016 Paper 1", "index": 46, "subject": "math", "type": "MCQ(multiple)", "question": "A solution curve of the differential equation $\\left(x^{2}+x y+4 x+2 y+4\\right) \\frac{d y}{d x}-y^{2}=0, x>0$, passes through the point $(1,3)$. Then the solution curve\n\n(A) intersects $y=x+2$ exactly at one point\n\n(B) intersects $y=x+2$ exactly at two points\n\n(C) intersects $y=(x+2)^{2}$\n\n(D) does NO'T intersect $y=(x+3)^{2}$", "gold": "AD" }, { "description": "JEE Adv 2016 Paper 1", "index": 47, "subject": "math", "type": "MCQ(multiple)", "question": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}, \\quad g: \\mathbb{R} \\rightarrow \\mathbb{R}$ and $h: \\mathbb{R} \\rightarrow \\mathbb{R}$ be differentiable functions such that $f(x)=x^{3}+3 x+2, g(f(x))=x$ and $h(g(g(x)))=x$ for all $x \\in \\mathbb{R}$. Then\n\n(A) $\\quad g^{\\prime}(2)=\\frac{1}{15}$\n\n(B) $h^{\\prime}(1)=666$\n\n(C) $h(0)=16$\n\n(D) $h(g(3))=36$", "gold": "BC" }, { "description": "JEE Adv 2016 Paper 1", "index": 48, "subject": "math", "type": "MCQ(multiple)", "question": "The circle $C_{1}: x^{2}+y^{2}=3$, with centre at $O$, intersects the parabola $x^{2}=2 y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_{1}$ at $P$ touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3}$, respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $2 \\sqrt{3}$ and centres $Q_{2}$ and $Q_{3}$, respectively. If $Q_{2}$ and $Q_{3}$ lie on the $y$-axis, then\n\n(A) $Q_{2} Q_{3}=12$\n\n(B) $\\quad R_{2} R_{3}=4 \\sqrt{6}$\n\n(C) area of the triangle $O R_{2} R_{3}$ is $6 \\sqrt{2}$\n\n(D) area of the triangle $P Q_{2} Q_{3}$ is $4 \\sqrt{2}$", "gold": "ABC" }, { "description": "JEE Adv 2016 Paper 1", "index": 49, "subject": "math", "type": "MCQ(multiple)", "question": "Let $R S$ be the diameter of the circle $x^{2}+y^{2}=1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$ ) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $R S$ at point $E$. Then the locus of $E$ passes through the point(s)\n\n(A) $\\left(\\frac{1}{3}, \\frac{1}{\\sqrt{3}}\\right)$\n\n(B) $\\left(\\frac{1}{4}, \\frac{1}{2}\\right)$\n\n(C) $\\left(\\frac{1}{3},-\\frac{1}{\\sqrt{3}}\\right)$\n\n(D) $\\left(\\frac{1}{4},-\\frac{1}{2}\\right)$", "gold": "AC" }, { "description": "JEE Adv 2016 Paper 1", "index": 50, "subject": "math", "type": "Integer", "question": "What is the total number of distinct $x \\in \\mathbb{R}$ for which $\\left|\\begin{array}{ccc}x & x^{2} & 1+x^{3} \\\\ 2 x & 4 x^{2} & 1+8 x^{3} \\\\ 3 x & 9 x^{2} & 1+27 x^{3}\\end{array}\\right|=10$?", "gold": "2" }, { "description": "JEE Adv 2016 Paper 1", "index": 51, "subject": "math", "type": "Integer", "question": "Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of $(1+x)^{2}+(1+x)^{3}+\\cdots+(1+x)^{49}+(1+m x)^{50}$ is $(3 n+1){ }^{51} C_{3}$ for some positive integer $n$. Then what is the value of $n$?", "gold": "5" }, { "description": "JEE Adv 2016 Paper 1", "index": 52, "subject": "math", "type": "Integer", "question": "What is the total number of distinct $x \\in[0,1]$ for which $\\int_{0}^{x} \\frac{t^{2}}{1+t^{4}} d t=2 x-1$?", "gold": "1" }, { "description": "JEE Adv 2016 Paper 1", "index": 53, "subject": "math", "type": "Integer", "question": "Let $\\alpha, \\beta \\in \\mathbb{R}$ be such that $\\lim _{x \\rightarrow 0} \\frac{x^{2} \\sin (\\beta x)}{\\alpha x-\\sin x}=1$.Then what is the value of $6(\\alpha+\\beta)$?", "gold": "7" }, { "description": "JEE Adv 2016 Paper 1", "index": 54, "subject": "math", "type": "Integer", "question": "Let $z=\\frac{-1+\\sqrt{3} i}{2}$, where $i=\\sqrt{-1}$, and $r, s \\in\\{1,2,3\\}$. Let $P=\\left[\\begin{array}{cc}(-z)^{r} & z^{2 s} \\\\ z^{2 s} & z^{r}\\end{array}\\right]$ and $I$ be the identity matrix of order 2 . Then what is the total number of ordered pairs $(r, s)$ for which $P^{2}=-I$?", "gold": "1" }, { "description": "JEE Adv 2016 Paper 2", "index": 1, "subject": "phy", "type": "MCQ", "question": "The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by\n\n\$\nE=\\frac{3}{5} \\frac{Z(Z-1) e^{2}}{4 \\pi \\varepsilon_{0} R}\n\$\n\nThe measured masses of the neutron, ${ }_{1}^{1} \\mathrm{H},{ }_{7}^{15} \\mathrm{~N}$ and ${ }_{8}^{15} \\mathrm{O}$ are $1.008665 \\mathrm{u}, 1.007825 \\mathrm{u}$, $15.000109 \\mathrm{u}$ and $15.003065 \\mathrm{u}$, respectively. Given that the radii of both the ${ }_{7}^{15} \\mathrm{~N}$ and ${ }_{8}^{15} \\mathrm{O}$ nuclei are same, $1 \\mathrm{u}=931.5 \\mathrm{MeV} / c^{2}$ ( $c$ is the speed of light) and $e^{2} /\\left(4 \\pi \\varepsilon_{0}\\right)=1.44 \\mathrm{MeV} \\mathrm{fm}$. Assuming that the difference between the binding energies of ${ }_{7}^{15} \\mathrm{~N}$ and ${ }_{8}^{15} \\mathrm{O}$ is purely due to the electrostatic energy, the radius of either of the nuclei is\n\n$\\left(1 \\mathrm{fm}=10^{-15} \\mathrm{~m}\\right)$\n\n(A) $2.85 \\mathrm{fm}$\n\n(B) $3.03 \\mathrm{fm}$\n\n(C) $3.42 \\mathrm{fm}$\n\n(D) $3.80 \\mathrm{fm}$", "gold": "C" }, { "description": "JEE Adv 2016 Paper 2", "index": 2, "subject": "phy", "type": "MCQ", "question": "An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?\n\n(A) 64\n\n(B) 90\n\n(C) 108\n\n(D) 120", "gold": "C" }, { "description": "JEE Adv 2016 Paper 2", "index": 3, "subject": "phy", "type": "MCQ", "question": "A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $P_{i}=10^{5} \\mathrm{~Pa}$ and volume $V_{i}=10^{-3} \\mathrm{~m}^{3}$ changes to a final state at $P_{f}=(1 / 32) \\times 10^{5} \\mathrm{~Pa}$ and $V_{f}=8 \\times 10^{-3} \\mathrm{~m}^{3}$ in an adiabatic quasi-static process, such that $P^{3} V^{5}=$ constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $P_{i}$ followed by an isochoric (isovolumetric) process at volume $V_{f}$. The amount of heat supplied to the system in the two-step process is approximately\n\n(A) $112 \\mathrm{~J}$\n\n(B) $294 \\mathrm{~J}$\n\n(C) $588 \\mathrm{~J}$\n\n(D) $813 \\mathrm{~J}$", "gold": "C" }, { "description": "JEE Adv 2016 Paper 2", "index": 4, "subject": "phy", "type": "MCQ", "question": "The ends $\\mathrm{Q}$ and $\\mathrm{R}$ of two thin wires, $\\mathrm{PQ}$ and RS, are soldered (joined) together. Initially each of the wires has a length of $1 \\mathrm{~m}$ at $10^{\\circ} \\mathrm{C}$. Now the end $P$ is maintained at $10^{\\circ} \\mathrm{C}$, while the end $\\mathrm{S}$ is heated and maintained at $400^{\\circ} \\mathrm{C}$. The system is thermally insulated from its surroundings. If the thermal conductivity of wire $\\mathrm{PQ}$ is twice that of the wire $R S$ and the coefficient of linear thermal expansion of $\\mathrm{PQ}$ is $1.2 \\times 10^{-5} \\mathrm{~K}^{-1}$, the change in length of the wire $P Q$ is\n\n(A) $0.78 \\mathrm{~mm}$\n\n(B) $0.90 \\mathrm{~mm}$\n\n(C) $1.56 \\mathrm{~mm}$\n\n(D) $2.34 \\mathrm{~mm}$", "gold": "A" }, { "description": "JEE Adv 2016 Paper 2", "index": 9, "subject": "phy", "type": "MCQ(multiple)", "question": "In an experiment to determine the acceleration due to gravity $g$, the formula used for the time period of a periodic motion is $T=2 \\pi \\sqrt{\\frac{7(R-r)}{5 g}}$. The values of $R$ and $r$ are measured to be $(60 \\pm 1) \\mathrm{mm}$ and $(10 \\pm 1) \\mathrm{mm}$, respectively. In five successive measurements, the time period is found to be $0.52 \\mathrm{~s}, 0.56 \\mathrm{~s}, 0.57 \\mathrm{~s}, 0.54 \\mathrm{~s}$ and $0.59 \\mathrm{~s}$. The least count of the watch used for the measurement of time period is $0.01 \\mathrm{~s}$. Which of the following statement(s) is(are) true?\n\n(A) The error in the measurement of $r$ is $10 \\%$\n\n(B) The error in the measurement of $T$ is $3.57 \\%$\n\n(C) The error in the measurement of $T$ is $2 \\%$\n\n(D) The error in the determined value of $g$ is $11 \\%$", "gold": "ABD" }, { "description": "JEE Adv 2016 Paper 2", "index": 10, "subject": "phy", "type": "MCQ(multiple)", "question": "Consider two identical galvanometers and two identical resistors with resistance $R$. If the internal resistance of the galvanometers $R_{\\mathrm{C}}1$ for $i=1,2, \\ldots, 101$. Suppose $\\log _{e} b_{1}, \\log _{e} b_{2}, \\ldots, \\log _{e} b_{101}$ are in Arithmetic Progression (A.P.) with the common difference $\\log _{e} 2$. Suppose $a_{1}, a_{2}, \\ldots, a_{101}$ are in A.P. such that $a_{1}=b_{1}$ and $a_{51}=b_{51}$. If $t=b_{1}+b_{2}+\\cdots+b_{51}$ and $s=a_{1}+a_{2}+\\cdots+a_{51}$, then\n\n(A) $s>t$ and $a_{101}>b_{101}$\n\n(B) $s>t$ and $a_{101}b_{101}$\n\n(D) $sf(2)$\n\n(D) $f(x)-f^{\\prime \\prime}(x)=0$ for at least one $x \\in \\mathbb{R}$", "gold": "AD" }, { "description": "JEE Adv 2016 Paper 2", "index": 46, "subject": "math", "type": "MCQ(multiple)", "question": "Let $f:\\left[-\\frac{1}{2}, 2\\right] \\rightarrow \\mathbb{R}$ and $g:\\left[-\\frac{1}{2}, 2\\right] \\rightarrow \\mathbb{R}$ be functions defined by $f(x)=\\left[x^{2}-3\\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \\in \\mathbb{R}$. Then\n\n(A) $f$ is discontinuous exactly at three points in $\\left[-\\frac{1}{2}, 2\\right]$\n\n(B) $f$ is discontinuous exactly at four points in $\\left[-\\frac{1}{2}, 2\\right]$\n\n(C) $g$ is NOT differentiable exactly at four points in $\\left(-\\frac{1}{2}, 2\\right)$\n\n(D) $g$ is NOT differentiable exactly at five points in $\\left(-\\frac{1}{2}, 2\\right)$", "gold": "BC" }, { "description": "JEE Adv 2016 Paper 2", "index": 47, "subject": "math", "type": "MCQ(multiple)", "question": "Let $a, b \\in \\mathbb{R}$ and $a^{2}+b^{2} \\neq 0$. Suppose $S=\\left\\{z \\in \\mathbb{C}: z=\\frac{1}{a+i b t}, t \\in \\mathbb{R}, t \\neq 0\\right\\}$, where $i=\\sqrt{-1}$. If $z=x+i y$ and $z \\in S$, then $(x, y)$ lies on\n\n(A) the circle with radius $\\frac{1}{2 a}$ and centre $\\left(\\frac{1}{2 a}, 0\\right)$ for $a>0, b \\neq 0$\n\n(B) the circle with radius $-\\frac{1}{2 a}$ and centre $\\left(-\\frac{1}{2 a}, 0\\right)$ for $a<0, b \\neq 0$\n\n(C) the $x$-axis for $a \\neq 0, b=0$\n\n(D) the $y$-axis for $a=0, b \\neq 0$", "gold": "ACD" }, { "description": "JEE Adv 2016 Paper 2", "index": 48, "subject": "math", "type": "MCQ(multiple)", "question": "Let $P$ be the point on the parabola $y^{2}=4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2}+y^{2}-4 x-16 y+64=0$. Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then\n\n(A) $S P=2 \\sqrt{5}$\n\n(B) $S Q: Q P=(\\sqrt{5}+1): 2$\n\n(C) the $x$-intercept of the normal to the parabola at $P$ is 6\n\n(D) the slope of the tangent to the circle at $Q$ is $\\frac{1}{2}$", "gold": "ACD" }, { "description": "JEE Adv 2016 Paper 2", "index": 49, "subject": "math", "type": "MCQ(multiple)", "question": "Let $a, \\lambda, \\mu \\in \\mathbb{R}$. Consider the system of linear equations\n\n\\n\\begin{aligned}\n\n& a x+2 y=\\lambda \\\\\n\n& 3 x-2 y=\\mu\n\n\\end{aligned}\n\\n\nWhich of the following statement(s) is(are) correct?\n\n(A) If $a=-3$, then the system has infinitely many solutions for all values of $\\lambda$ and $\\mu$\n\n(B) If $a \\neq-3$, then the system has a unique solution for all values of $\\lambda$ and $\\mu$\n\n(C) If $\\lambda+\\mu=0$, then the system has infinitely many solutions for $a=-3$\n\n(D) If $\\lambda+\\mu \\neq 0$, then the system has no solution for $\\alpha=-3$", "gold": "BCD" }, { "description": "JEE Adv 2016 Paper 2", "index": 50, "subject": "math", "type": "MCQ(multiple)", "question": "Let $\\hat{u}=u_{1} \\hat{i}+u_{2} \\hat{j}+u_{3} \\hat{k}$ be a unit vector in $\\mathbb{R}^{3}$ and $\\hat{w}=\\frac{1}{\\sqrt{6}}(\\hat{i}+\\hat{j}+2 \\hat{k})$. Given that there exists a vector $\\vec{v}$ in $\\mathbb{R}^{3}$ such that $|\\hat{u} \\times \\vec{v}|=1$ and $\\hat{w} \\cdot(\\hat{u} \\times \\vec{v})=1$. Which of the following statement\u00eds) is(are) correct?\n\n(A) There is exactly one choice for such $\\vec{v}$\n\n(B) There are infinitely many choices for such $\\vec{v}$\n\n(C) If $\\hat{u}$ lies in the $x y$-plane then $\\left|u_{1}\\right|=\\left|u_{2}\\right|$\n\n(D) If $\\hat{u}$ lies in the $x z$-plane then $2\\left|u_{1}\\right|=\\left|u_{3}\\right|$", "gold": "BC" }, { "description": "JEE Adv 2017 Paper 1", "index": 1, "subject": "phy", "type": "MCQ(multiple)", "question": "A flat plate is moving normal to its plane through a gas under the action of a constant force $F$. The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?\n\n[A] The pressure difference between the leading and trailing faces of the plate is proportional to $u v$\n\n[B] The resistive force experienced by the plate is proportional to $v$\n\n[C] The plate will continue to move with constant non-zero acceleration, at all times\n\n[D] At a later time the external force $F$ balances the resistive force", "gold": "ABD" }, { "description": "JEE Adv 2017 Paper 1", "index": 4, "subject": "phy", "type": "MCQ(multiple)", "question": "A human body has a surface area of approximately $1 \\mathrm{~m}^{2}$. The normal body temperature is $10 \\mathrm{~K}$ above the surrounding room temperature $T_{0}$. Take the room temperature to be $T_{0}=300 \\mathrm{~K}$. For $T_{0}=300 \\mathrm{~K}$, the value of $\\sigma T_{0}^{4}=460 \\mathrm{Wm}^{-2}$ (where $\\sigma$ is the StefanBoltzmann constant). Which of the following options is/are correct?\n\n[A] The amount of energy radiated by the body in 1 second is close to 60 Joules\n\n[B] If the surrounding temperature reduces by a small amount $\\Delta T_{0} \\ll T_{0}$, then to maintain the same body temperature the same (living) human being needs to radiate $\\Delta W=4 \\sigma T_{0}^{3} \\Delta T_{0}$ more energy per unit time\n\n[C] Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation\n\n[D] If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths", "gold": "C" }, { "description": "JEE Adv 2017 Paper 1", "index": 7, "subject": "phy", "type": "MCQ(multiple)", "question": "For an isosceles prism of angle $A$ and refractive index $\\mu$, it is found that the angle of minimum deviation $\\delta_{m}=A$. Which of the following options is/are correct?\n\n[A] For the angle of incidence $i_{1}=A$, the ray inside the prism is parallel to the base of the prism\n\n[B] For this prism, the refractive index $\\mu$ and the angle of prism $A$ are related as $A=\\frac{1}{2} \\cos ^{-1}\\left(\\frac{\\mu}{2}\\right)$\n\n[C] At minimum deviation, the incident angle $i_{1}$ and the refracting angle $r_{1}$ at the first refracting surface are related by $r_{1}=\\left(i_{1} / 2\\right)$\n\n[D] For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is $i_{1}=\\sin ^{-1}\\left[\\sin A \\sqrt{4 \\cos ^{2} \\frac{A}{2}-1}-\\cos A\\right]$", "gold": "ACD" }, { "description": "JEE Adv 2017 Paper 1", "index": 8, "subject": "phy", "type": "Integer", "question": "A drop of liquid of radius $\\mathrm{R}=10^{-2} \\mathrm{~m}$ having surface tension $\\mathrm{S}=\\frac{0.1}{4 \\pi} \\mathrm{Nm}^{-1}$ divides itself into $K$ identical drops. In this process the total change in the surface energy $\\Delta U=10^{-3} \\mathrm{~J}$. If $K=10^{\\alpha}$ then what is the value of $\\alpha$?", "gold": "6" }, { "description": "JEE Adv 2017 Paper 1", "index": 9, "subject": "phy", "type": "Integer", "question": "An electron in a hydrogen atom undergoes a transition from an orbit with quantum number $n_{i}$ to another with quantum number $n_{f} . V_{i}$ and $V_{f}$ are respectively the initial and final potential energies of the electron. If $\\frac{V_{i}}{V_{f}}=6.25$, then what is the smallest possible $n_{f}$?", "gold": "5" }, { "description": "JEE Adv 2017 Paper 1", "index": 11, "subject": "phy", "type": "Integer", "question": "A stationary source emits sound of frequency $f_{0}=492 \\mathrm{~Hz}$. The sound is reflected by a large car approaching the source with a speed of $2 \\mathrm{~ms}^{-1}$. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is $330 \\mathrm{~ms}^{-1}$ and the car reflects the sound at the frequency it has received).", "gold": "6" }, { "description": "JEE Adv 2017 Paper 1", "index": 12, "subject": "phy", "type": "Integer", "question": "${ }^{131} \\mathrm{I}$ is an isotope of Iodine that $\\beta$ decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with ${ }^{131} \\mathrm{I}$ is injected into the blood of a person. The activity of the amount of ${ }^{131} \\mathrm{I}$ injected was $2.4 \\times 10^{5}$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, $2.5 \\mathrm{ml}$ of blood is drawn from the person's body, and gives an activity of $115 \\mathrm{~Bq}$. What is the total volume of blood in the person's body, in liters is approximately? (you may use $e^{x} \\approx 1+x$ for $|x| \\ll 1$ and $\\ln 2 \\approx 0.7$ ).", "gold": "5" }, { "description": "JEE Adv 2017 Paper 1", "index": 19, "subject": "chem", "type": "MCQ(multiple)", "question": "An ideal gas is expanded from $\\left(\\mathrm{p}_{1}, \\mathrm{~V}_{1}, \\mathrm{~T}_{1}\\right)$ to $\\left(\\mathrm{p}_{2}, \\mathrm{~V}_{2}, \\mathrm{~T}_{2}\\right)$ under different conditions. The correct statement(s) among the following is(are)\n\n[A] The work done on the gas is maximum when it is compressed irreversibly from $\\left(\\mathrm{p}_{2}, \\mathrm{~V}_{2}\\right)$ to $\\left(\\mathrm{p}_{1}, \\mathrm{~V}_{1}\\right)$ against constant pressure $\\mathrm{p}_{1}$\n\n[B] If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic\n\n[C] The work done by the gas is less when it is expanded reversibly from $\\mathrm{V}_{1}$ to $\\mathrm{V}_{2}$ under adiabatic conditions as compared to that when expanded reversibly from $V_{1}$ to $\\mathrm{V}_{2}$ under isothermal conditions\n\n[D] The change in internal energy of the gas is (i) zero, if it is expanded reversibly with $\\mathrm{T}_{1}=\\mathrm{T}_{2}$, and (ii) positive, if it is expanded reversibly under adiabatic conditions with $\\mathrm{T}_{1} \\neq \\mathrm{T}_{2}$", "gold": "ABC" }, { "description": "JEE Adv 2017 Paper 1", "index": 21, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) about the oxoacids, $\\mathrm{HClO}_{4}$ and $\\mathrm{HClO}$, is(are)\n\n[A] The central atom in both $\\mathrm{HClO}_{4}$ and $\\mathrm{HClO}$ is $s p^{3}$ hybridized\n\n[B] $\\mathrm{HClO}_{4}$ is more acidic than $\\mathrm{HClO}$ because of the resonance stabilization of its anion\n\n[C] $\\mathrm{HClO}_{4}$ is formed in the reaction between $\\mathrm{Cl}_{2}$ and $\\mathrm{H}_{2} \\mathrm{O}$\n\n[D] The conjugate base of $\\mathrm{HClO}_{4}$ is weaker base than $\\mathrm{H}_{2} \\mathrm{O}$", "gold": "ABD" }, { "description": "JEE Adv 2017 Paper 1", "index": 22, "subject": "chem", "type": "MCQ(multiple)", "question": "The colour of the $\\mathrm{X}_{2}$ molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to\n\n[A] the physical state of $\\mathrm{X}_{2}$ at room temperature changes from gas to solid down the group\n\n[B] decrease in ionization energy down the group\n\n[C] decrease in $\\pi^{*}-\\sigma^{*}$ gap down the group\n\n[D] decrease in HOMO-LUMO gap down the group", "gold": "CD" }, { "description": "JEE Adv 2017 Paper 1", "index": 23, "subject": "chem", "type": "MCQ(multiple)", "question": "Addition of excess aqueous ammonia to a pink coloured aqueous solution of $\\mathrm{MCl}_{2} \\cdot 6 \\mathrm{H}_{2} \\mathrm{O}$ $(\\mathbf{X})$ and $\\mathrm{NH}_{4} \\mathrm{Cl}$ gives an octahedral complex $\\mathbf{Y}$ in the presence of air. In aqueous solution, complex $\\mathbf{Y}$ behaves as 1:3 electrolyte. The reaction of $\\mathbf{X}$ with excess $\\mathrm{HCl}$ at room temperature results in the formation of a blue coloured complex $\\mathbf{Z}$. The calculated spin only magnetic moment of $\\mathbf{X}$ and $\\mathbf{Z}$ is 3.87 B.M., whereas it is zero for complex $\\mathbf{Y}$.\n\nAmong the following options, which statement(s) is(are) correct?\n\n[A] Addition of silver nitrate to $\\mathbf{Y}$ gives only two equivalents of silver chloride\n\n[B] The hybridization of the central metal ion in $\\mathbf{Y}$ is $\\mathrm{d}^{2} \\mathrm{sp}^{3}$\n\n[C] $\\mathbf{Z}$ is a tetrahedral complex\n\n[D] When $\\mathbf{X}$ and $\\mathbf{Z}$ are in equilibrium at $0^{\\circ} \\mathrm{C}$, the colour of the solution is pink", "gold": "BCD" }, { "description": "JEE Adv 2017 Paper 1", "index": 26, "subject": "chem", "type": "Integer", "question": "A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of $400 \\mathrm{pm}$. If the density of the substance in the crystal is $8 \\mathrm{~g} \\mathrm{~cm}^{-3}$, then the number of atoms present in $256 \\mathrm{~g}$ of the crystal is $N \\times 10^{24}$. What is the value of $N$?", "gold": "2" }, { "description": "JEE Adv 2017 Paper 1", "index": 27, "subject": "chem", "type": "Integer", "question": "The conductance of a $0.0015 \\mathrm{M}$ aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is $120 \\mathrm{~cm}$ with an area of cross section of $1 \\mathrm{~cm}^{2}$. The conductance of this solution was found to be $5 \\times 10^{-7} \\mathrm{~S}$. The $\\mathrm{pH}$ of the solution is 4 . The value of limiting molar conductivity $\\left(\\Lambda_{m}^{o}\\right)$ of this weak monobasic acid in aqueous solution is $Z \\times 10^{2} \\mathrm{~S} \\mathrm{~cm}^{-1} \\mathrm{~mol}^{-1}$. What is the value of $Z$?", "gold": "6" }, { "description": "JEE Adv 2017 Paper 1", "index": 28, "subject": "chem", "type": "Integer", "question": "What is the sum of the number of lone pairs of electrons on each central atom in the following species?\n\n$\\left[\\mathrm{TeBr}_{6}\\right]^{2-},\\left[\\mathrm{BrF}_{2}\\right]^{+}, \\mathrm{SNF}_{3}$, and $\\left[\\mathrm{XeF}_{3}\\right]^{-}$\n\n(Atomic numbers: $\\mathrm{N}=7, \\mathrm{~F}=9, \\mathrm{~S}=16, \\mathrm{Br}=35, \\mathrm{Te}=52, \\mathrm{Xe}=54$ )", "gold": "6" }, { "description": "JEE Adv 2017 Paper 1", "index": 29, "subject": "chem", "type": "Integer", "question": "Among $\\mathrm{H}_{2}, \\mathrm{He}_{2}{ }^{+}, \\mathrm{Li}_{2}, \\mathrm{Be}_{2}, \\mathrm{~B}_{2}, \\mathrm{C}_{2}, \\mathrm{~N}_{2}, \\mathrm{O}_{2}^{-}$, and $\\mathrm{F}_{2}$, what is the number of diamagnetic species? (Atomic numbers: $\\mathrm{H}=1, \\mathrm{He}=2, \\mathrm{Li}=3, \\mathrm{Be}=4, \\mathrm{~B}=5, \\mathrm{C}=6, \\mathrm{~N}=7, \\mathrm{O}=8, \\mathrm{~F}=9$ )", "gold": "5" }, { "description": "JEE Adv 2017 Paper 1", "index": 37, "subject": "math", "type": "MCQ(multiple)", "question": "If $2 x-y+1=0$ is a tangent to the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{16}=1$, then which of the following CANNOT be sides of a right angled triangle?\n\n[A] $a, 4,1$\n\n[B] $a, 4,2$\n\n[C] $2 a, 8,1$\n\n[D] $2 a, 4,1$", "gold": "ABC" }, { "description": "JEE Adv 2017 Paper 1", "index": 38, "subject": "math", "type": "MCQ(multiple)", "question": "If a chord, which is not a tangent, of the parabola $y^{2}=16 x$ has the equation $2 x+y=p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p, h$ and $k$ ?\n\n[A] $p=-2, h=2, k=-4$\n\n[B] $p=-1, h=1, k=-3$\n\n[C] $p=2, h=3, k=-4$\n\n[D] $p=5, h=4, k=-3$", "gold": "C" }, { "description": "JEE Adv 2017 Paper 1", "index": 40, "subject": "math", "type": "MCQ(multiple)", "question": "Let $f: \\mathbb{R} \\rightarrow(0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$ ?\n\n[A] $x^{9}-f(x)$\n\n[B] $x-\\int_{0}^{\\frac{\\pi}{2}-x} f(t) \\cos t d t$\n\n[C] e^{x}-\\int_{0}^{x} f(t) \\sin t d t$\n\n[D] f(x)+\\int_{0}^{\\frac{\\pi}{2}} f(t) \\sin t d t$", "gold": "AB" }, { "description": "JEE Adv 2017 Paper 1", "index": 41, "subject": "math", "type": "MCQ(multiple)", "question": "Which of the following is(are) NOT the square of a $3 \\times 3$ matrix with real entries?\n\n[A]$\\left[\\begin{array}{lll}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{array}\\right]\n\n[B]$\\left[\\begin{array}{ccc}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1\\end{array}\\right]$\n\n[C]$\\left[\\begin{array}{ccc}1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -1\\end{array}\\right]\n\n[D]$\\left[\\begin{array}{ccc}-1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -1\\end{array}\\right]$", "gold": "BD" }, { "description": "JEE Adv 2017 Paper 1", "index": 42, "subject": "math", "type": "MCQ(multiple)", "question": "Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y \\neq 0$. If the complex number $z=x+i y$ satisfies $\\operatorname{Im}\\left(\\frac{a z+b}{z+1}\\right)=\\mathrm{y}$, then which of the following is(are) possible value(s) of $x ?$\n\n[A]$-1+\\sqrt{1-y^{2}}$\n\n[B]$-1-\\sqrt{1-y^{2}}$\n\n[C]$1+\\sqrt{1+y^{2}}$\n\n[D]$1-\\sqrt{1+y^{2}}$", "gold": "AB" }, { "description": "JEE Adv 2017 Paper 1", "index": 43, "subject": "math", "type": "MCQ(multiple)", "question": "Let $X$ and $Y$ be two events such that $P(X)=\\frac{1}{3}, P(X \\mid Y)=\\frac{1}{2}$ and $P(Y \\mid X)=\\frac{2}{5}$. Then\n\n[A] $P(Y)=\\frac{4}{15}$\n\n[B] $P\\left(X^{\\prime} \\mid Y\\right)=\\frac{1}{2}$\n\n[C] \\quad P(X \\cap Y)=\\frac{1}{5}$\n\n[D]$P(X \\cup Y)=\\frac{2}{5}$", "gold": "AB" }, { "description": "JEE Adv 2017 Paper 1", "index": 44, "subject": "math", "type": "Integer", "question": "For how many values of$p$, the circle$x^{2}+y^{2}+2 x+4 y-p=0$and the coordinate axes have exactly three common points?", "gold": "2" }, { "description": "JEE Adv 2017 Paper 1", "index": 45, "subject": "math", "type": "Integer", "question": "Let$f: \\mathbb{R} \\rightarrow \\mathbb{R}$be a differentiable function such that$f(0)=0, f\\left(\\frac{\\pi}{2}\\right)=3$and$f^{\\prime}(0)=1$. If\n\n\$\ng(x)=\\int_{x}^{\\frac{\\pi}{2}}\\left[f^{\\prime}(t) \\operatorname{cosec} t-\\cot t \\operatorname{cosec} t f(t)\\right] d t\n\$\n\nfor$x \\in\\left(0, \\frac{\\pi}{2}\\right]$, then what is the$\\lim _{x \\rightarrow 0} g(x)$?", "gold": "2" }, { "description": "JEE Adv 2017 Paper 1", "index": 46, "subject": "math", "type": "Integer", "question": "For a real number$\\alpha$, if the system\n\n\$\n\\left[\\begin{array}{ccc}\n\n1 & \\alpha & \\alpha^{2} \\\\\n\n\\alpha & 1 & \\alpha \\\\\n\n\\alpha^{2} & \\alpha & 1\n\n\\end{array}\\right]\\left[\\begin{array}{l}\n\nx \\\\\n\ny \\\\\n\nz\n\n\\end{array}\\right]=\\left[\\begin{array}{r}\n\n1 \\\\\n\n-1 \\\\\n\n1\n\n\\end{array}\\right]\n\$\n\nof linear equations, has infinitely many solutions, then what is the value of$1+\\alpha+\\alpha^{2}$?", "gold": "1" }, { "description": "JEE Adv 2017 Paper 1", "index": 47, "subject": "math", "type": "Integer", "question": "Words of length 10 are formed using the letters$A, B, C, D, E, F, G, H, I, J$. Let$x$be the number of such words where no letter is repeated; and let$y$be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, what is the value of$\\frac{y}{9 x}$?", "gold": "5" }, { "description": "JEE Adv 2017 Paper 1", "index": 48, "subject": "math", "type": "Integer", "question": "The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?", "gold": "6" }, { "description": "JEE Adv 2017 Paper 2", "index": 1, "subject": "phy", "type": "MCQ", "question": "Consider an expanding sphere of instantaneous radius$R$whose total mass remains constant. The expansion is such that the instantaneous density$\\rho$remains uniform throughout the volume. The rate of fractional change in density$\\left(\\frac{1}{\\rho} \\frac{d \\rho}{d t}\\right)$is constant. The velocity$v$of any point on the surface of the expanding sphere is proportional to\n\n[A]$R$\n\n[B]$R^{3}$\n\n[C]$\\frac{1}{R}$\n\n[D]$R^{2 / 3}$", "gold": "A" }, { "description": "JEE Adv 2017 Paper 2", "index": 3, "subject": "phy", "type": "MCQ", "question": "A photoelectric material having work-function$\\phi_{0}$is illuminated with light of wavelength$\\lambda\\left(\\lambda<\\frac{h c}{\\phi_{0}}\\right)$. The fastest photoelectron has a de Broglie wavelength$\\lambda_{d}$. A change in wavelength of the incident light by$\\Delta \\lambda$results in a change$\\Delta \\lambda_{d}$in$\\lambda_{d}$. Then the ratio$\\Delta \\lambda_{d} / \\Delta \\lambda$is proportional to\n\n$[\\mathrm{A}] \\quad \\lambda_{d} / \\lambda$\n\n$[\\mathrm{B}] \\quad \\lambda_{d}^{2} / \\lambda^{2}$\n\n$[\\mathrm{C}] \\lambda_{d}^{3} / \\lambda$\n\n$[\\mathrm{D}] \\lambda_{d}^{3} / \\lambda^{2}$", "gold": "D" }, { "description": "JEE Adv 2017 Paper 2", "index": 6, "subject": "phy", "type": "MCQ", "question": "A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is$3 \\times 10^{5}$times heavier than the Earth and is at a distance$2.5 \\times 10^{4}$times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is$v_{e}=11.2 \\mathrm{~km} \\mathrm{~s}^{-1}$. The minimum initial velocity$\\left(v_{S}\\right)$required for the rocket to be able to leave the Sun-Earth system is closest to\n\n(Ignore the rotation and revolution of the Earth and the presence of any other planet)\n\n$[\\mathrm{A}] \\quad v_{S}=22 \\mathrm{~km} \\mathrm{~s}^{-1}$\n\n$[\\mathrm{B}] v_{S}=42 \\mathrm{~km} \\mathrm{~s}^{-1}$\n\n$[\\mathrm{C}] \\quad v_{S}=62 \\mathrm{~km} \\mathrm{~s}^{-1}$\n\n[D]$v_{S}=72 \\mathrm{~km} \\mathrm{~s}^{-1}$", "gold": "B" }, { "description": "JEE Adv 2017 Paper 2", "index": 7, "subject": "phy", "type": "MCQ", "question": "A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is$\\delta T=0.01$seconds and he measures the depth of the well to be$L=20$meters. Take the acceleration due to gravity$g=10 \\mathrm{~ms}^{-2}$and the velocity of sound is$300 \\mathrm{~ms}^{-1}$. Then the fractional error in the measurement,$\\delta L / L$, is closest to\n\n[A]$0.2 \\%$\n\n[B]$1 \\%$\n\n[C]$3 \\%$\n\n[D]$5 \\%$", "gold": "B" }, { "description": "JEE Adv 2017 Paper 2", "index": 9, "subject": "phy", "type": "MCQ(multiple)", "question": "The instantaneous voltages at three terminals marked$X, Y$and$Zare given by\n\n\\n\\begin{aligned}\n\n& V_{X}=V_{0} \\sin \\omega t, \\\\\n\n& V_{Y}=V_{0} \\sin \\left(\\omega t+\\frac{2 \\pi}{3}\\right) \\text { and } \\\\\n\n& V_{Z}=V_{0} \\sin \\left(\\omega t+\\frac{4 \\pi}{3}\\right) .\n\n\\end{aligned}\n\\n\nAn ideal voltmeter is configured to read\\mathrm{rms}$value of the potential difference between its terminals. It is connected between points$X$and$Y$and then between$Y$and$Z$. The reading(s) of the voltmeter will be\n\n[A]$\\quad V_{X Y}^{r m s}=V_{0} \\sqrt{\\frac{3}{2}}$\n\n[B]$\\quad V_{Y Z}^{r m s}=V_{0} \\sqrt{\\frac{1}{2}}$\n\n[C]$\\quad V_{X Y}^{r m s}=V_{0}$\n\n[D] independent of the choice of the two terminals", "gold": "AD" }, { "description": "JEE Adv 2017 Paper 2", "index": 20, "subject": "chem", "type": "MCQ", "question": "For the following cell,\n\n\$\n\\mathrm{Zn}(s)\\left|\\mathrm{ZnSO}_{4}(a q) \\| \\mathrm{CuSO}_{4}(a q)\\right| \\mathrm{Cu}(s)\n\$\n\nwhen the concentration of$\\mathrm{Zn}^{2+}$is 10 times the concentration of$\\mathrm{Cu}^{2+}$, the expression for$\\Delta G\\left(\\right.$in$\\left.\\mathrm{J} \\mathrm{mol}^{-1}\\right)$is\n\n[$\\mathrm{F}$is Faraday constant;$\\mathrm{R}$is gas constant;$\\mathrm{T}$is temperature;$E^{o}($cell$)=1.1 \\mathrm{~V}$]\n\n[A]$1.1 \\mathrm{~F}$\n\n[B]$2.303 \\mathrm{RT}-2.2 \\mathrm{~F}$\n\n[C]$2.303 \\mathrm{RT}+1.1 \\mathrm{~F}$\n\n[D]-2.2 \\mathrm{~F}$", "gold": "B" }, { "description": "JEE Adv 2017 Paper 2", "index": 21, "subject": "chem", "type": "MCQ", "question": "The standard state Gibbs free energies of formation of $\\mathrm{C}$ (graphite) and $C$ (diamond) at $\\mathrm{T}=298 \\mathrm{~K}$ are\n\n\$\n\\begin{gathered}\n\n\\Delta_{f} G^{o}[\\mathrm{C}(\\text { graphite })]=0 \\mathrm{~kJ} \\mathrm{~mol} \\\\\n\n\\Delta_{f} G^{o}[\\mathrm{C}(\\text { diamond })]=2.9 \\mathrm{~kJ} \\mathrm{~mol}^{-1} .\n\n\\end{gathered}\n\$\n\nThe standard state means that the pressure should be 1 bar, and substance should be pure at a given temperature. The conversion of graphite [ C(graphite)] to diamond [ C(diamond)] reduces its volume by $2 \\times 10^{-6} \\mathrm{~m}^{3} \\mathrm{~mol}^{-1}$. If $\\mathrm{C}$ (graphite) is converted to $\\mathrm{C}$ (diamond) isothermally at $\\mathrm{T}=298 \\mathrm{~K}$, the pressure at which $\\mathrm{C}$ (graphite) is in equilibrium with $\\mathrm{C}($ diamond), is\n\n[Useful information: $1 \\mathrm{~J}=1 \\mathrm{~kg} \\mathrm{~m}^{2} \\mathrm{~s}^{-2} ; 1 \\mathrm{~Pa}=1 \\mathrm{~kg} \\mathrm{~m}^{-1} \\mathrm{~s}^{-2} ; 1$ bar $=10^{5} \\mathrm{~Pa}$ ]\n\n[A] 14501 bar\n\n[B] 58001 bar\n\n[C] 1450 bar\n\n[D] 29001 bar", "gold": "A" }, { "description": "JEE Adv 2017 Paper 2", "index": 22, "subject": "chem", "type": "MCQ", "question": "Which of the following combination will produce $\\mathrm{H}_{2}$ gas?\n\n[A] Fe metal and conc. $\\mathrm{HNO}_{3}$\n\n[B] Cu metal and conc. $\\mathrm{HNO}_{3}$\n\n[C] $\\mathrm{Zn}$ metal and $\\mathrm{NaOH}(\\mathrm{aq})$\n\n[D] Au metal and $\\mathrm{NaCN}(\\mathrm{aq})$ in the presence of air", "gold": "C" }, { "description": "JEE Adv 2017 Paper 2", "index": 23, "subject": "chem", "type": "MCQ", "question": "The order of the oxidation state of the phosphorus atom in $\\mathrm{H}_{3} \\mathrm{PO}_{2}, \\mathrm{H}_{3} \\mathrm{PO}_{4}, \\mathrm{H}_{3} \\mathrm{PO}_{3}$, and $\\mathrm{H}_{4} \\mathrm{P}_{2} \\mathrm{O}_{6}$ is\n\n[A] $\\mathrm{H}_{3} \\mathrm{PO}_{3}>\\mathrm{H}_{3} \\mathrm{PO}_{2}>\\mathrm{H}_{3} \\mathrm{PO}_{4}>\\mathrm{H}_{4} \\mathrm{P}_{2} \\mathrm{O}_{6}$\n\n[B] $\\mathrm{H}_{3} \\mathrm{PO}_{4}>\\mathrm{H}_{3} \\mathrm{PO}_{2}>\\mathrm{H}_{3} \\mathrm{PO}_{3}>\\mathrm{H}_{4} \\mathrm{P}_{2} \\mathrm{O}_{6}$\n\n[C] $\\mathrm{H}_{3} \\mathrm{PO}_{4}>\\mathrm{H}_{4} \\mathrm{P}_{2} \\mathrm{O}_{6}>\\mathrm{H}_{3} \\mathrm{PO}_{3}>\\mathrm{H}_{3} \\mathrm{PO}_{2}$\n\n[D] $\\mathrm{H}_{3} \\mathrm{PO}_{2}>\\mathrm{H}_{3} \\mathrm{PO}_{3}>\\mathrm{H}_{4} \\mathrm{P}_{2} \\mathrm{O}_{6}>\\mathrm{H}_{3} \\mathrm{PO}_{4}$", "gold": "C" }, { "description": "JEE Adv 2017 Paper 2", "index": 26, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) about surface properties is(are)\n\n[A] Adsorption is accompanied by decrease in enthalpy and decrease in entropy of the system\n\n[B] The critical temperatures of ethane and nitrogen are $563 \\mathrm{~K}$ and $126 \\mathrm{~K}$, respectively. The adsorption of ethane will be more than that of nitrogen on same amount of activated charcoal at a given temperature\n\n[C] Cloud is an emulsion type of colloid in which liquid is dispersed phase and gas is dispersion medium\n\n[D] Brownian motion of colloidal particles does not depend on the size of the particles but depends on viscosity of the solution", "gold": "AB" }, { "description": "JEE Adv 2017 Paper 2", "index": 27, "subject": "chem", "type": "MCQ(multiple)", "question": "For a reaction taking place in a container in equilibrium with its surroundings, the effect of temperature on its equilibrium constant $K$ in terms of change in entropy is described by\n\n[A] With increase in temperature, the value of $K$ for exothermic reaction decreases because the entropy change of the system is positive\n\n[B] With increase in temperature, the value of $K$ for endothermic reaction increases because unfavourable change in entropy of the surroundings decreases\n\n[C] With increase in temperature, the value of $K$ for endothermic reaction increases because the entropy change of the system is negative\n\n[D] With increase in temperature, the value of $K$ for exothermic reaction decreases because favourable change in entropy of the surroundings decreases", "gold": "BD" }, { "description": "JEE Adv 2017 Paper 2", "index": 28, "subject": "chem", "type": "MCQ(multiple)", "question": "In a bimolecular reaction, the steric factor $P$ was experimentally determined to be 4.5 . The correct option(s) among the following is(are)\n\n[A] The activation energy of the reaction is unaffected by the value of the steric factor\n\n[B] Experimentally determined value of frequency factor is higher than that predicted by Arrhenius equation\n\n[C] Since $\\mathrm{P}=4.5$, the reaction will not proceed unless an effective catalyst is used\n\n[D] The value of frequency factor predicted by Arrhenius equation is higher than that determined experimentally", "gold": "AB" }, { "description": "JEE Adv 2017 Paper 2", "index": 30, "subject": "chem", "type": "MCQ(multiple)", "question": "Among the following, the correct statement(s) is(are)\n\n[A] $\\mathrm{Al}\\left(\\mathrm{CH}_{3}\\right)_{3}$ has the three-centre two-electron bonds in its dimeric structure\n\n[B] $\\mathrm{BH}_{3}$ has the three-centre two-electron bonds in its dimeric structure\n\n[C] $\\mathrm{AlCl}_{3}$ has the three-centre two-electron bonds in its dimeric structure\n\n[D] The Lewis acidity of $\\mathrm{BCl}_{3}$ is greater than that of $\\mathrm{AlCl}_{3}$", "gold": "ABD" }, { "description": "JEE Adv 2017 Paper 2", "index": 31, "subject": "chem", "type": "MCQ(multiple)", "question": "The option(s) with only amphoteric oxides is(are)\n\n[A] $\\mathrm{Cr}_{2} \\mathrm{O}_{3}, \\mathrm{BeO}, \\mathrm{SnO}, \\mathrm{SnO}_{2}$\n\n[B] $\\mathrm{Cr}_{2} \\mathrm{O}_{3}, \\mathrm{CrO}, \\mathrm{SnO}, \\mathrm{PbO}$\n\n[C] $\\mathrm{NO}, \\mathrm{B}_{2} \\mathrm{O}_{3}, \\mathrm{PbO}, \\mathrm{SnO}_{2}$\n\n[D] $\\mathrm{ZnO}, \\mathrm{Al}_{2} \\mathrm{O}_{3}, \\mathrm{PbO}, \\mathrm{PbO}_{2}$", "gold": "AD" }, { "description": "JEE Adv 2017 Paper 2", "index": 37, "subject": "math", "type": "MCQ", "question": "The equation of the plane passing through the point $(1,1,1)$ and perpendicular to the planes $2 x+y-2 z=5$ and $3 x-6 y-2 z=7$, is\n\n[A] $14 x+2 y-15 z=1$\n\n[B] $14 x-2 y+15 z=27$\n\n[C] $\\quad 14 x+2 y+15 z=31$\n\n[D] $-14 x+2 y+15 z=3$", "gold": "C" }, { "description": "JEE Adv 2017 Paper 2", "index": 38, "subject": "math", "type": "MCQ", "question": "Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that\n\n\$\n\\overrightarrow{O P} \\cdot \\overrightarrow{O Q}+\\overrightarrow{O R} \\cdot \\overrightarrow{O S}=\\overrightarrow{O R} \\cdot \\overrightarrow{O P}+\\overrightarrow{O Q} \\cdot \\overrightarrow{O S}=\\overrightarrow{O Q} \\cdot \\overrightarrow{O R}+\\overrightarrow{O P} \\cdot \\overrightarrow{O S}\n\$\n\nThen the triangle $P Q R$ has $S$ as its\n\n[A] centroid\n\n[B] circumcentre\n\n[C] incentre\n\n[D] orthocenter", "gold": "D" }, { "description": "JEE Adv 2017 Paper 2", "index": 39, "subject": "math", "type": "MCQ", "question": "If $y=y(x)$ satisfies the differential equation\n\$\n8 \\sqrt{x}(\\sqrt{9+\\sqrt{x}}) d y=(\\sqrt{4+\\sqrt{9+\\sqrt{x}}})^{-1} d x, \\quad x>0\n\$\n\nand $y(0)=\\sqrt{7}$, then $y(256)=$\n\n[A] 3\n\n[B] 9\n\n[C] 16\n\n[D] 80", "gold": "A" }, { "description": "JEE Adv 2017 Paper 2", "index": 40, "subject": "math", "type": "MCQ", "question": "If $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ is a twice differentiable function such that $f^{\\prime \\prime}(x)>0$ for all $x \\in \\mathbb{R}$, and $f\\left(\\frac{1}{2}\\right)=\\frac{1}{2}, f(1)=1$, then\n\n[A] $f^{\\prime}(1) \\leq 0$\n\n[B] $01$", "gold": "D" }, { "description": "JEE Adv 2017 Paper 2", "index": 41, "subject": "math", "type": "MCQ", "question": "How many $3 \\times 3$ matrices $M$ with entries from $\\{0,1,2\\}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is $5 ?$\n\n[A] 126\n\n[B] 198\n\n[C] 162\n\n[D] 135", "gold": "B" }, { "description": "JEE Adv 2017 Paper 2", "index": 42, "subject": "math", "type": "MCQ", "question": "Let $S=\\{1,2,3, \\ldots, 9\\}$. For $k=1,2, \\ldots, 5$, let $N_{k}$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=$\n\n[A] 210\n\n[B] 252\n\n[C] 125\n\n[D] 126", "gold": "D" }, { "description": "JEE Adv 2017 Paper 2", "index": 43, "subject": "math", "type": "MCQ", "question": "Three randomly chosen nonnegative integers $x, y$ and $z$ are found to satisfy the equation $x+y+z=10$. Then the probability that $z$ is even, is\n\n[A] $\\frac{36}{55}$\n\n[B] $\\frac{6}{11}$\n\n[C] $\\frac{1}{2}$\n\n[D] $\\frac{5}{11}$", "gold": "B" }, { "description": "JEE Adv 2017 Paper 2", "index": 46, "subject": "math", "type": "MCQ(multiple)", "question": "If $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ is a differentiable function such that $f^{\\prime}(x)>2 f(x)$ for all $x \\in \\mathbb{R}$, and $f(0)=1$, then\n\n[A] $f(x)$ is increasing in $(0, \\infty)$\n\n[B] $f(x)$ is decreasing in $(0, \\infty)$\n\n[C] $\\quad f(x)>e^{2 x}$ in $(0, \\infty)$\n\n[D] $f^{\\prime}(x)\\log _{e} 99$\n\n[B] $I<\\log _{e} 99$\n\n[C] $I<\\frac{49}{50}$\n\n[D] $I>\\frac{49}{50}$", "gold": "BD" }, { "description": "JEE Adv 2018 Paper 1", "index": 1, "subject": "phy", "type": "MCQ(multiple)", "question": "The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $V(r)=k r^{2} / 2$, where $k$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $R$ about the point $O$. If $v$ is the speed of the particle and $L$ is the magnitude of its angular momentum about $O$, which of the following statements is (are) true?\n\n(A) $v=\\sqrt{\\frac{k}{2 m}} R$\n\n(B) $v=\\sqrt{\\frac{k}{m}} R$\n\n(C) $L=\\sqrt{m k} R^{2}$\n\n(D) $L=\\sqrt{\\frac{m k}{2}} R^{2}$", "gold": "BC" }, { "description": "JEE Adv 2018 Paper 1", "index": 2, "subject": "phy", "type": "MCQ(multiple)", "question": "Consider a body of mass $1.0 \\mathrm{~kg}$ at rest at the origin at time $t=0$. A force $\\vec{F}=(\\alpha t \\hat{i}+\\beta \\hat{j})$ is applied on the body, where $\\alpha=1.0 \\mathrm{Ns}^{-1}$ and $\\beta=1.0 \\mathrm{~N}$. The torque acting on the body about the origin at time $t=1.0 \\mathrm{~s}$ is $\\vec{\\tau}$. Which of the following statements is (are) true?\n\n(A) $|\\vec{\\tau}|=\\frac{1}{3} N m$\n\n(B) The torque $\\vec{\\tau}$ is in the direction of the unit vector $+\\hat{k}$\n\n(C) The velocity of the body at $t=1 s$ is $\\vec{v}=\\frac{1}{2}(\\hat{i}+2 \\hat{j}) m s^{-1}$\n\n(D) The magnitude of displacement of the body at $t=1 s$ is $\\frac{1}{6} m$", "gold": "AC" }, { "description": "JEE Adv 2018 Paper 1", "index": 3, "subject": "phy", "type": "MCQ(multiple)", "question": "A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\\sigma$. The angle of contact between water and the wall of the capillary tube is $\\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?\n\n(A) For a given material of the capillary tube, $h$ decreases with increase in $r$\n\n(B) For a given material of the capillary tube, $h$ is independent of $\\sigma$\n\n(C) If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases\n\n(D) $h$ is proportional to contact angle $\\theta$", "gold": "AC" }, { "description": "JEE Adv 2018 Paper 1", "index": 5, "subject": "phy", "type": "MCQ(multiple)", "question": "Two infinitely long straight wires lie in the $x y$-plane along the lines $x= \\pm R$. The wire located at $x=+R$ carries a constant current $I_{1}$ and the wire located at $x=-R$ carries a constant current $I_{2}$. A circular loop of radius $R$ is suspended with its centre at $(0,0, \\sqrt{3} R)$ and in a plane parallel to the $x y$-plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\\hat{j}$ direction. Which of the following statements regarding the magnetic field $\\vec{B}$ is (are) true?\n\n(A) If $I_{1}=I_{2}$, then $\\vec{B}$ cannot be equal to zero at the origin $(0,0,0)$\n\n(B) If $I_{1}>0$ and $I_{2}<0$, then $\\vec{B}$ can be equal to zero at the origin $(0,0,0)$\n\n(C) If $I_{1}<0$ and $I_{2}>0$, then $\\vec{B}$ can be equal to zero at the origin $(0,0,0)$\n\n(D) If $I_{1}=I_{2}$, then the $z$-component of the magnetic field at the centre of the loop is $\\left(-\\frac{\\mu_{0} I}{2 R}\\right)$", "gold": "ABD" }, { "description": "JEE Adv 2018 Paper 1", "index": 7, "subject": "phy", "type": "Numeric", "question": "Two vectors $\\vec{A}$ and $\\vec{B}$ are defined as $\\vec{A}=a \\hat{i}$ and $\\vec{B}=a(\\cos \\omega t \\hat{i}+\\sin \\omega t \\hat{j})$, where $a$ is a constant and $\\omega=\\pi / 6 \\mathrm{rads}^{-1}$. If $|\\vec{A}+\\vec{B}|=\\sqrt{3}|\\vec{A}-\\vec{B}|$ at time $t=\\tau$ for the first time, what is the value of $\\tau$, in seconds?", "gold": "2" }, { "description": "JEE Adv 2018 Paper 1", "index": 8, "subject": "phy", "type": "Numeric", "question": "Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0 \\mathrm{~ms}^{-1}$ and the man behind walks at a speed $2.0 \\mathrm{~m} \\mathrm{~s}^{-1}$. A third man is standing at a height $12 \\mathrm{~m}$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $1430 \\mathrm{~Hz}$. The speed of sound in air is $330 \\mathrm{~m} \\mathrm{~s}^{-1}$. At the instant, when the moving men are $10 \\mathrm{~m}$ apart, the stationary man is equidistant from them. What is the frequency of beats in $\\mathrm{Hz}$, heard by the stationary man at this instant?", "gold": "5" }, { "description": "JEE Adv 2018 Paper 1", "index": 9, "subject": "phy", "type": "Numeric", "question": "A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $60^{\\circ}$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $(2-\\sqrt{3}) / \\sqrt{10} s$, then what is the height of the top of the inclined plane, in metres?\n\nTake $g=10 m s^{-2}$.", "gold": "0.75" }, { "description": "JEE Adv 2018 Paper 1", "index": 13, "subject": "phy", "type": "Numeric", "question": "Sunlight of intensity $1.3 \\mathrm{~kW} \\mathrm{~m}^{-2}$ is incident normally on a thin convex lens of focal length $20 \\mathrm{~cm}$. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. What is the average intensity of light, in $\\mathrm{kW} \\mathrm{m}{ }^{-2}$, at a distance $22 \\mathrm{~cm}$ from the lens on the other side?", "gold": "130" }, { "description": "JEE Adv 2018 Paper 1", "index": 19, "subject": "chem", "type": "MCQ(multiple)", "question": "The compound(s) which generate(s) $\\mathrm{N}_{2}$ gas upon thermal decomposition below $300^{\\circ} \\mathrm{C}$ is (are)\n\n(A) $\\mathrm{NH}_{4} \\mathrm{NO}_{3}$\n\n(B) $\\left(\\mathrm{NH}_{4}\\right)_{2} \\mathrm{Cr}_{2} \\mathrm{O}_{7}$\n\n(C) $\\mathrm{Ba}\\left(\\mathrm{N}_{3}\\right)_{2}$\n\n(D) $\\mathrm{Mg}_{3} \\mathrm{~N}_{2}$", "gold": "BC" }, { "description": "JEE Adv 2018 Paper 1", "index": 20, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: $\\mathrm{Fe}=26, \\mathrm{Ni}=28$ )\n\n(A) Total number of valence shell electrons at metal centre in $\\mathrm{Fe}(\\mathrm{CO})_{5}$ or $\\mathrm{Ni}(\\mathrm{CO})_{4}$ is 16\n\n(B) These are predominantly low spin in nature\n\n(C) Metal-carbon bond strengthens when the oxidation state of the metal is lowered\n\n(D) The carbonyl C-O bond weakens when the oxidation state of the metal is increased", "gold": "BC" }, { "description": "JEE Adv 2018 Paper 1", "index": 21, "subject": "chem", "type": "MCQ(multiple)", "question": "Based on the compounds of group 15 elements, the correct statement(s) is (are)\n\n(A) $\\mathrm{Bi}_{2} \\mathrm{O}_{5}$ is more basic than $\\mathrm{N}_{2} \\mathrm{O}_{5}$\n\n(B) $\\mathrm{NF}_{3}$ is more covalent than $\\mathrm{BiF}_{3}$\n\n(C) $\\mathrm{PH}_{3}$ boils at lower temperature than $\\mathrm{NH}_{3}$\n\n(D) The $\\mathrm{N}-\\mathrm{N}$ single bond is stronger than the $\\mathrm{P}-\\mathrm{P}$ single bond", "gold": "ABC" }, { "description": "JEE Adv 2018 Paper 1", "index": 25, "subject": "chem", "type": "Numeric", "question": "Among the species given below, what is the total number of diamagnetic species?\n\n$\\mathrm{H}$ atom, $\\mathrm{NO}_{2}$ monomer, $\\mathrm{O}_{2}^{-}$(superoxide), dimeric sulphur in vapour phase,\n\n$\\mathrm{Mn}_{3} \\mathrm{O}_{4},\\left(\\mathrm{NH}_{4}\\right)_{2}\\left[\\mathrm{FeCl}_{4}\\right],\\left(\\mathrm{NH}_{4}\\right)_{2}\\left[\\mathrm{NiCl}_{4}\\right], \\mathrm{K}_{2} \\mathrm{MnO}_{4}, \\mathrm{~K}_{2} \\mathrm{CrO}_{4}$", "gold": "1" }, { "description": "JEE Adv 2018 Paper 1", "index": 26, "subject": "chem", "type": "Numeric", "question": "The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by $\\mathrm{NiCl}_{2} \\cdot 6 \\mathrm{H}_{2} \\mathrm{O}$ to form a stable coordination compound. Assume that both the reactions are $100 \\%$ complete. If $1584 \\mathrm{~g}$ of ammonium sulphate and $952 \\mathrm{~g}$ of $\\mathrm{NiCl}_{2} .6 \\mathrm{H}_{2} \\mathrm{O}$ are used in the preparation, what is the combined weight (in grams) of gypsum and the nickelammonia coordination compound thus produced?\n\n(Atomic weights in $\\mathrm{g} \\mathrm{mol}^{-1}: \\mathrm{H}=1, \\mathrm{~N}=14, \\mathrm{O}=16, \\mathrm{~S}=32, \\mathrm{Cl}=35.5, \\mathrm{Ca}=40, \\mathrm{Ni}=59$ )", "gold": "2992" }, { "description": "JEE Adv 2018 Paper 1", "index": 27, "subject": "chem", "type": "Numeric", "question": "Consider an ionic solid $\\mathbf{M X}$ with $\\mathrm{NaCl}$ structure. Construct a new structure (Z) whose unit cell is constructed from the unit cell of $\\mathbf{M X}$ following the sequential instructions given below. Neglect the charge balance.\n\n(i) Remove all the anions (X) except the central one\n\n(ii) Replace all the face centered cations (M) by anions (X)\n\n(iii) Remove all the corner cations (M)\n\n(iv) Replace the central anion (X) with cation (M)\n\nWhat is the value of $\\left(\\frac{\\text { number of anions }}{\\text { number of cations }}\\right)$ in $\\mathbf{Z}$?", "gold": "3" }, { "description": "JEE Adv 2018 Paper 1", "index": 28, "subject": "chem", "type": "Numeric", "question": "For the electrochemical cell,\n\n\$\n\\operatorname{Mg}(\\mathrm{s})\\left|\\mathrm{Mg}^{2+}(\\mathrm{aq}, 1 \\mathrm{M}) \\| \\mathrm{Cu}^{2+}(\\mathrm{aq}, 1 \\mathrm{M})\\right| \\mathrm{Cu}(\\mathrm{s})\n\$\n\nthe standard emf of the cell is $2.70 \\mathrm{~V}$ at $300 \\mathrm{~K}$. When the concentration of $\\mathrm{Mg}^{2+}$ is changed to $\\boldsymbol{x} \\mathrm{M}$, the cell potential changes to $2.67 \\mathrm{~V}$ at $300 \\mathrm{~K}$. What is the value of $\\boldsymbol{x}$?\n\n(given, $\\frac{F}{R}=11500 \\mathrm{~K} \\mathrm{~V}^{-1}$, where $F$ is the Faraday constant and $R$ is the gas constant, $\\ln (10)=2.30)$", "gold": "10" }, { "description": "JEE Adv 2018 Paper 1", "index": 30, "subject": "chem", "type": "Numeric", "question": "Liquids $\\mathbf{A}$ and $\\mathbf{B}$ form ideal solution over the entire range of composition. At temperature $\\mathrm{T}$, equimolar binary solution of liquids $\\mathbf{A}$ and $\\mathbf{B}$ has vapour pressure 45 Torr. At the same temperature, a new solution of $\\mathbf{A}$ and $\\mathbf{B}$ having mole fractions $x_{A}$ and $x_{B}$, respectively, has vapour pressure of 22.5 Torr. What is the value of $x_{A} / x_{B}$ in the new solution? (given that the vapour pressure of pure liquid $\\mathbf{A}$ is 20 Torr at temperature $\\mathrm{T}$ )", "gold": "19" }, { "description": "JEE Adv 2018 Paper 1", "index": 31, "subject": "chem", "type": "Numeric", "question": "The solubility of a salt of weak acid (AB) at $\\mathrm{pH} 3$ is $\\mathbf{Y} \\times 10^{-3} \\mathrm{~mol} \\mathrm{~L}^{-1}$. The value of $\\mathbf{Y}$ is (Given that the value of solubility product of $\\mathbf{A B}\\left(K_{s p}\\right)=2 \\times 10^{-10}$ and the value of ionization constant of $\\left.\\mathbf{H B}\\left(K_{a}\\right)=1 \\times 10^{-8}\\right)$", "gold": "4.47" }, { "description": "JEE Adv 2018 Paper 1", "index": 37, "subject": "math", "type": "MCQ(multiple)", "question": "For a non-zero complex number $z$, let $\\arg (z)$ denote the principal argument with $-\\pi<\\arg (z) \\leq \\pi$. Then, which of the following statement(s) is (are) FALSE?\n\n\\end{itemize}\n\n(A) $\\arg (-1-i)=\\frac{\\pi}{4}$, where $i=\\sqrt{-1}$\n\n(B) The function $f: \\mathbb{R} \\rightarrow(-\\pi, \\pi]$, defined by $f(t)=\\arg (-1+i t)$ for all $t \\in \\mathbb{R}$, is continuous at all points of $\\mathbb{R}$, where $i=\\sqrt{-1}$\n\n(C) For any two non-zero complex numbers $z_{1}$ and $z_{2}$,\n\n\$\n\\arg \\left(\\frac{z_{1}}{z_{2}}\\right)-\\arg \\left(z_{1}\\right)+\\arg \\left(z_{2}\\right)\n\$\n\nis an integer multiple of $2 \\pi$\n\n(D) For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$, the locus of the point $z$ satisfying the condition\n\n\$\n\\arg \\left(\\frac{\\left(z-z_{1}\\right)\\left(z_{2}-z_{3}\\right)}{\\left(z-z_{3}\\right)\\left(z_{2}-z_{1}\\right)}\\right)=\\pi\n\$\n\nlies on a straight line", "gold": "ABD" }, { "description": "JEE Adv 2018 Paper 1", "index": 38, "subject": "math", "type": "MCQ(multiple)", "question": "In a triangle $P Q R$, let $\\angle P Q R=30^{\\circ}$ and the sides $P Q$ and $Q R$ have lengths $10 \\sqrt{3}$ and 10 , respectively. Then, which of the following statement(s) is (are) TRUE?\n\n(A) $\\angle Q P R=45^{\\circ}$\n\n(B) The area of the triangle $P Q R$ is $25 \\sqrt{3}$ and $\\angle Q R P=120^{\\circ}$\n\n(C) The radius of the incircle of the triangle $P Q R$ is $10 \\sqrt{3}-15$\n\n(D) The area of the circumcircle of the triangle $P Q R$ is $100 \\pi$", "gold": "BCD" }, { "description": "JEE Adv 2018 Paper 1", "index": 39, "subject": "math", "type": "MCQ(multiple)", "question": "Let $P_{1}: 2 x+y-z=3$ and $P_{2}: x+2 y+z=2$ be two planes. Then, which of the following statement(s) is (are) TRUE?\n\n(A) The line of intersection of $P_{1}$ and $P_{2}$ has direction ratios $1,2,-1$\n\n(B) The line\n\n\$\n\\frac{3 x-4}{9}=\\frac{1-3 y}{9}=\\frac{z}{3}\n\$\n\nis perpendicular to the line of intersection of $P_{1}$ and $P_{2}$\n\n(C) The acute angle between $P_{1}$ and $P_{2}$ is $60^{\\circ}$\n\n(D) If $P_{3}$ is the plane passing through the point $(4,2,-2)$ and perpendicular to the line of intersection of $P_{1}$ and $P_{2}$, then the distance of the point $(2,1,1)$ from the plane $P_{3}$ is $\\frac{2}{\\sqrt{3}}$", "gold": "CD" }, { "description": "JEE Adv 2018 Paper 1", "index": 40, "subject": "math", "type": "MCQ(multiple)", "question": "For every twice differentiable function $f: \\mathbb{R} \\rightarrow[-2,2]$ with $(f(0))^{2}+\\left(f^{\\prime}(0)\\right)^{2}=85$, which of the following statement(s) is (are) TRUE?\n\n(A) There exist $r, s \\in \\mathbb{R}$, where $r1-\\log _{\\mathrm{e}} 2$\n\n(C) $g(1)>1-\\log _{\\mathrm{e}} 2$\n\n(D) $g(1)<1-\\log _{\\mathrm{e}} 2$", "gold": "BC" }, { "description": "JEE Adv 2018 Paper 1", "index": 43, "subject": "math", "type": "Numeric", "question": "What is the value of\n\n\$\n\\left(\\left(\\log _{2} 9\\right)^{2}\\right)^{\\frac{1}{\\log _{2}\\left(\\log _{2} 9\\right)}} \\times(\\sqrt{7})^{\\frac{1}{\\log _{4} 7}}\n\$?", "gold": "8" }, { "description": "JEE Adv 2018 Paper 1", "index": 44, "subject": "math", "type": "Numeric", "question": "What is the number of 5 digit numbers which are divisible by 4 , with digits from the set $\\{1,2,3,4,5\\}$ and the repetition of digits is allowed?", "gold": "625" }, { "description": "JEE Adv 2018 Paper 1", "index": 45, "subject": "math", "type": "Numeric", "question": "Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11, \\ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23, \\ldots$. Then, what is the number of elements in the set $X \\cup Y$?", "gold": "3748" }, { "description": "JEE Adv 2018 Paper 1", "index": 46, "subject": "math", "type": "Numeric", "question": "What is the number of real solutions of the equation\n\n\$\n\\sin ^{-1}\\left(\\sum_{i=1}^{\\infty} x^{i+1}-x \\sum_{i=1}^{\\infty}\\left(\\frac{x}{2}\\right)^{i}\\right)=\\frac{\\pi}{2}-\\cos ^{-1}\\left(\\sum_{i=1}^{\\infty}\\left(-\\frac{x}{2}\\right)^{i}-\\sum_{i=1}^{\\infty}(-x)^{i}\\right)\n\$\n\nlying in the interval $\\left(-\\frac{1}{2}, \\frac{1}{2}\\right)$ is\n\n(Here, the inverse trigonometric functions $\\sin ^{-1} x$ and $\\cos ^{-1} x$ assume values in $\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]$ and $[0, \\pi]$, respectively.)", "gold": "2" }, { "description": "JEE Adv 2018 Paper 1", "index": 47, "subject": "math", "type": "Numeric", "question": "For each positive integer $n$, let\n\n\$\ny_{n}=\\frac{1}{n}((n+1)(n+2) \\cdots(n+n))^{\\frac{1}{n}}\n\$\n\nFor $x \\in \\mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. If $\\lim _{n \\rightarrow \\infty} y_{n}=L$, then what is the value of $[L]$?", "gold": "1" }, { "description": "JEE Adv 2018 Paper 1", "index": 48, "subject": "math", "type": "Numeric", "question": "Let $\\vec{a}$ and $\\vec{b}$ be two unit vectors such that $\\vec{a} \\cdot \\vec{b}=0$. For some $x, y \\in \\mathbb{R}$, let $\\vec{c}=x \\vec{a}+y \\vec{b}+(\\vec{a} \\times \\vec{b})$. If $|\\vec{c}|=2$ and the vector $\\vec{c}$ is inclined the same angle $\\alpha$ to both $\\vec{a}$ and $\\vec{b}$, then what is the value of $8 \\cos ^{2} \\alpha$?", "gold": "3" }, { "description": "JEE Adv 2018 Paper 1", "index": 49, "subject": "math", "type": "Numeric", "question": "Let $a, b, c$ be three non-zero real numbers such that the equation\n\n\$\n\\sqrt{3} a \\cos x+2 b \\sin x=c, x \\in\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]\n\$\n\nhas two distinct real roots $\\alpha$ and $\\beta$ with $\\alpha+\\beta=\\frac{\\pi}{3}$. Then, what is the value of $\\frac{b}{a}$?", "gold": "0.5" }, { "description": "JEE Adv 2018 Paper 1", "index": 50, "subject": "math", "type": "Numeric", "question": "A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P(0,0), Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $F_{2}$ takes away the region which lies between the side $P Q$ and a curve of the form $y=x^{n}(n>1)$. If the area of the region taken away by the farmer $F_{2}$ is exactly $30 \\%$ of the area of $\\triangle P Q R$, then what is the value of $n$?", "gold": "4" }, { "description": "JEE Adv 2018 Paper 2", "index": 1, "subject": "phy", "type": "MCQ(multiple)", "question": "A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the $x$-axis. Its kinetic energy $K$ changes with time as $d K / d t=\\gamma t$, where $\\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?\n\n(A) The force applied on the particle is constant\n\n(B) The speed of the particle is proportional to time\n\n(C) The distance of the particle from the origin increases linearly with time\n\n(D) The force is conservative", "gold": "ABD" }, { "description": "JEE Adv 2018 Paper 2", "index": 2, "subject": "phy", "type": "MCQ(multiple)", "question": "Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_{0}$. Which of the following statements is (are) true?\n\n(A) The resistive force of liquid on the plate is inversely proportional to $h$\n\n(B) The resistive force of liquid on the plate is independent of the area of the plate\n\n(C) The tangential (shear) stress on the floor of the tank increases with $u_{0}$\n\n(D) The tangential (shear) stress on the plate varies linearly with the viscosity $\\eta$ of the liquid", "gold": "ACD" }, { "description": "JEE Adv 2018 Paper 2", "index": 5, "subject": "phy", "type": "MCQ(multiple)", "question": "In a radioactive decay chain, ${ }_{90}^{232} \\mathrm{Th}$ nucleus decays to ${ }_{82}^{212} \\mathrm{~Pb}$ nucleus. Let $N_{\\alpha}$ and $N_{\\beta}$ be the number of $\\alpha$ and $\\beta^{-}$particles, respectively, emitted in this decay process. Which of the following statements is (are) true?\n\n(A) $N_{\\alpha}=5$\n\n(B) $N_{\\alpha}=6$\n\n(C) $N_{\\beta}=2$\n\n(D) $N_{\\beta}=4$", "gold": "AC" }, { "description": "JEE Adv 2018 Paper 2", "index": 6, "subject": "phy", "type": "MCQ(multiple)", "question": "In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500 \\mathrm{~Hz}$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $50.7 \\mathrm{~cm}$ and $83.9 \\mathrm{~cm}$. Which of the following statements is (are) true?\n\n(A) The speed of sound determined from this experiment is $332 \\mathrm{~ms}^{-1}$\n\n(B) The end correction in this experiment is $0.9 \\mathrm{~cm}$\n\n(C) The wavelength of the sound wave is $66.4 \\mathrm{~cm}$\n\n(D) The resonance at $50.7 \\mathrm{~cm}$ corresponds to the fundamental harmonic", "gold": "AC" }, { "description": "JEE Adv 2018 Paper 2", "index": 7, "subject": "phy", "type": "Numeric", "question": "A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4 \\mathrm{~kg}$ is at rest on this surface. An impulse of $1.0 \\mathrm{~N}$ is applied to the block at time $t=0$ so that it starts moving along the $x$-axis with a velocity $v(t)=v_{0} e^{-t / \\tau}$, where $v_{0}$ is a constant and $\\tau=4 \\mathrm{~s}$. What is the displacement of the block, in metres, at $t=\\tau$? Take $e^{-1}=0.37$", "gold": "6.3" }, { "description": "JEE Adv 2018 Paper 2", "index": 8, "subject": "phy", "type": "Numeric", "question": "A ball is projected from the ground at an angle of $45^{\\circ}$ with the horizontal surface. It reaches a maximum height of $120 \\mathrm{~m}$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30^{\\circ}$ with the horizontal surface. What is the maximum height it reaches after the bounce, in metres?", "gold": "30" }, { "description": "JEE Adv 2018 Paper 2", "index": 9, "subject": "phy", "type": "Numeric", "question": "A particle, of mass $10^{-3} \\mathrm{~kg}$ and charge $1.0 \\mathrm{C}$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\\vec{E}(t)=E_{0} \\sin \\omega t \\hat{i}$, where $E_{0}=1.0 \\mathrm{~N}^{-1}$ and $\\omega=10^{3} \\mathrm{rad} \\mathrm{s}^{-1}$. Consider the effect of only the electrical force on the particle. Then what is the maximum speed, in $m s^{-1}$, attained by the particle at subsequent times?", "gold": "2" }, { "description": "JEE Adv 2018 Paper 2", "index": 10, "subject": "phy", "type": "Numeric", "question": "A moving coil galvanometer has 50 turns and each turn has an area $2 \\times 10^{-4} \\mathrm{~m}^{2}$. The magnetic field produced by the magnet inside the galvanometer is $0.02 T$. The torsional constant of the suspension wire is $10^{-4} \\mathrm{~N} \\mathrm{~m} \\mathrm{rad}{ }^{-1}$. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $0.2 \\mathrm{rad}$. The resistance of the coil of the galvanometer is $50 \\Omega$. This galvanometer is to be converted into an ammeter capable of measuring current in the range $0-1.0 \\mathrm{~A}$. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. What is the value of this shunt resistance, in ohms?", "gold": "5.56" }, { "description": "JEE Adv 2018 Paper 2", "index": 11, "subject": "phy", "type": "Numeric", "question": "A steel wire of diameter $0.5 \\mathrm{~mm}$ and Young's modulus $2 \\times 10^{11} \\mathrm{~N} \\mathrm{~m}^{-2}$ carries a load of mass $M$. The length of the wire with the load is $1.0 \\mathrm{~m}$. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count $1.0 \\mathrm{~mm}$, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by $1.2 \\mathrm{~kg}$, what is the vernier scale division which coincides with a main scale division? Take $g=10 \\mathrm{~ms}^{-2}$ and $\\pi=3.2$", "gold": "3" }, { "description": "JEE Adv 2018 Paper 2", "index": 12, "subject": "phy", "type": "Numeric", "question": "One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100 \\mathrm{~K}$ and the universal gas constant $R=8.0 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$, what is the decrease in its internal energy, in Joule?", "gold": "900" }, { "description": "JEE Adv 2018 Paper 2", "index": 13, "subject": "phy", "type": "Numeric", "question": "In a photoelectric experiment a parallel beam of monochromatic light with power of $200 \\mathrm{~W}$ is incident on a perfectly absorbing cathode of work function $6.25 \\mathrm{eV}$. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is $100 \\%$. A potential difference of $500 \\mathrm{~V}$ is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $F=n \\times 10^{-4} N$ due to the impact of the electrons. What is the value of $n$?\n\nMass of the electron $m_{e}=9 \\times 10^{-31} \\mathrm{~kg}$ and $1.0 \\mathrm{eV}=1.6 \\times 10^{-19} \\mathrm{~J}$.", "gold": "24" }, { "description": "JEE Adv 2018 Paper 2", "index": 14, "subject": "phy", "type": "Numeric", "question": "Consider a hydrogen-like ionized atom with atomic number $Z$ with a single electron. In the emission spectrum of this atom, the photon emitted in the $n=2$ to $n=1$ transition has energy $74.8 \\mathrm{eV}$ higher than the photon emitted in the $n=3$ to $n=2$ transition. The ionization energy of the hydrogen atom is $13.6 \\mathrm{eV}$. What is the value of $Z$?", "gold": "3" }, { "description": "JEE Adv 2018 Paper 2", "index": 19, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct option(s) regarding the complex $\\left[\\mathrm{Co}(\\mathrm{en})\\left(\\mathrm{NH}_{3}\\right)_{3}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)\\right]^{3+}$ (en $=\\mathrm{H}_{2} \\mathrm{NCH}_{2} \\mathrm{CH}_{2} \\mathrm{NH}_{2}$ ) is (are)\n\n(A) It has two geometrical isomers\n\n(B) It will have three geometrical isomers if bidentate 'en' is replaced by two cyanide ligands\n\n(C) It is paramagnetic\n\n(D) It absorbs light at longer wavelength as compared to $\\left[\\mathrm{Co}(\\mathrm{en})\\left(\\mathrm{NH}_{3}\\right)_{4}\\right]^{3+}$", "gold": "ABD" }, { "description": "JEE Adv 2018 Paper 2", "index": 20, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct option(s) to distinguish nitrate salts of $\\mathrm{Mn}^{2+}$ and $\\mathrm{Cu}^{2+}$ taken separately is (are)\n\n(A) $\\mathrm{Mn}^{2+}$ shows the characteristic green colour in the flame test\n\n(B) Only $\\mathrm{Cu}^{2+}$ shows the formation of precipitate by passing $\\mathrm{H}_{2} \\mathrm{~S}$ in acidic medium\n\n(C) Only $\\mathrm{Mn}^{2+}$ shows the formation of precipitate by passing $\\mathrm{H}_{2} \\mathrm{~S}$ in faintly basic medium\n\n(D) $\\mathrm{Cu}^{2+} / \\mathrm{Cu}$ has higher reduction potential than $\\mathrm{Mn}^{2+} / \\mathrm{Mn}$ (measured under similar conditions)", "gold": "BD" }, { "description": "JEE Adv 2018 Paper 2", "index": 25, "subject": "chem", "type": "Numeric", "question": "What is the total number of compounds having at least one bridging oxo group among the molecules given below?\n\n$\\mathrm{N}_{2} \\mathrm{O}_{3}, \\mathrm{~N}_{2} \\mathrm{O}_{5}, \\mathrm{P}_{4} \\mathrm{O}_{6}, \\mathrm{P}_{4} \\mathrm{O}_{7}, \\mathrm{H}_{4} \\mathrm{P}_{2} \\mathrm{O}_{5}, \\mathrm{H}_{5} \\mathrm{P}_{3} \\mathrm{O}_{10}, \\mathrm{H}_{2} \\mathrm{~S}_{2} \\mathrm{O}_{3}, \\mathrm{H}_{2} \\mathrm{~S}_{2} \\mathrm{O}_{5}$", "gold": "6" }, { "description": "JEE Adv 2018 Paper 2", "index": 26, "subject": "chem", "type": "Numeric", "question": "Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of air is stopped, but the heating is continued in a closed furnace such that the contents undergo self-reduction. What is the weight (in $\\mathrm{kg}$ ) of $\\mathrm{Pb}$ produced per $\\mathrm{kg}$ of $\\mathrm{O}_{2}$ consumed?\n\n(Atomic weights in $\\mathrm{g} \\mathrm{mol}^{-1}: \\mathrm{O}=16, \\mathrm{~S}=32, \\mathrm{~Pb}=207$ )", "gold": "6.47" }, { "description": "JEE Adv 2018 Paper 2", "index": 27, "subject": "chem", "type": "Numeric", "question": "To measure the quantity of $\\mathrm{MnCl}_{2}$ dissolved in an aqueous solution, it was completely converted to $\\mathrm{KMnO}_{4}$ using the reaction, $\\mathrm{MnCl}_{2}+\\mathrm{K}_{2} \\mathrm{~S}_{2} \\mathrm{O}_{8}+\\mathrm{H}_{2} \\mathrm{O} \\rightarrow \\mathrm{KMnO}_{4}+\\mathrm{H}_{2} \\mathrm{SO}_{4}+\\mathrm{HCl}$ (equation not balanced).\n\nFew drops of concentrated $\\mathrm{HCl}$ were added to this solution and gently warmed. Further, oxalic acid (225 mg) was added in portions till the colour of the permanganate ion disappeared. The quantity of $\\mathrm{MnCl}_{2}$ (in $\\mathrm{mg}$ ) present in the initial solution is\n\n(Atomic weights in $\\mathrm{g} \\mathrm{mol}^{-1}: \\mathrm{Mn}=55, \\mathrm{Cl}=35.5$ )", "gold": "126" }, { "description": "JEE Adv 2018 Paper 2", "index": 30, "subject": "chem", "type": "Numeric", "question": "The surface of copper gets tarnished by the formation of copper oxide. $\\mathrm{N}_{2}$ gas was passed to prevent the oxide formation during heating of copper at $1250 \\mathrm{~K}$. However, the $\\mathrm{N}_{2}$ gas contains 1 mole $\\%$ of water vapour as impurity. The water vapour oxidises copper as per the reaction given below:\n\n$2 \\mathrm{Cu}(\\mathrm{s})+\\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) \\rightarrow \\mathrm{Cu}_{2} \\mathrm{O}(\\mathrm{s})+\\mathrm{H}_{2}(\\mathrm{~g})$\n\n$p_{\\mathrm{H}_{2}}$ is the minimum partial pressure of $\\mathrm{H}_{2}$ (in bar) needed to prevent the oxidation at $1250 \\mathrm{~K}$. What is the value of $\\ln \\left(p_{\\mathrm{H}_{2}}\\right)$?\n\n(Given: total pressure $=1$ bar, $R$ (universal gas constant $)=8 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}, \\ln (10)=2.3 \\cdot \\mathrm{Cu}(\\mathrm{s})$ and $\\mathrm{Cu}_{2} \\mathrm{O}(\\mathrm{s})$ are mutually immiscible.\n\nAt $1250 \\mathrm{~K}: 2 \\mathrm{Cu}(\\mathrm{s})+1 / 2 \\mathrm{O}_{2}(\\mathrm{~g}) \\rightarrow \\mathrm{Cu}_{2} \\mathrm{O}(\\mathrm{s}) ; \\Delta G^{\\theta}=-78,000 \\mathrm{~J} \\mathrm{~mol}^{-1}$\n\n\$\n\\mathrm{H}_{2}(\\mathrm{~g})+1 / 2 \\mathrm{O}_{2}(\\mathrm{~g}) \\rightarrow \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ; \\quad \\Delta G^{\\theta}=-1,78,000 \\mathrm{~J} \\mathrm{~mol}^{-1} ; G \\text { is the Gibbs energy) }\n\$", "gold": "-14.6" }, { "description": "JEE Adv 2018 Paper 2", "index": 31, "subject": "chem", "type": "Numeric", "question": "Consider the following reversible reaction,\n\n\$\n\\mathrm{A}(\\mathrm{g})+\\mathrm{B}(\\mathrm{g}) \\rightleftharpoons \\mathrm{AB}(\\mathrm{g})\n\$\n\nThe activation energy of the backward reaction exceeds that of the forward reaction by $2 R T$ (in $\\mathrm{J} \\mathrm{mol}^{-1}$ ). If the pre-exponential factor of the forward reaction is 4 times that of the reverse reaction, what is the absolute value of $\\Delta G^{\\theta}$ (in $\\mathrm{J} \\mathrm{mol}^{-1}$ ) for the reaction at $300 \\mathrm{~K}$?\n\n(Given; $\\ln (2)=0.7, R T=2500 \\mathrm{~J} \\mathrm{~mol}^{-1}$ at $300 \\mathrm{~K}$ and $G$ is the Gibbs energy)", "gold": "8500" }, { "description": "JEE Adv 2018 Paper 2", "index": 32, "subject": "chem", "type": "Numeric", "question": "Consider an electrochemical cell: $\\mathrm{A}(\\mathrm{s})\\left|\\mathrm{A}^{\\mathrm{n}+}(\\mathrm{aq}, 2 \\mathrm{M}) \\| \\mathrm{B}^{2 \\mathrm{n}+}(\\mathrm{aq}, 1 \\mathrm{M})\\right| \\mathrm{B}(\\mathrm{s})$. The value of $\\Delta H^{\\theta}$ for the cell reaction is twice that of $\\Delta G^{\\theta}$ at $300 \\mathrm{~K}$. If the emf of the cell is zero, what is the $\\Delta S^{\\ominus}$ (in $\\mathrm{J} \\mathrm{K}^{-1} \\mathrm{~mol}^{-1}$ ) of the cell reaction per mole of $\\mathrm{B}$ formed at $300 \\mathrm{~K}$?\n\n(Given: $\\ln (2)=0.7, R$ (universal gas constant) $=8.3 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1} . H, S$ and $G$ are enthalpy, entropy and Gibbs energy, respectively.)\n", "gold": "-11.62" }, { "description": "JEE Adv 2018 Paper 2", "index": 37, "subject": "math", "type": "MCQ(multiple)", "question": "For any positive integer $n$, define $f_{n}:(0, \\infty) \\rightarrow \\mathbb{R}$ as\n\n\$\nf_{n}(x)=\\sum_{j=1}^{n} \\tan ^{-1}\\left(\\frac{1}{1+(x+j)(x+j-1)}\\right) \\text { for all } x \\in(0, \\infty)\n\$\n\n(Here, the inverse trigonometric function $\\tan ^{-1} x$ assumes values in $\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$. ) Then, which of the following statement(s) is (are) TRUE?\n\n(A) $\\sum_{j=1}^{5} \\tan ^{2}\\left(f_{j}(0)\\right)=55$\n\n(B) $\\sum_{j=1}^{10}\\left(1+f_{j}^{\\prime}(0)\\right) \\sec ^{2}\\left(f_{j}(0)\\right)=10$\n\n(C) For any fixed positive integer $n, \\lim _{x \\rightarrow \\infty} \\tan \\left(f_{n}(x)\\right)=\\frac{1}{n}$\n\n(D) For any fixed positive integer $n$, $\\lim _{x \\rightarrow \\infty} \\sec ^{2}\\left(f_{n}(x)\\right)=1$", "gold": "D" }, { "description": "JEE Adv 2018 Paper 2", "index": 41, "subject": "math", "type": "MCQ(multiple)", "question": "Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+i y$ $(x, y \\in \\mathbb{R}, i=\\sqrt{-1})$ of the equation $s z+t \\bar{z}+r=0$, where $\\bar{z}=x-i y$. Then, which of the following statement(s) is (are) TRUE?\n\n(A) If $L$ has exactly one element, then $|s| \\neq|t|$\n\n(B) If $|s|=|t|$, then $L$ has infinitely many elements\n\n(C) The number of elements in $L \\cap\\{z:|z-1+i|=5\\}$ is at most 2\n\n(D) If $L$ has more than one element, then $L$ has infinitely many elements", "gold": "ACD" }, { "description": "JEE Adv 2018 Paper 2", "index": 43, "subject": "math", "type": "Numeric", "question": "What is the value of the integral\n\n\$\n\\int_{0}^{\\frac{1}{2}} \\frac{1+\\sqrt{3}}{\\left((x+1)^{2}(1-x)^{6}\\right)^{\\frac{1}{4}}} d x\n\$?", "gold": "2" }, { "description": "JEE Adv 2018 Paper 2", "index": 44, "subject": "math", "type": "Numeric", "question": "Let $P$ be a matrix of order $3 \\times 3$ such that all the entries in $P$ are from the set $\\{-1,0,1\\}$. Then, what is the maximum possible value of the determinant of $P$?", "gold": "4" }, { "description": "JEE Adv 2018 Paper 2", "index": 45, "subject": "math", "type": "Numeric", "question": "Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\\alpha$ is the number of one-one functions from $X$ to $Y$ and $\\beta$ is the number of onto functions from $Y$ to $X$, then what is the value of $\\frac{1}{5 !}(\\beta-\\alpha)$?", "gold": "119" }, { "description": "JEE Adv 2018 Paper 2", "index": 46, "subject": "math", "type": "Numeric", "question": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation\n\n\$\n\\frac{d y}{d x}=(2+5 y)(5 y-2)\n\$\n\nthen what is the value of $\\lim _{x \\rightarrow-\\infty} f(x)$?", "gold": "0.4" }, { "description": "JEE Adv 2018 Paper 2", "index": 47, "subject": "math", "type": "Numeric", "question": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation\n\n\$\nf(x+y)=f(x) f^{\\prime}(y)+f^{\\prime}(x) f(y) \\text { for all } x, y \\in \\mathbb{R} .\n\$\n\nThen, the value of $\\log _{e}(f(4))$ is", "gold": "2" }, { "description": "JEE Adv 2018 Paper 2", "index": 48, "subject": "math", "type": "Numeric", "question": "Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $P Q$ is perpendicular to the plane $x+y=3$ and the mid-point of $P Q$ lies in the plane $x+y=3$ ) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be 5 . If $R$ is the image of $P$ in the $x y$-plane, then what is the length of $P R$?", "gold": "8" }, { "description": "JEE Adv 2018 Paper 2", "index": 49, "subject": "math", "type": "Numeric", "question": "Consider the cube in the first octant with sides $O P, O Q$ and $O R$ of length 1 , along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0,0,0)$ is the origin. Let $S\\left(\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}\\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $O T$. If $\\vec{p}=\\overrightarrow{S P}, \\vec{q}=\\overrightarrow{S Q}, \\vec{r}=\\overrightarrow{S R}$ and $\\vec{t}=\\overrightarrow{S T}$, then what is the value of $|(\\vec{p} \\times \\vec{q}) \\times(\\vec{r} \\times \\vec{t})|$?", "gold": "0.5" }, { "description": "JEE Adv 2018 Paper 2", "index": 50, "subject": "math", "type": "Numeric", "question": "Let\n\n\$\nX=\\left({ }^{10} C_{1}\\right)^{2}+2\\left({ }^{10} C_{2}\\right)^{2}+3\\left({ }^{10} C_{3}\\right)^{2}+\\cdots+10\\left({ }^{10} C_{10}\\right)^{2}\n\$\n\nwhere ${ }^{10} C_{r}, r \\in\\{1,2, \\cdots, 10\\}$ denote binomial coefficients. Then, what is the value of $\\frac{1}{1430} X$?", "gold": "646" }, { "description": "JEE Adv 2019 Paper 1", "index": 1, "subject": "phy", "type": "MCQ", "question": "Consider a spherical gaseous cloud of mass density $\\rho(r)$ in free space where $r$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass $m$ moving in circular orbits about the common center with the same kinetic energy $K$. The force acting on the particles is their mutual gravitational force. If $\\rho(r)$ is constant in time, the particle number density $n(r)=\\rho(r) / m$ is\n\n[ $G$ is universal gravitational constant]\n\n(A) $\\frac{K}{2 \\pi r^{2} m^{2} G}$\n\n(B) $\\frac{K}{\\pi r^{2} m^{2} G}$\n\n(C) $\\frac{3 K}{\\pi r^{2} m^{2} G}$\n\n(D) $\\frac{K}{6 \\pi r^{2} m^{2} G}$", "gold": "A" }, { "description": "JEE Adv 2019 Paper 1", "index": 2, "subject": "phy", "type": "MCQ", "question": "A thin spherical insulating shell of radius $R$ carries a uniformly distributed charge such that the potential at its surface is $V_{0}$. A hole with a small area $\\alpha 4 \\pi R^{2}(\\alpha \\ll 1)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?\n\n(A) The potential at the center of the shell is reduced by $2 \\alpha V_{0}$\n\n(B) The magnitude of electric field at the center of the shell is reduced by $\\frac{\\alpha V_{0}}{2 R}$\n\n(C) The ratio of the potential at the center of the shell to that of the point at $\\frac{1}{2} R$ from center towards the hole will be $\\frac{1-\\alpha}{1-2 \\alpha}$\n\n(D) The magnitude of electric field at a point, located on a line passing through the hole and shell's center, on a distance $2 R$ from the center of the spherical shell will be reduced by $\\frac{\\alpha V_{0}}{2 R}$", "gold": "C" }, { "description": "JEE Adv 2019 Paper 1", "index": 3, "subject": "phy", "type": "MCQ", "question": "A current carrying wire heats a metal rod. The wire provides a constant power $(P)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $(T)$ in the metal rod changes with time $(t)$ as\n\n\$\nT(t)=T_{0}\\left(1+\\beta t^{\\frac{1}{4}}\\right)\n\$\n\nwhere $\\beta$ is a constant with appropriate dimension while $T_{0}$ is a constant with dimension of temperature. The heat capacity of the metal is,\n\n(A) $\\frac{4 P\\left(T(t)-T_{0}\\right)^{3}}{\\beta^{4} T_{0}^{4}}$\n\n(B) $\\frac{4 P\\left(T(t)-T_{0}\\right)^{4}}{\\beta^{4} T_{0}^{5}}$\n\n(C) $\\frac{4 P\\left(T(t)-T_{0}\\right)^{2}}{\\beta^{4} T_{0}^{3}}$\n\n(D) $\\frac{4 P\\left(T(t)-T_{0}\\right)}{\\beta^{4} T_{0}^{2}}$", "gold": "A" }, { "description": "JEE Adv 2019 Paper 1", "index": 4, "subject": "phy", "type": "MCQ", "question": "In a radioactive sample, ${ }_{19}^{40} \\mathrm{~K}$ nuclei either decay into stable ${ }_{20}^{40} \\mathrm{Ca}$ nuclei with decay constant $4.5 \\times 10^{-10}$ per year or into stable ${ }_{18}^{40} \\mathrm{Ar}$ nuclei with decay constant $0.5 \\times 10^{-10}$ per year. Given that in this sample all the stable ${ }_{20}^{40} \\mathrm{Ca}$ and ${ }_{18}^{40} \\mathrm{Ar}$ nuclei are produced by the ${ }_{19}^{40} \\mathrm{~K}$ nuclei only. In time $t \\times 10^{9}$ years, if the ratio of the sum of stable ${ }_{20}^{40} \\mathrm{Ca}$ and ${ }_{18}^{40} \\mathrm{Ar}$ nuclei to the radioactive ${ }_{19}^{40} \\mathrm{~K}$ nuclei is 99 , the value of $t$ will be,\n\n[Given: $\\ln 10=2.3]$\n\n(A) 1.15\n\n(B) 9.2\n\n(C) 2.3\n\n(D) 4.6", "gold": "B" }, { "description": "JEE Adv 2019 Paper 1", "index": 8, "subject": "phy", "type": "MCQ(multiple)", "question": "A charged shell of radius $R$ carries a total charge $Q$. Given $\\Phi$ as the flux of electric field through a closed cylindrical surface of height $h$, radius $r$ and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?\n\n$\\left[\\epsilon_{0}\\right.$ is the permittivity of free space]\n\n(A) If $h>2 R$ and $r>R$ then $\\Phi=\\mathrm{Q} / \\epsilon_{0}$\n\n(B) If $h<8 R / 5$ and $r=3 R / 5$ then $\\Phi=0$\n\n(C) If $h>2 R$ and $r=3 R / 5$ then $\\Phi=Q / 5 \\epsilon_{0}$\n\n(D) If $h>2 R$ and $r=4 R / 5$ then $\\Phi=\\mathrm{Q} / 5 \\epsilon_{0}$", "gold": "ABC" }, { "description": "JEE Adv 2019 Paper 1", "index": 11, "subject": "phy", "type": "MCQ(multiple)", "question": "Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$, which of the following statement(s) is/are correct?\n\n(A) The dimension of linear momentum is $L^{-1}$\n\n(B) The dimension of energy is $L^{-2}$\n\n(C) The dimension of force is $L^{-3}$\n\n(D) The dimension of power is $L^{-5}$", "gold": "ABC" }, { "description": "JEE Adv 2019 Paper 1", "index": 12, "subject": "phy", "type": "MCQ(multiple)", "question": "Two identical moving coil galvanometers have $10 \\Omega$ resistance and full scale deflection at $2 \\mu \\mathrm{A}$ current. One of them is converted into a voltmeter of $100 \\mathrm{mV}$ full scale reading and the other into an Ammeter of $1 \\mathrm{~mA}$ full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with $R=1000$ $\\Omega$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?\n\n(A) The resistance of the Voltmeter will be $100 \\mathrm{k} \\Omega$\n\n(B) The resistance of the Ammeter will be $0.02 \\Omega$ (round off to $2^{\\text {nd }}$ decimal place)\n\n(C) The measured value of $R$ will be $978 \\Omega10^{3}\\right)$, the capacitance $C$ is $\\alpha\\left(\\frac{K \\epsilon_{0} A}{d \\ln 2}\\right)$. What will be the value of $\\alpha$? $\\left[\\epsilon_{0}\\right.$ is the permittivity of free space]", "gold": "1" }, { "description": "JEE Adv 2019 Paper 1", "index": 19, "subject": "chem", "type": "MCQ", "question": "The green colour produced in the borax bead test of a chromium(III) salt is due to\n\n(A) $\\mathrm{Cr}\\left(\\mathrm{BO}_{2}\\right)_{3}$\n\n(B) $\\mathrm{Cr}_{2}\\left(\\mathrm{~B}_{4} \\mathrm{O}_{7}\\right)_{3}$\n\n(C) $\\mathrm{Cr}_{2} \\mathrm{O}_{3}$\n\n(D) $\\mathrm{CrB}$", "gold": "C" }, { "description": "JEE Adv 2019 Paper 1", "index": 20, "subject": "chem", "type": "MCQ", "question": "Calamine, malachite, magnetite and cryolite, respectively, are\n\n(A) $\\mathrm{ZnSO}_{4}, \\mathrm{CuCO}_{3}, \\mathrm{Fe}_{2} \\mathrm{O}_{3}, \\mathrm{AlF}_{3}$\n\n(B) $\\mathrm{ZnSO}_{4}, \\mathrm{Cu}(\\mathrm{OH})_{2}, \\mathrm{Fe}_{3} \\mathrm{O}_{4}, \\mathrm{Na}_{3} \\mathrm{AlF}_{6}$\n\n(C) $\\mathrm{ZnCO}_{3}, \\mathrm{CuCO}_{3} \\cdot \\mathrm{Cu}(\\mathrm{OH})_{2}, \\mathrm{Fe}_{3} \\mathrm{O}_{4}, \\mathrm{Na}_{3} \\mathrm{AlF}_{6}$\n\n(D) $\\mathrm{ZnCO}_{3}, \\mathrm{CuCO}_{3}, \\mathrm{Fe}_{2} \\mathrm{O}_{3}, \\mathrm{Na}_{3} \\mathrm{AlF}_{6}$", "gold": "C" }, { "description": "JEE Adv 2019 Paper 1", "index": 23, "subject": "chem", "type": "MCQ(multiple)", "question": "A tin chloride $\\mathrm{Q}$ undergoes the following reactions (not balanced)\n\n$\\mathrm{Q}+\\mathrm{Cl}^{-} \\rightarrow \\mathrm{X}$\n\n$\\mathrm{Q}+\\mathrm{Me}_{3} \\mathrm{~N} \\rightarrow \\mathrm{Y}$\n\n$\\mathbf{Q}+\\mathrm{CuCl}_{2} \\rightarrow \\mathbf{Z}+\\mathrm{CuCl}$\n\n$\\mathrm{X}$ is a monoanion having pyramidal geometry. Both $\\mathrm{Y}$ and $\\mathrm{Z}$ are neutral compounds.\n\nChoose the correct option(s)\n\n(A) The central atom in $\\mathrm{X}$ is $s p^{3}$ hybridized\n\n(B) There is a coordinate bond in $\\mathrm{Y}$\n\n(C) The oxidation state of the central atom in $\\mathrm{Z}$ is +2\n\n(D) The central atom in $\\mathrm{Z}$ has one lone pair of electrons", "gold": "AB" }, { "description": "JEE Adv 2019 Paper 1", "index": 24, "subject": "chem", "type": "MCQ(multiple)", "question": "Fusion of $\\mathrm{MnO}_{2}$ with $\\mathrm{KOH}$ in presence of $\\mathrm{O}_{2}$ produces a salt W. Alkaline solution of $\\mathbf{W}$ upon electrolytic oxidation yields another salt $\\mathrm{X}$. The manganese containing ions present in $\\mathbf{W}$ and $\\mathbf{X}$, respectively, are $\\mathbf{Y}$ and $\\mathbf{Z}$. Correct statement(s) is(are)\n\n(A) In aqueous acidic solution, $\\mathrm{Y}$ undergoes disproportionation reaction to give $\\mathrm{Z}$ and $\\mathrm{MnO}_{2}$\n\n(B) Both $\\mathrm{Y}$ and $\\mathrm{Z}$ are coloured and have tetrahedral shape\n\n(C) $\\mathrm{Y}$ is diamagnetic in nature while $\\mathrm{Z}$ is paramagnetic\n\n(D) In both $\\mathrm{Y}$ and $\\mathrm{Z}, \\pi$-bonding occurs between $p$-orbitals of oxygen and $d$-orbitals of manganese", "gold": "ABD" }, { "description": "JEE Adv 2019 Paper 1", "index": 25, "subject": "chem", "type": "MCQ(multiple)", "question": "Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.\n\n(A) $2 \\mathrm{H}_{2}(\\mathrm{~g})+\\mathrm{O}_{2}(\\mathrm{~g}) \\rightarrow 2 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{l})$\n\n(B) $2 \\mathrm{C}(\\mathrm{g})+3 \\mathrm{H}_{2}(\\mathrm{~g}) \\rightarrow \\mathrm{C}_{2} \\mathrm{H}_{6}(\\mathrm{~g})$\n\n(C) $\\frac{3}{2} \\mathrm{O}_{2}(\\mathrm{~g}) \\rightarrow \\mathrm{O}_{3}(\\mathrm{~g})$\n\n(D) $\\frac{1}{8} \\mathrm{~S}_{8}(\\mathrm{~s})+\\mathrm{O}_{2}(\\mathrm{~g}) \\rightarrow \\mathrm{SO}_{2}(\\mathrm{~g})$", "gold": "CD" }, { "description": "JEE Adv 2019 Paper 1", "index": 26, "subject": "chem", "type": "MCQ(multiple)", "question": "Which of the following statement(s) is(are) correct regarding the root mean square speed ( $\\left.u_{rms}\\right)$ and average translational kinetic energy ( $\\left.\\varepsilon_{\\text {av }}\\right)$ of a molecule in a gas at equilibrium?\n\n(A) $u_{rms}$ is doubled when its temperature is increased four times\n\n(B) $\\varepsilon_{av}}$ is doubled when its temperature is increased four times\n\n(C) $\\varepsilon_{av}$ at a given temperature does not depend on its molecular mass\n\n(D) $u_{rms}$ is inversely proportional to the square root of its molecular mass", "gold": "ACD" }, { "description": "JEE Adv 2019 Paper 1", "index": 27, "subject": "chem", "type": "MCQ(multiple)", "question": "Each of the following options contains a set of four molecules. Identify the option(s) where all four molecules possess permanent dipole moment at room temperature.\n\n(A) $\\mathrm{BeCl}_{2}, \\mathrm{CO}_{2}, \\mathrm{BCl}_{3}, \\mathrm{CHCl}_{3}$\n\n(B) $\\mathrm{NO}_{2}, \\mathrm{NH}_{3}, \\mathrm{POCl}_{3}, \\mathrm{CH}_{3} \\mathrm{Cl}$\n\n(C) $\\mathrm{BF}_{3}, \\mathrm{O}_{3}, \\mathrm{SF}_{6}, \\mathrm{XeF}_{6}$\n\n(D) $\\mathrm{SO}_{2}, \\mathrm{C}_{6} \\mathrm{H}_{5} \\mathrm{Cl}, \\mathrm{H}_{2} \\mathrm{Se}, \\mathrm{BrF}_{5}$", "gold": "BD" }, { "description": "JEE Adv 2019 Paper 1", "index": 28, "subject": "chem", "type": "MCQ(multiple)", "question": "In the decay sequence,\n\n${ }_{92}^{238} \\mathrm{U} \\stackrel{-\\mathrm{x}_{1}}{\\longrightarrow}{ }_{90}^{234} \\mathrm{Th} \\stackrel{-\\mathrm{x}_{2}}{\\longrightarrow}{ }_{91}^{234} \\mathrm{~Pa} \\stackrel{-\\mathrm{x}_{3}}{\\longrightarrow}{ }^{234} \\mathbf{Z} \\stackrel{-\\mathrm{x}_{4}}{\\longrightarrow}{ }_{90}^{230} \\mathrm{Th}$\n\n$\\mathrm{x}_{1}, \\mathrm{x}_{2}, \\mathrm{x}_{3}$ and $\\mathrm{x}_{4}$ are particles/radiation emitted by the respective isotopes. The correct option(s) is $($ are $)$\n\n(A) $x_{1}$ will deflect towards negatively charged plate\n\n(B) $\\mathrm{x}_{2}$ is $\\beta^{-}$\n\n(C) $x_{3}$ is $\\gamma$-ray\n\n(D) $\\mathbf{Z}$ is an isotope of uranium", "gold": "ABD" }, { "description": "JEE Adv 2019 Paper 1", "index": 29, "subject": "chem", "type": "MCQ(multiple)", "question": "Which of the following statement(s) is(are) true?\n\n(A) Monosaccharides cannot be hydrolysed to give polyhydroxy aldehydes and ketones\n\n(B) Oxidation of glucose with bromine water gives glutamic acid\n\n(C) Hydrolysis of sucrose gives dextrorotatory glucose and laevorotatory fructose\n\n(D) The two six-membered cyclic hemiacetal forms of $\\mathrm{D}-(+)$-glucose are called anomers", "gold": "ACD" }, { "description": "JEE Adv 2019 Paper 1", "index": 31, "subject": "chem", "type": "Numeric", "question": "Among $\\mathrm{B}_{2} \\mathrm{H}_{6}, \\mathrm{~B}_{3} \\mathrm{~N}_{3} \\mathrm{H}_{6}, \\mathrm{~N}_{2} \\mathrm{O}, \\mathrm{N}_{2} \\mathrm{O}_{4}, \\mathrm{H}_{2} \\mathrm{~S}_{2} \\mathrm{O}_{3}$ and $\\mathrm{H}_{2} \\mathrm{~S}_{2} \\mathrm{O}_{8}$, what is the total number of molecules containing covalent bond between two atoms of the same kind?", "gold": "4" }, { "description": "JEE Adv 2019 Paper 1", "index": 32, "subject": "chem", "type": "Numeric", "question": "At $143 \\mathrm{~K}$, the reaction of $\\mathrm{XeF}_{4}$ with $\\mathrm{O}_{2} \\mathrm{~F}_{2}$ produces a xenon compound $\\mathrm{Y}$. What is the total number of lone pair(s) of electrons present on the whole molecule of $\\mathrm{Y}$?", "gold": "19" }, { "description": "JEE Adv 2019 Paper 1", "index": 34, "subject": "chem", "type": "Numeric", "question": "On dissolving $0.5 \\mathrm{~g}$ of a non-volatile non-ionic solute to $39 \\mathrm{~g}$ of benzene, its vapor pressure decreases from $650 \\mathrm{~mm} \\mathrm{Hg}$ to $640 \\mathrm{~mm} \\mathrm{Hg}$. What is the depression of freezing point of benzene (in $\\mathrm{K}$ ) upon addition of the solute? (Given data: Molar mass and the molal freezing point depression constant of benzene are $78 \\mathrm{~g}$ $\\mathrm{mol}^{-1}$ and $5.12 \\mathrm{~K} \\mathrm{~kg} \\mathrm{~mol}^{-1}$, respectively)", "gold": "1.02" }, { "description": "JEE Adv 2019 Paper 1", "index": 35, "subject": "chem", "type": "Numeric", "question": "Consider the kinetic data given in the following table for the reaction $\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow$ Product.\n\n\\begin{center}\n\n\\begin{tabular}{|c|c|c|c|c|}\n\n\\hline\n\nExperiment No. & $\\begin{array}{c}{[\\mathrm{A}]} \\\\ \\left(\\mathrm{mol} \\mathrm{dm}^{-3}\\right)\\end{array}$ & $\\begin{array}{c}{[\\mathrm{B}]} \\\\ \\left(\\mathrm{mol} \\mathrm{dm}^{-3}\\right)\\end{array}$ & $\\begin{array}{c}{[\\mathrm{C}]} \\\\ \\left(\\mathrm{mol} \\mathrm{dm}^{-3}\\right)\\end{array}$ & $\\begin{array}{c}\\text { Rate of reaction } \\\\ \\left(\\mathrm{mol} \\mathrm{dm}^{-3} \\mathrm{~s}^{-1}\\right)\\end{array}$ \\\\\n\n\\hline\n\n1 & 0.2 & 0.1 & 0.1 & $6.0 \\times 10^{-5}$ \\\\\n\n\\hline\n\n2 & 0.2 & 0.2 & 0.1 & $6.0 \\times 10^{-5}$ \\\\\n\n\\hline\n\n3 & 0.2 & 0.1 & 0.2 & $1.2 \\times 10^{-4}$ \\\\\n\n\\hline\n\n4 & 0.3 & 0.1 & 0.1 & $9.0 \\times 10^{-5}$ \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}\n\nThe rate of the reaction for $[\\mathrm{A}]=0.15 \\mathrm{~mol} \\mathrm{dm}^{-3},[\\mathrm{~B}]=0.25 \\mathrm{~mol} \\mathrm{dm}^{-3}$ and $[\\mathrm{C}]=0.15 \\mathrm{~mol} \\mathrm{dm}^{-3}$ is found to be $\\mathbf{Y} \\times 10^{-5} \\mathrm{~mol} \\mathrm{dm}^{-3} \\mathrm{~s}^{-1}$. What is the value of $\\mathbf{Y}$?", "gold": "6.75" }, { "description": "JEE Adv 2019 Paper 1", "index": 37, "subject": "math", "type": "MCQ", "question": "Let $S$ be the set of all complex numbers $Z$ satisfying $|z-2+i| \\geq \\sqrt{5}$. If the complex number $Z_{0}$ is such that $\\frac{1}{\\left|Z_{0}-1\\right|}$ is the maximum of the set $\\left\\{\\frac{1}{|z-1|}: z \\in S\\right\\}$, then the principal argument of $\\frac{4-z_{0}-\\overline{z_{0}}}{Z_{0}-\\overline{z_{0}}+2 i}$ is\n\n(A) $-\\frac{\\pi}{2}$\n\n(B) $\\frac{\\pi}{4}$\n\n(C) $\\frac{\\pi}{2}$\n\n(D) $\\frac{3 \\pi}{4}$", "gold": "A" }, { "description": "JEE Adv 2019 Paper 1", "index": 38, "subject": "math", "type": "MCQ", "question": "Let\n\n\$\nM=\\left[\\begin{array}{cc}\n\n\\sin ^{4} \\theta & -1-\\sin ^{2} \\theta \\\\\n\n1+\\cos ^{2} \\theta & \\cos ^{4} \\theta\n\n\\end{array}\\right]=\\alpha I+\\beta M^{-1}\n\$\n\nwhere $\\alpha=\\alpha(\\theta)$ and $\\beta=\\beta(\\theta)$ are real numbers, and $I$ is the $2 \\times 2$ identity matrix. If\n\n$\\alpha^{*}$ is the minimum of the set $\\{\\alpha(\\theta): \\theta \\in[0,2 \\pi)\\}$ and\n\n$\\beta^{*}$ is the minimum of the set $\\{\\beta(\\theta): \\theta \\in[0,2 \\pi)\\}$\n\nthen the value of $\\alpha^{*}+\\beta^{*}$ is\n\n(A) $-\\frac{37}{16}$\n\n(B) $-\\frac{31}{16}$\n\n(C) $-\\frac{29}{16}$\n\n(D) $-\\frac{17}{16}$", "gold": "C" }, { "description": "JEE Adv 2019 Paper 1", "index": 39, "subject": "math", "type": "MCQ", "question": "A line $y=m x+1$ intersects the circle $(x-3)^{2}+(y+2)^{2}=25$ at the points $P$ and $Q$. If the midpoint of the line segment $P Q$ has $x$-coordinate $-\\frac{3}{5}$, then which one of the following options is correct?\n\n(A) $-3 \\leq m<-1$\n\n(B) $2 \\leq m<4$\n\n(C) $4 \\leq m<6$\n\n(D) $6 \\leq m<8$", "gold": "B" }, { "description": "JEE Adv 2019 Paper 1", "index": 40, "subject": "math", "type": "MCQ", "question": "The area of the region $\\left\\{(x, y): x y \\leq 8,1 \\leq y \\leq x^{2}\\right\\}$ is\n\n(A) $16 \\log _{e} 2-\\frac{14}{3}$\n\n(B) $8 \\log _{e} 2-\\frac{14}{3}$\n\n(C) $16 \\log _{e} 2-6$\n\n(D) $8 \\log _{e} 2-\\frac{7}{3}$", "gold": "A" }, { "description": "JEE Adv 2019 Paper 1", "index": 42, "subject": "math", "type": "MCQ(multiple)", "question": "Let\n\n\$\nM=\\left[\\begin{array}{lll}\n\n0 & 1 & a \\\\\n\n1 & 2 & 3 \\\\\n\n3 & b & 1\n\n\\end{array}\\right] \\quad \\text { and adj } M=\\left[\\begin{array}{rrr}\n\n-1 & 1 & -1 \\\\\n\n8 & -6 & 2 \\\\\n\n-5 & 3 & -1\n\n\\end{array}\\right]\n\$\n\nwhere $a$ and $b$ are real numbers. Which of the following options is/are correct?\n\n(A) $a+b=3$\n\n(B) $(\\operatorname{adj} M)^{-1}+\\operatorname{adj} M^{-1}=-M$\n\n(C) $\\operatorname{det}\\left(\\operatorname{adj} M^{2}\\right)=81$\n\n(D) If $M\\left[\\begin{array}{l}\\alpha \\\\ \\beta \\\\ \\gamma\\end{array}\\right]=\\left[\\begin{array}{l}1 \\\\ 2 \\\\ 3\\end{array}\\right]$, then $\\alpha-\\beta+\\gamma=3$", "gold": "ABD" }, { "description": "JEE Adv 2019 Paper 1", "index": 43, "subject": "math", "type": "MCQ(multiple)", "question": "There are three bags $B_{1}, B_{2}$ and $B_{3}$. The bag $B_{1}$ contains 5 red and 5 green balls, $B_{2}$ contains 3 red and 5 green balls, and $B_{3}$ contains 5 red and 3 green balls. Bags $B_{1}, B_{2}$ and $B_{3}$ have probabilities $\\frac{3}{10}, \\frac{3}{10}$ and $\\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?\n\n(A) Probability that the chosen ball is green, given that the selected bag is $B_{3}$, equals $\\frac{3}{8}$\n\n(B) Probability that the chosen ball is green equals $\\frac{39}{80}$\n\n(C) Probability that the selected bag is $B_{3}$, given that the chosen ball is green, equals $\\frac{5}{13}$\n\n(D) Probability that the selected bag is $B_{3}$ and the chosen ball is green equals $\\frac{3}{10}$", "gold": "AB" }, { "description": "JEE Adv 2019 Paper 1", "index": 44, "subject": "math", "type": "MCQ(multiple)", "question": "In a non-right-angled triangle $\\triangle P Q R$, let $p, q, r$ denote the lengths of the sides opposite to the angles at $P, Q, R$ respectively. The median from $R$ meets the side $P Q$ at $S$, the perpendicular from $P$ meets the side $Q R$ at $E$, and $R S$ and $P E$ intersect at $O$. If $p=\\sqrt{3}, q=1$, and the radius of the circumcircle of the $\\triangle P Q R$ equals 1 , then which of the following options is/are correct?\n\n(A) Length of $R S=\\frac{\\sqrt{7}}{2}$\n\n(B) Area of $\\triangle S O E=\\frac{\\sqrt{3}}{12}$\n\n(C) Length of $O E=\\frac{1}{6}$\n\n(D) Radius of incircle of $\\triangle P Q R=\\frac{\\sqrt{3}}{2}(2-\\sqrt{3})$", "gold": "ACD" }, { "description": "JEE Adv 2019 Paper 1", "index": 45, "subject": "math", "type": "MCQ(multiple)", "question": "Define the collections $\\left\\{E_{1}, E_{2}, E_{3}, \\ldots\\right\\}$ of ellipses and $\\left\\{R_{1}, R_{2}, R_{3}, \\ldots\\right\\}$ of rectangles as follows:\n\n$E_{1}: \\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1$\n\n$R_{1}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_{1}$;\n\n$E_{n}:$ ellipse $\\frac{x^{2}}{a_{n}^{2}}+\\frac{y^{2}}{b_{n}^{2}}=1$ of largest area inscribed in $R_{n-1}, n>1$;\n\n$R_{n}:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_{n}, n>1$.\n\nThen which of the following options is/are correct?\n\n(A) The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal\n\n(B) $\\quad \\sum_{n=1}^{N}\\left(\\right.$ area of $\\left.R_{n}\\right)<24$, for each positive integer $N$\n\n(C) The length of latus rectum of $E_{9}$ is $\\frac{1}{6}$\n\n(D) The distance of a focus from the centre in $E_{9}$ is $\\frac{\\sqrt{5}}{32}$", "gold": "BC" }, { "description": "JEE Adv 2019 Paper 1", "index": 47, "subject": "math", "type": "MCQ(multiple)", "question": "Let $\\Gamma$ denote a curve $y=y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\\Gamma$ at a point $P$ intersect the $y$-axis at $Y_{P}$. If $P Y_{P}$ has length 1 for each point $P$ on $\\Gamma$, then which of the following options is/are correct?\n\n(A) $y=\\log _{e}\\left(\\frac{1+\\sqrt{1-x^{2}}}{x}\\right)-\\sqrt{1-x^{2}}$\n\n(B) $x y^{\\prime}+\\sqrt{1-x^{2}}=0$\n\n(C) $y=-\\log _{e}\\left(\\frac{1+\\sqrt{1-x^{2}}}{x}\\right)+\\sqrt{1-x^{2}}$\n\n(D) $x y^{\\prime}-\\sqrt{1-x^{2}}=0$", "gold": "AB" }, { "description": "JEE Adv 2019 Paper 1", "index": 48, "subject": "math", "type": "MCQ(multiple)", "question": "Let $L_{1}$ and $L_{2}$ denote the lines\n\n\$\n\\vec{r}=\\hat{i}+\\lambda(-\\hat{i}+2 \\hat{j}+2 \\hat{k}), \\lambda \\in \\mathbb{R}\n\$\n and \n\$\\vec{r}=\\mu(2 \\hat{i}-\\hat{j}+2 \\hat{k}), \\mu \\in \\mathbb{R}\n\$\n\nrespectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, then which of the following options describe(s) $L_{3}$ ?\n\n(A) $\\vec{r}=\\frac{2}{9}(4 \\hat{i}+\\hat{j}+\\hat{k})+t(2 \\hat{i}+2 \\hat{j}-\\hat{k}), t \\in \\mathbb{R}$\n\n(B) $\\vec{r}=\\frac{2}{9}(2 \\hat{i}-\\hat{j}+2 \\hat{k})+t(2 \\hat{i}+2 \\hat{j}-\\hat{k}), t \\in \\mathbb{R}$\n\n(C) $\\vec{r}=\\frac{1}{3}(2 \\hat{i}+\\hat{k})+t(2 \\hat{i}+2 \\hat{j}-\\hat{k}), t \\in \\mathbb{R}$\n\n(D) $\\vec{r}=t(2 \\hat{i}+2 \\hat{j}-\\hat{k}), t \\in \\mathbb{R}$", "gold": "ABC" }, { "description": "JEE Adv 2019 Paper 1", "index": 49, "subject": "math", "type": "Numeric", "question": "Let $\\omega \\neq 1$ be a cube root of unity. Then what is the minimum of the set\n\n\$\n\\left\\{\\left|a+b \\omega+c \\omega^{2}\\right|^{2}: a, b, c \\text { distinct non-zero integers }\\right\\}\n\$ equal?", "gold": "3" }, { "description": "JEE Adv 2019 Paper 1", "index": 50, "subject": "math", "type": "Numeric", "question": "Let $A P(a ; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d>0$. If\n\n\$\nA P(1 ; 3) \\cap A P(2 ; 5) \\cap A P(3 ; 7)=A P(a ; d)\n\$\n\nthen what does $a+d$ equal?", "gold": "157" }, { "description": "JEE Adv 2019 Paper 1", "index": 51, "subject": "math", "type": "Numeric", "question": "Let $S$ be the sample space of all $3 \\times 3$ matrices with entries from the set $\\{0,1\\}$. Let the events $E_{1}$ and $E_{2}$ be given by\n\n\\n\\begin{aligned}\n\n& E_{1}=\\{A \\in S: \\operatorname{det} A=0\\} \\text { and } \\\\\n\n& E_{2}=\\{A \\in S: \\text { sum of entries of } A \\text { is } 7\\} .\n\n\\end{aligned}\n\\n\nIf a matrix is chosen at random from $S$, then what is the conditional probability $P\\left(E_{1} \\mid E_{2}\\right)$?", "gold": "0.5" }, { "description": "JEE Adv 2019 Paper 1", "index": 52, "subject": "math", "type": "Numeric", "question": "Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\\Gamma_{A}$ and $\\Gamma_{B}$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\\Gamma_{A}$ and $\\Gamma_{B}$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then what is the length of the line segment $A C$?", "gold": "10" }, { "description": "JEE Adv 2019 Paper 1", "index": 53, "subject": "math", "type": "Numeric", "question": "If\n\n\$\nI=\\frac{2}{\\pi} \\int_{-\\pi / 4}^{\\pi / 4} \\frac{d x}{\\left(1+e^{\\sin x}\\right)(2-\\cos 2 x)}\n\$\n\nthen what does $27 I^{2}$ equal?", "gold": "4" }, { "description": "JEE Adv 2019 Paper 1", "index": 54, "subject": "math", "type": "Numeric", "question": "Three lines are given by\n\n\$\n\\vec{r} & =\\lambda \\hat{i}, \\lambda \\in \\mathbb{R}\n\$\n\$\\vec{r} & =\\mu(\\hat{i}+\\hat{j}), \\mu \\in \\mathbb{R}\n\$\n \$\n\\vec{r} =v(\\hat{i}+\\hat{j}+\\hat{k}), v \\in \\mathbb{R}.\n\$\n\nLet the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $\\triangle$ then what is the value of $(6 \\Delta)^{2}$?", "gold": "0.75" }, { "description": "JEE Adv 2019 Paper 2", "index": 1, "subject": "phy", "type": "MCQ(multiple)", "question": "A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle $60^{\\circ}$ with vertical?\n\n[ $g$ is the acceleration due to gravity]\n\n(A) The angular speed of the rod will be $\\sqrt{\\frac{3 g}{2 L}}$\n\n(B) The angular acceleration of the rod will be $\\frac{2 g}{L}$\n\n(C) The radial acceleration of the rod's center of mass will be $\\frac{3 g}{4}$\n\n(D) The normal reaction force from the floor on the rod will be $\\frac{M g}{16}$", "gold": "ACD" }, { "description": "JEE Adv 2019 Paper 2", "index": 5, "subject": "phy", "type": "MCQ(multiple)", "question": "A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure $P_{0}$, volume $V_{0}$, and temperature $T_{0}$. If the gas mixture is adiabatically compressed to a volume $V_{0} / 4$, then the correct statement(s) is/are, (Given $2^{1.2}=2.3 ; 2^{3.2}=9.2 ; R$ is gas constant)\n\n(A) The work $|W|$ done during the process is $13 R T_{0}$\n\n(B) The average kinetic energy of the gas mixture after compression is in between $18 R T_{0}$ and $19 R T_{0}$\n\n(C) The final pressure of the gas mixture after compression is in between $9 P_{0}$ and $10 P_{0}$\n\n(D) Adiabatic constant of the gas mixture is 1.6", "gold": "ACD" }, { "description": "JEE Adv 2019 Paper 2", "index": 8, "subject": "phy", "type": "MCQ(multiple)", "question": "A free hydrogen atom after absorbing a photon of wavelength $\\lambda_{a}$ gets excited from the state $n=1$ to the state $n=4$. Immediately after that the electron jumps to $n=m$ state by emitting a photon of wavelength $\\lambda_{e}$. Let the change in momentum of atom due to the absorption and the emission are $\\Delta p_{a}$ and $\\Delta p_{e}$, respectively. If $\\lambda_{a} / \\lambda_{e}=\\frac{1}{5}$, which of the option(s) is/are correct?\n\n[Use $h c=1242 \\mathrm{eV} \\mathrm{nm} ; 1 \\mathrm{~nm}=10^{-9} \\mathrm{~m}, h$ and $c$ are Planck's constant and speed of light, respectively]\n\n(A) $m=2$\n\n(B) $\\lambda_{e}=418 \\mathrm{~nm}$\n\n(C) $\\Delta p_{a} / \\Delta p_{e}=\\frac{1}{2}$\n\n(D) The ratio of kinetic energy of the electron in the state $n=m$ to the state $n=1$ is $\\frac{1}{4}$", "gold": "AD" }, { "description": "JEE Adv 2019 Paper 2", "index": 13, "subject": "phy", "type": "Numeric", "question": "Suppose a ${ }_{88}^{226} R a$ nucleus at rest and in ground state undergoes $\\alpha$-decay to a ${ }_{86}^{222} R n$ nucleus in its excited state. The kinetic energy of the emitted $\\alpha$ particle is found to be $4.44 \\mathrm{MeV}$. ${ }_{86}^{222} R n$ nucleus then goes to its ground state by $\\gamma$-decay. What is the energy of the emitted $\\gamma$ photon is $\\mathrm{keV}$?\n\n[Given: atomic mass of ${ }_{88}^{226} R a=226.005 \\mathrm{u}$, atomic mass of ${ }_{86}^{222} R n=222.000 \\mathrm{u}$, atomic mass of $\\alpha$ particle $=4.000 \\mathrm{u}, 1 \\mathrm{u}=931 \\mathrm{MeV} / \\mathrm{c}^{2}, \\mathrm{c}$ is speed of the light]", "gold": "135" }, { "description": "JEE Adv 2019 Paper 2", "index": 14, "subject": "phy", "type": "Numeric", "question": "An optical bench has $1.5 \\mathrm{~m}$ long scale having four equal divisions in each $\\mathrm{cm}$. While measuring the focal length of a convex lens, the lens is kept at $75 \\mathrm{~cm}$ mark of the scale and the object pin is kept at $45 \\mathrm{~cm}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135 \\mathrm{~cm}$ mark. In this experiment, what is the percentage error in the measurement of the focal length of the lens?", "gold": "0.69" }, { "description": "JEE Adv 2019 Paper 2", "index": 19, "subject": "chem", "type": "MCQ(multiple)", "question": "The cyanide process of gold extraction involves leaching out gold from its ore with $\\mathrm{CN}^{-}$in the presence of $\\mathbf{Q}$ in water to form $\\mathrm{R}$. Subsequently, $\\mathrm{R}$ is treated with $\\mathrm{T}$ to obtain $\\mathrm{Au}$ and $\\mathrm{Z}$. Choose the correct option(s)\n\n(A) $\\mathrm{Q}$ is $\\mathrm{O}_{2}$\n\n(B) $\\mathrm{T}$ is $\\mathrm{Zn}$\n\n(C) $\\mathrm{Z}$ is $\\left[\\mathrm{Zn}(\\mathrm{CN})_{4}\\right]^{2-}$\n\n(D) $\\mathrm{R}$ is $\\left[\\mathrm{Au}(\\mathrm{CN})_{4}\\right]^{-}$", "gold": "ABC" }, { "description": "JEE Adv 2019 Paper 2", "index": 20, "subject": "chem", "type": "MCQ(multiple)", "question": "With reference to aqua regia, choose the correct option(s)\n\n(A) Aqua regia is prepared by mixing conc. $\\mathrm{HCl}$ and conc. $\\mathrm{HNO}_{3}$ in $3: 1(v / v)$ ratio\n\n(B) Reaction of gold with aqua regia produces an anion having Au in +3 oxidation state\n\n(C) Reaction of gold with aqua regia produces $\\mathrm{NO}_{2}$ in the absence of air\n\n(D) The yellow colour of aqua regia is due to the presence of $\\mathrm{NOCl}$ and $\\mathrm{Cl}_{2}$", "gold": "ABD" }, { "description": "JEE Adv 2019 Paper 2", "index": 21, "subject": "chem", "type": "MCQ(multiple)", "question": "Consider the following reactions (unbalanced)\n\n$\\mathrm{Zn}+$ hot conc. $\\mathrm{H}_{2} \\mathrm{SO}_{4} \\rightarrow \\mathrm{G}+\\mathrm{R}+\\mathrm{X}$\n\n$\\mathrm{Zn}+$ conc. $\\mathrm{NaOH} \\rightarrow \\mathrm{T}+\\mathbf{Q}$\n\n$\\mathbf{G}+\\mathrm{H}_{2} \\mathrm{~S}+\\mathrm{NH}_{4} \\mathrm{OH} \\rightarrow \\mathbf{Z}$ (a precipitate) $+\\mathbf{X}+\\mathrm{Y}$\n\nChoose the correct option(s)\n\n(A) $\\mathrm{Z}$ is dirty white in colour\n\n(B) The oxidation state of $\\mathrm{Zn}$ in $\\mathrm{T}$ is +1\n\n(C) $\\mathrm{R}$ is a V-shaped molecule\n\n(D) Bond order of $\\mathbf{Q}$ is 1 in its ground state", "gold": "ACD" }, { "description": "JEE Adv 2019 Paper 2", "index": 22, "subject": "chem", "type": "MCQ(multiple)", "question": "The ground state energy of hydrogen atom is $-13.6 \\mathrm{eV}$. Consider an electronic state $\\Psi$ of $\\mathrm{He}^{+}$ whose energy, azimuthal quantum number and magnetic quantum number are $-3.4 \\mathrm{eV}, 2$ and 0 , respectively. Which of the following statement(s) is(are) true for the state $\\Psi$ ?\n\n(A) It is a $4 d$ state\n\n(B) It has 2 angular nodes\n\n(C) It has 3 radial nodes\n\n(D) The nuclear charge experienced by the electron in this state is less than $2 e$, where $e$ is the magnitude of the electronic charge", "gold": "AB" }, { "description": "JEE Adv 2019 Paper 2", "index": 26, "subject": "chem", "type": "MCQ(multiple)", "question": "Choose the correct option(s) from the following\n\n(A) Natural rubber is polyisoprene containing trans alkene units\n\n(B) Nylon-6 has amide linkages\n\n(C) Teflon is prepared by heating tetrafluoroethene in presence of a persulphate catalyst at high pressure\n\n(D) Cellulose has only $\\alpha$-D-glucose units that are joined by glycosidic linkages", "gold": "BC" }, { "description": "JEE Adv 2019 Paper 2", "index": 27, "subject": "chem", "type": "Numeric", "question": "What is the amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. $\\mathrm{HNO}_{3}$ to a compound with the highest oxidation state of sulphur?\n\n(Given data: Molar mass of water $=18 \\mathrm{~g} \\mathrm{~mol}^{-1}$ )", "gold": "288" }, { "description": "JEE Adv 2019 Paper 2", "index": 28, "subject": "chem", "type": "Numeric", "question": "What is the total number of cis $\\mathrm{N}-\\mathrm{Mn}-\\mathrm{Cl}$ bond angles (that is, $\\mathrm{Mn}-\\mathrm{N}$ and $\\mathrm{Mn}-\\mathrm{Cl}$ bonds in cis positions) present in a molecule of cis-[Mn(en $\\left.)_{2} \\mathrm{Cl}_{2}\\right]$ complex?\n\n(en $=\\mathrm{NH}_{2} \\mathrm{CH}_{2} \\mathrm{CH}_{2} \\mathrm{NH}_{2}$ )", "gold": "6" }, { "description": "JEE Adv 2019 Paper 2", "index": 29, "subject": "chem", "type": "Numeric", "question": "The decomposition reaction $2 \\mathrm{~N}_{2} \\mathrm{O}_{5}(g) \\stackrel{\\Delta}{\\rightarrow} 2 \\mathrm{~N}_{2} \\mathrm{O}_{4}(g)+\\mathrm{O}_{2}(g)$ is started in a closed cylinder under isothermal isochoric condition at an initial pressure of $1 \\mathrm{~atm}$. After $\\mathrm{Y} \\times 10^{3} \\mathrm{~s}$, the pressure inside the cylinder is found to be $1.45 \\mathrm{~atm}$. If the rate constant of the reaction is $5 \\times 10^{-4} \\mathrm{~s}^{-1}$, assuming ideal gas behavior, what is the value of $\\mathrm{Y}$?", "gold": "2.3" }, { "description": "JEE Adv 2019 Paper 2", "index": 30, "subject": "chem", "type": "Numeric", "question": "The mole fraction of urea in an aqueous urea solution containing $900 \\mathrm{~g}$ of water is 0.05 . If the density of the solution is $1.2 \\mathrm{~g} \\mathrm{~cm}^{-3}$, what is the molarity of urea solution? (Given data: Molar masses of urea and water are $60 \\mathrm{~g} \\mathrm{~mol}^{-1}$ and $18 \\mathrm{~g} \\mathrm{~mol}^{-1}$, respectively)", "gold": "2.98" }, { "description": "JEE Adv 2019 Paper 2", "index": 32, "subject": "chem", "type": "Numeric", "question": "What is the total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula $\\mathrm{C}_{4} \\mathrm{H}_{8} \\mathrm{O}$?", "gold": "10" }, { "description": "JEE Adv 2019 Paper 2", "index": 37, "subject": "math", "type": "MCQ(multiple)", "question": "Let\n\n\\n\\begin{aligned}\n\n& P_{1}=I=\\left[\\begin{array}{lll}\n\n1 & 0 & 0 \\\\\n\n0 & 1 & 0 \\\\\n\n0 & 0 & 1\n\n\\end{array}\\right], \\quad P_{2}=\\left[\\begin{array}{lll}\n\n1 & 0 & 0 \\\\\n\n0 & 0 & 1 \\\\\n\n0 & 1 & 0\n\n\\end{array}\\right], \\quad P_{3}=\\left[\\begin{array}{lll}\n\n0 & 1 & 0 \\\\\n\n1 & 0 & 0 \\\\\n\n0 & 0 & 1\n\n\\end{array}\\right], \\\\\n\n& P_{4}=\\left[\\begin{array}{lll}\n\n0 & 1 & 0 \\\\\n\n0 & 0 & 1 \\\\\n\n1 & 0 & 0\n\n\\end{array}\\right], \\quad P_{5}=\\left[\\begin{array}{lll}\n\n0 & 0 & 1 \\\\\n\n1 & 0 & 0 \\\\\n\n0 & 1 & 0\n\n\\end{array}\\right], \\quad P_{6}=\\left[\\begin{array}{ccc}\n\n0 & 0 & 1 \\\\\n\n0 & 1 & 0 \\\\\n\n1 & 0 & 0\n\n\\end{array}\\right] \\\\\n\n& \\text { and } X=\\sum_{k=1}^{6} P_{k}\\left[\\begin{array}{lll}\n\n2 & 1 & 3 \\\\\n\n1 & 0 & 2 \\\\\n\n3 & 2 & 1\n\n\\end{array}\\right] P_{k}^{T}\n\n\\end{aligned}\n\\n\nwhere $P_{k}^{T}$ denotes the transpose of the matrix $P_{k}$. Then which of the following options is/are correct?\n\n(A) If $X\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1\\end{array}\\right]=\\alpha\\left[\\begin{array}{l}1 \\\\ 1 \\\\ 1\\end{array}\\right]$, then $\\alpha=30$\n\n(B) $X$ is a symmetric matrix\n\n(C) The sum of diagonal entries of $X$ is 18\n\n(D) $X-30 I$ is an invertible matrix", "gold": "ABC" }, { "description": "JEE Adv 2019 Paper 2", "index": 40, "subject": "math", "type": "MCQ(multiple)", "question": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a function. We say that $f$ has\n\nPROPERTY 1 if $\\lim _{h \\rightarrow 0} \\frac{f(h)-f(0)}{\\sqrt{|h|}}$ exists and is finite, and\n\nPROPERTY 2 if $\\lim _{h \\rightarrow 0} \\frac{f(h)-f(0)}{h^{2}}$ exists and is finite.\n\nThen which of the following options is/are correct?\n\n(A) $f(x)=|x|$ has PROPERTY 1\n\n(B) $f(x)=x^{2 / 3}$ has PROPERTY 1\n\n(C) $f(x)=x|x|$ has PROPERTY 2\n\n(D) $f(x)=\\sin x$ has PROPERTY 2", "gold": "AB" }, { "description": "JEE Adv 2019 Paper 2", "index": 41, "subject": "math", "type": "MCQ(multiple)", "question": "Let\n\n\$\nf(x)=\\frac{\\sin \\pi x}{x^{2}}, \\quad x>0\n\$\n\nLet $x_{1}2$ for every $n$\n\n(C) $\\quad x_{n} \\in\\left(2 n, 2 n+\\frac{1}{2}\\right)$ for every $n$\n\n(D) $\\left|x_{n}-y_{n}\\right|>1$ for every $n$", "gold": "BCD" }, { "description": "JEE Adv 2019 Paper 2", "index": 42, "subject": "math", "type": "MCQ(multiple)", "question": "For $a \\in \\mathbb{R},|a|>1$, let\n\n\$\n\\lim _{n \\rightarrow \\infty}\\left(\\frac{1+\\sqrt[3]{2}+\\cdots+\\sqrt[3]{n}}{n^{7 / 3}\\left(\\frac{1}{(a n+1)^{2}}+\\frac{1}{(a n+2)^{2}}+\\cdots+\\frac{1}{(a n+n)^{2}}\\right)}\\right)=54\n\$\n\nThen the possible value(s) of $a$ is/are\n\n(A) -9\n\n(B) -6\n\n(C) 7\n\n(D) 8", "gold": "AD" }, { "description": "JEE Adv 2019 Paper 2", "index": 43, "subject": "math", "type": "MCQ(multiple)", "question": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define\n\n\$\nF(x)=\\int_{0}^{x} f(t) d t, \\quad x>0 .\n\$\n\nThen which of the following options is/are correct?\n\n(A) $F$ has a local minimum at $x=1$\n\n(B) $F$ has a local maximum at $x=2$\n\n(C) $F$ has two local maxima and one local minimum in $(0, \\infty)$\n\n(D) $\\quad F(x) \\neq 0$ for all $x \\in(0,5)$", "gold": "ABD" }, { "description": "JEE Adv 2019 Paper 2", "index": 44, "subject": "math", "type": "MCQ(multiple)", "question": "Three lines\n\n\\n\\begin{aligned}\n\nL_{1}: & \\vec{r}=\\lambda \\hat{i}, \\lambda \\in \\mathbb{R}, \\\\\n\nL_{2}: & \\vec{r}=\\hat{k}+\\mu \\hat{j}, \\mu \\in \\mathbb{R} \\text { and } \\\\\n\nL_{3}: & \\vec{r}=\\hat{i}+\\hat{j}+v \\hat{k}, \\quad v \\in \\mathbb{R}\n\n\\end{aligned}\n\\n\nare given. For which point(s) $Q$ on $L_{2}$ can we find a point $P$ on $L_{1}$ and a point $R$ on $L_{3}$ so that $P, Q$ and $R$ are collinear?\n\n(A) $\\hat{k}-\\frac{1}{2} \\hat{j}$\n\n(B) $\\hat{k}$\n\n(C) $\\hat{k}+\\frac{1}{2} \\hat{j}$\n\n(D) $\\hat{k}+\\hat{j}$", "gold": "AC" }, { "description": "JEE Adv 2019 Paper 2", "index": 45, "subject": "math", "type": "Numeric", "question": "Suppose\n\n\$\n\\operatorname{det}\\left[\\begin{array}{cc}\n\n\\sum_{k=0}^{n} k & \\sum_{k=0}^{n}{ }^{n} C_{k} k^{2} \\\\\n\n\\sum_{k=0}^{n}{ }^{n} C_{k} k & \\sum_{k=0}^{n}{ }^{n} C_{k} 3^{k}\n\n\\end{array}\\right]=0\n\$\n\nholds for some positive integer $n$. Then what does $\\sum_{k=0}^{n} \\frac{{ }^{n} C_{k}}{k+1}$?", "gold": "6.2" }, { "description": "JEE Adv 2019 Paper 2", "index": 46, "subject": "math", "type": "Numeric", "question": "Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then what is the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats?", "gold": "30" }, { "description": "JEE Adv 2019 Paper 2", "index": 47, "subject": "math", "type": "Numeric", "question": "Let $|X|$ denote the number of elements in a set $X$. Let $S=\\{1,2,3,4,5,6\\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then what is the number of ordered pairs $(A, B)$ such that $1 \\leq|B|<|A|$?", "gold": "422" }, { "description": "JEE Adv 2019 Paper 2", "index": 48, "subject": "math", "type": "Numeric", "question": "What is the value of\n\n\$\n\\sec ^{-1}\\left(\\frac{1}{4} \\sum_{k=0}^{10} \\sec \\left(\\frac{7 \\pi}{12}+\\frac{k \\pi}{2}\\right) \\sec \\left(\\frac{7 \\pi}{12}+\\frac{(k+1) \\pi}{2}\\right)\\right)\n\$\n\nin the interval $\\left[-\\frac{\\pi}{4}, \\frac{3 \\pi}{4}\\right]$?", "gold": "0" }, { "description": "JEE Adv 2019 Paper 2", "index": 49, "subject": "math", "type": "Numeric", "question": "What is the value of the integral\n\n\n\$\n\\int_{0}^{\\pi / 2} \\frac{3 \\sqrt{\\cos \\theta}}{(\\sqrt{\\cos \\theta}+\\sqrt{\\sin \\theta})^{5}} d \\theta\n\$?", "gold": "0.5" }, { "description": "JEE Adv 2020 Paper 1", "index": 7, "subject": "phy", "type": "MCQ(multiple)", "question": "A particle of mass $m$ moves in circular orbits with potential energy $V(r)=F r$, where $F$ is a positive constant and $r$ is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by $R$ and its speed and energy are denoted by $v$ and $E$, respectively, then for the $n^{\\text {th }}$ orbit (here $h$ is the Planck's constant)\n\n(A) $R \\propto n^{1 / 3}$ and $v \\propto n^{2 / 3}$\n\n(B) $R \\propto n^{2 / 3}$ and $\\mathrm{v} \\propto n^{1 / 3}$\n\n(C) $E=\\frac{3}{2}\\left(\\frac{n^{2} h^{2} F^{2}}{4 \\pi^{2} m}\\right)^{1 / 3}$\n\n(D) $E=2\\left(\\frac{n^{2} h^{2} F^{2}}{4 \\pi^{2} m}\\right)^{1 / 3}$", "gold": "BC" }, { "description": "JEE Adv 2020 Paper 1", "index": 8, "subject": "phy", "type": "MCQ(multiple)", "question": "The filament of a light bulb has surface area $64 \\mathrm{~mm}^{2}$. The filament can be considered as a black body at temperature $2500 \\mathrm{~K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \\mathrm{~m}$. Assume the pupil of the eyes of the observer to be circular with radius $3 \\mathrm{~mm}$. Then\n\n(Take Stefan-Boltzmann constant $=5.67 \\times 10^{-8} \\mathrm{Wm}^{-2} \\mathrm{~K}^{-4}$, Wien's displacement constant $=$ $2.90 \\times 10^{-3} \\mathrm{~m}-\\mathrm{K}$, Planck's constant $=6.63 \\times 10^{-34} \\mathrm{Js}$, speed of light in vacuum $=3.00 \\times$ $\\left.10^{8} \\mathrm{~ms}^{-1}\\right)$\n\n(A) power radiated by the filament is in the range $642 \\mathrm{~W}$ to $645 \\mathrm{~W}$\n\n(B) radiated power entering into one eye of the observer is in the range $3.15 \\times 10^{-8} \\mathrm{~W}$ to\n\n\$\n3.25 \\times 10^{-8} \\mathrm{~W}\n\$\n\n(C) the wavelength corresponding to the maximum intensity of light is $1160 \\mathrm{~nm}$\n\n(D) taking the average wavelength of emitted radiation to be $1740 \\mathrm{~nm}$, the total number of photons entering per second into one eye of the observer is in the range $2.75 \\times 10^{11}$ to $2.85 \\times 10^{11}$", "gold": "BCD" }, { "description": "JEE Adv 2020 Paper 1", "index": 9, "subject": "phy", "type": "MCQ(multiple)", "question": "Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: [position $]=\\left[X^{\\alpha}\\right]$; [speed $]=\\left[X^{\\beta}\\right]$; [acceleration $]=\\left[X^{p}\\right]$; $[$ linear momentum $]=\\left[X^{q}\\right] ;$ force $]=\\left[X^{r}\\right]$. Then\n\n(A) $\\alpha+p=2 \\beta$\n\n(B) $p+q-r=\\beta$\n\n(C) $p-q+r=\\alpha$\n\n(D) $p+q+r=\\beta$", "gold": "AB" }, { "description": "JEE Adv 2020 Paper 1", "index": 13, "subject": "phy", "type": "Numeric", "question": "Put a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \\mathrm{~cm}$ and the right one at $90.00 \\mathrm{~cm}$. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_{R}$ from the center $(50.00 \\mathrm{~cm})$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32 , respectively, what is the value of $x_{R}$ (in $\\mathrm{cm})$?", "gold": "25.6" }, { "description": "JEE Adv 2020 Paper 1", "index": 16, "subject": "phy", "type": "Numeric", "question": "Consider one mole of helium gas enclosed in a container at initial pressure $P_{1}$ and volume $V_{1}$. It expands isothermally to volume $4 V_{1}$. After this, the gas expands adiabatically and its volume becomes $32 V_{1}$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\\text {iso }}$ and $W_{\\text {adia }}$, respectively. If the ratio $\\frac{W_{\\text {iso }}}{W_{\\text {adia }}}=f \\ln 2$, then what is the value of $f$?", "gold": "1.77" }, { "description": "JEE Adv 2020 Paper 1", "index": 17, "subject": "phy", "type": "Numeric", "question": "A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \\mathrm{~ms}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \\mathrm{~ms}^{-1}$, what is the smallest value of the percentage change required in the length of the pipe?", "gold": "0.62" }, { "description": "JEE Adv 2020 Paper 1", "index": 18, "subject": "phy", "type": "Numeric", "question": "A circular disc of radius $R$ carries surface charge density $\\sigma(r)=\\sigma_{0}\\left(1-\\frac{r}{R}\\right)$, where $\\sigma_{0}$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $\\phi_{0}$. Electric flux through another spherical surface of radius $\\frac{R}{4}$ and concentric with the disc is $\\phi$. Then what is the ratio $\\frac{\\phi_{0}}{\\phi}$?", "gold": "6.4" }, { "description": "JEE Adv 2020 Paper 1", "index": 20, "subject": "chem", "type": "MCQ", "question": "Which of the following liberates $\\mathrm{O}_{2}$ upon hydrolysis?\n\n(A) $\\mathrm{Pb}_{3} \\mathrm{O}_{4}$\n\n(B) $\\mathrm{KO}_{2}$\n\n(C) $\\mathrm{Na}_{2} \\mathrm{O}_{2}$\n\n(D) $\\mathrm{Li}_{2} \\mathrm{O}_{2}$", "gold": "B" }, { "description": "JEE Adv 2020 Paper 1", "index": 21, "subject": "chem", "type": "MCQ", "question": "A colorless aqueous solution contains nitrates of two metals, $\\mathbf{X}$ and $\\mathbf{Y}$. When it was added to an aqueous solution of $\\mathrm{NaCl}$, a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue $\\mathbf{P}$ and a solution $\\mathbf{Q}$. The residue $\\mathbf{P}$ was soluble in aq. $\\mathrm{NH}_{3}$ and also in excess sodium thiosulfate. The hot solution $\\mathbf{Q}$ gave a yellow precipitate with KI. The metals $\\mathbf{X}$ and $\\mathbf{Y}$, respectively, are\n\n(A) Ag and $\\mathrm{Pb}$\n\n(B) Ag and Cd\n\n(C) $\\mathrm{Cd}$ and $\\mathrm{Pb}$\n\n(D) Cd and Zn", "gold": "A" }, { "description": "JEE Adv 2020 Paper 1", "index": 25, "subject": "chem", "type": "MCQ(multiple)", "question": "In thermodynamics, the $P-V$ work done is given by\n\n\$\nw=-\\int d V P_{\\mathrm{ext}}\n\$\n\nFor a system undergoing a particular process, the work done is,\n\nThis equation is applicable to a\n\n\$\nw=-\\int d V\\left(\\frac{R T}{V-b}-\\frac{a}{V^{2}}\\right)\n\$\n\n(A) system that satisfies the van der Waals equation of state.\n\n(B) process that is reversible and isothermal.\n\n(C) process that is reversible and adiabatic.\n\n(D) process that is irreversible and at constant pressure.", "gold": "ABC" }, { "description": "JEE Adv 2020 Paper 1", "index": 28, "subject": "chem", "type": "MCQ(multiple)", "question": "Choose the correct statement(s) among the following:\n\n(A) $\\left[\\mathrm{FeCl}_{4}\\right]^{-}$has tetrahedral geometry.\n\n(B) $\\left[\\mathrm{Co}(\\mathrm{en})\\left(\\mathrm{NH}_{3}\\right)_{2} \\mathrm{Cl}_{2}\\right]^{+}$has 2 geometrical isomers.\n\n(C) $\\left[\\mathrm{FeCl}_{4}\\right]^{-}$has higher spin-only magnetic moment than $\\left[\\mathrm{Co}(\\mathrm{en})\\left(\\mathrm{NH}_{3}\\right)_{2} \\mathrm{Cl}_{2}\\right]^{+}$.\n\n(D) The cobalt ion in $\\left[\\mathrm{Co}(\\mathrm{en})\\left(\\mathrm{NH}_{3}\\right)_{2} \\mathrm{Cl}_{2}\\right]^{+}$has $\\mathrm{sp}^{3} d^{2}$ hybridization.", "gold": "AC" }, { "description": "JEE Adv 2020 Paper 1", "index": 29, "subject": "chem", "type": "MCQ(multiple)", "question": "With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s).\n\n(A) The hypochlorite ion is the strongest conjugate base.\n\n(B) The molecular shape of only chlorate ion is influenced by the lone pair of electrons of $\\mathrm{Cl}$.\n\n(C) The hypochlorite and chlorate ions disproportionate to give rise to identical set of ions.\n\n(D) The hypochlorite ion oxidizes the sulfite ion.", "gold": "ABD" }, { "description": "JEE Adv 2020 Paper 1", "index": 31, "subject": "chem", "type": "Numeric", "question": "$5.00 \\mathrm{~mL}$ of $0.10 \\mathrm{M}$ oxalic acid solution taken in a conical flask is titrated against $\\mathrm{NaOH}$ from a burette using phenolphthalein indicator. The volume of $\\mathrm{NaOH}$ required for the appearance of permanent faint pink color is tabulated below for five experiments. What is the concentration, in molarity, of the $\\mathrm{NaOH}$ solution?\n\n\\begin{center}\n\n\\begin{tabular}{|c|c|}\n\n\\hline\n\nExp. No. & Vol. of NaOH (mL) \\\\\n\n\\hline\n\n$\\mathbf{1}$ & 12.5 \\\\\n\n\\hline\n\n$\\mathbf{2}$ & 10.5 \\\\\n\n\\hline\n\n$\\mathbf{3}$ & 9.0 \\\\\n\n\\hline\n\n$\\mathbf{4}$ & 9.0 \\\\\n\n\\hline\n\n$\\mathbf{5}$ & 9.0 \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}", "gold": "0.11" }, { "description": "JEE Adv 2020 Paper 1", "index": 33, "subject": "chem", "type": "Numeric", "question": "Consider a 70\\% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and $298 \\mathrm{~K}$. Its cell reaction is\n\n\$\n\\mathrm{H}_{2}(g)+\\frac{1}{2} \\mathrm{O}_{2}(g) \\rightarrow \\mathrm{H}_{2} \\mathrm{O}(l)\n\$\n\nThe work derived from the cell on the consumption of $1.0 \\times 10^{-3} \\mathrm{~mol} \\mathrm{of}_{2}(g)$ is used to compress $1.00 \\mathrm{~mol}$ of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K) of the ideal gas?\n\nThe standard reduction potentials for the two half-cells are given below.\n\n\$\n\\begin{gathered}\n\n\\mathrm{O}_{2}(g)+4 \\mathrm{H}^{+}(a q)+4 e^{-} \\rightarrow 2 \\mathrm{H}_{2} \\mathrm{O}(l), \\quad E^{0}=1.23 \\mathrm{~V}, \\\\\n\n2 \\mathrm{H}^{+}(a q)+2 e^{-} \\rightarrow \\mathrm{H}_{2}(g), \\quad E^{0}=0.00 \\mathrm{~V}\n\n\\end{gathered}\n\$\n\nUse $F=96500 \\mathrm{C} \\mathrm{mol}^{-1}, R=8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$.", "gold": "13.32" }, { "description": "JEE Adv 2020 Paper 1", "index": 34, "subject": "chem", "type": "Numeric", "question": "Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters (L) produced at $300 \\mathrm{~K}$ and $1.0 \\mathrm{~atm}$ pressure, when $5.4 \\mathrm{~g}$ of aluminium and $50.0 \\mathrm{~mL}$ of $5.0 \\mathrm{M}$ sulfuric acid are combined for the reaction?\n\n(Use molar mass of aluminium as $27.0 \\mathrm{~g} \\mathrm{~mol}^{-1}, R=0.082 \\mathrm{~atm} \\mathrm{~L} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$ )", "gold": "6.15" }, { "description": "JEE Adv 2020 Paper 1", "index": 35, "subject": "chem", "type": "Numeric", "question": "${ }_{92}^{238} \\mathrm{U}$ is known to undergo radioactive decay to form ${ }_{82}^{206} \\mathrm{~Pb}$ by emitting alpha and beta particles. A rock initially contained $68 \\times 10^{-6} \\mathrm{~g}$ of ${ }_{92}^{238} \\mathrm{U}$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} \\mathrm{U}$ to ${ }_{82}^{206} \\mathrm{~Pb}$ in three half-lives is $Z \\times 10^{18}$, then what is the value of $Z$ ?", "gold": "1.2" }, { "description": "JEE Adv 2020 Paper 1", "index": 37, "subject": "math", "type": "MCQ", "question": "Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of\n\n\$\na c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)\n\$\n\nis\n\n(A) 0\n\n(B) 8000\n\n(C) 8080\n\n(D) 16000", "gold": "D" }, { "description": "JEE Adv 2020 Paper 1", "index": 38, "subject": "math", "type": "MCQ", "question": "If the function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ is defined by $f(x)=|x|(x-\\sin x)$, then which of the following statements is TRUE?\n\n(A) $f$ is one-one, but NOT onto\n\n(B) $f$ is onto, but NOT one-one\n\n(C) $f$ is BOTH one-one and onto\n\n(D) $f$ is NEITHER one-one NOR onto", "gold": "C" }, { "description": "JEE Adv 2020 Paper 1", "index": 39, "subject": "math", "type": "MCQ", "question": "Let the functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ and $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ be defined by\n\n\$\nf(x)=e^{x-1}-e^{-|x-1|} \\quad \\text { and } \\quad g(x)=\\frac{1}{2}\\left(e^{x-1}+e^{1-x}\\right)\n\$\n\nThen the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is\n\n(A) $(2-\\sqrt{3})+\\frac{1}{2}\\left(e-e^{-1}\\right)$\n\n(B) $(2+\\sqrt{3})+\\frac{1}{2}\\left(e-e^{-1}\\right)$\n\n(C) $(2-\\sqrt{3})+\\frac{1}{2}\\left(e+e^{-1}\\right)$\n\n(D) $(2+\\sqrt{3})+\\frac{1}{2}\\left(e+e^{-1}\\right)$", "gold": "A" }, { "description": "JEE Adv 2020 Paper 1", "index": 40, "subject": "math", "type": "MCQ", "question": "Let $a, b$ and $\\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^{2}=4 \\lambda x$, and suppose the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is\n\n(A) $\\frac{1}{\\sqrt{2}}$\n\n(B) $\\frac{1}{2}$\n\n(C) $\\frac{1}{3}$\n\n(D) $\\frac{2}{5}$", "gold": "A" }, { "description": "JEE Adv 2020 Paper 1", "index": 41, "subject": "math", "type": "MCQ", "question": "Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\\frac{2}{3}$ and $\\frac{1}{3}$, respectively. Suppose $\\alpha$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $\\beta$ is the number of heads that appear when $C_{2}$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^{2}-\\alpha x+\\beta$ are real and equal, is\n\n(A) $\\frac{40}{81}$\n\n(B) $\\frac{20}{81}$\n\n(C) $\\frac{1}{2}$\n\n(D) $\\frac{1}{4}$", "gold": "B" }, { "description": "JEE Adv 2020 Paper 1", "index": 42, "subject": "math", "type": "MCQ", "question": "Consider all rectangles lying in the region\n\n\$\n\\left\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R}: 0 \\leq x \\leq \\frac{\\pi}{2} \\text { and } 0 \\leq y \\leq 2 \\sin (2 x)\\right\\}\n\$\n\nand having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is\n\n(A) $\\frac{3 \\pi}{2}$\n\n(B) $\\pi$\n\n(C) $\\frac{\\pi}{2 \\sqrt{3}}$\n\n(D) $\\frac{\\pi \\sqrt{3}}{2}$", "gold": "C" }, { "description": "JEE Adv 2020 Paper 1", "index": 43, "subject": "math", "type": "MCQ(multiple)", "question": "Let the function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \\sin x$ and let $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ be an arbitrary function. Let $f g: \\mathbb{R} \\rightarrow \\mathbb{R}$ be the product function defined by $(f g)(x)=f(x) g(x)$. Then which of the following statements is/are TRUE?\n\n(A) If $g$ is continuous at $x=1$, then $f g$ is differentiable at $x=1$\n\n(B) If $f g$ is differentiable at $x=1$, then $g$ is continuous at $x=1$\n\n(C) If $g$ is differentiable at $x=1$, then $f g$ is differentiable at $x=1$\n\n(D) If $f g$ is differentiable at $x=1$, then $g$ is differentiable at $x=1$", "gold": "AC" }, { "description": "JEE Adv 2020 Paper 1", "index": 44, "subject": "math", "type": "MCQ(multiple)", "question": "Let $M$ be a $3 \\times 3$ invertible matrix with real entries and let $I$ denote the $3 \\times 3$ identity matrix. If $M^{-1}=\\operatorname{adj}(\\operatorname{adj} M)$, then which of the following statements is/are ALWAYS TRUE?\n\n(A) $M=I$\n\n(B) $\\operatorname{det} M=1$\n\n(C) $M^{2}=I$\n\n(D) $(\\operatorname{adj} M)^{2}=I$", "gold": "BCD" }, { "description": "JEE Adv 2020 Paper 1", "index": 45, "subject": "math", "type": "MCQ(multiple)", "question": "Let $S$ be the set of all complex numbers $z$ satisfying $\\left|z^{2}+z+1\\right|=1$. Then which of the following statements is/are TRUE?\n\n(A) $\\left|z+\\frac{1}{2}\\right| \\leq \\frac{1}{2}$ for all $z \\in S$\n\n(B) $|z| \\leq 2$ for all $z \\in S$\n\n(C) $\\left|z+\\frac{1}{2}\\right| \\geq \\frac{1}{2}$ for all $z \\in S$\n\n(D) The set $S$ has exactly four elements", "gold": "BC" }, { "description": "JEE Adv 2020 Paper 1", "index": 46, "subject": "math", "type": "MCQ(multiple)", "question": "Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$, respectively. If\n\n\$\n\\tan \\frac{X}{2}+\\tan \\frac{Z}{2}=\\frac{2 y}{x+y+z}\n\$\n\nthen which of the following statements is/are TRUE?\n\n(A) $2 Y=X+Z$\n\n(B) $Y=X+Z$\n\n(C) $\\tan \\frac{x}{2}=\\frac{x}{y+z}$\n\n(D) $x^{2}+z^{2}-y^{2}=x z$", "gold": "BC" }, { "description": "JEE Adv 2020 Paper 1", "index": 47, "subject": "math", "type": "MCQ(multiple)", "question": "Let $L_{1}$ and $L_{2}$ be the following straight lines.\n\n\$\nL_{1}: \\frac{x-1}{1}=\\frac{y}{-1}=\\frac{z-1}{3} \\text { and } L_{2}: \\frac{x-1}{-3}=\\frac{y}{-1}=\\frac{z-1}{1}\n\$\n\nSuppose the straight line\n\n\$\nL: \\frac{x-\\alpha}{l}=\\frac{y-1}{m}=\\frac{z-\\gamma}{-2}\n\$\n\nlies in the plane containing $L_{1}$ and $L_{2}$, and passes through the point of intersection of $L_{1}$ and $L_{2}$. If the line $L$ bisects the acute angle between the lines $L_{1}$ and $L_{2}$, then which of the following statements is/are TRUE?\n\n(A) $\\alpha-\\gamma=3$\n\n(B) $l+m=2$\n\n(C) $\\alpha-\\gamma=1$\n\n(D) $l+m=0$", "gold": "AB" }, { "description": "JEE Adv 2020 Paper 1", "index": 48, "subject": "math", "type": "MCQ(multiple)", "question": "Which of the following inequalities is/are TRUE?\n\n(A) $\\int_{0}^{1} x \\cos x d x \\geq \\frac{3}{8}$\n\n(B) $\\int_{0}^{1} x \\sin x d x \\geq \\frac{3}{10}$\n\n(C) $\\int_{0}^{1} x^{2} \\cos x d x \\geq \\frac{1}{2}$\n\n(D) $\\int_{0}^{1} x^{2} \\sin x d x \\geq \\frac{2}{9}$", "gold": "ABD" }, { "description": "JEE Adv 2020 Paper 1", "index": 49, "subject": "math", "type": "Numeric", "question": "Let $m$ be the minimum possible value of $\\log _{3}\\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1}+y_{2}+y_{3}=9$. Let $M$ be the maximum possible value of $\\left(\\log _{3} x_{1}+\\log _{3} x_{2}+\\log _{3} x_{3}\\right)$, where $x_{1}, x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$. Then what is the value of $\\log _{2}\\left(m^{3}\\right)+\\log _{3}\\left(M^{2}\\right)$?", "gold": "8" }, { "description": "JEE Adv 2020 Paper 1", "index": 50, "subject": "math", "type": "Numeric", "question": "Let $a_{1}, a_{2}, a_{3}, \\ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_{1}, b_{2}, b_{3}, \\ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_{1}=b_{1}=c$, then what is the number of all possible values of $c$, for which the equality\n\n\$\n2\\left(a_{1}+a_{2}+\\cdots+a_{n}\\right)=b_{1}+b_{2}+\\cdots+b_{n}\n\$\n\nholds for some positive integer $n$?", "gold": "1" }, { "description": "JEE Adv 2020 Paper 1", "index": 51, "subject": "math", "type": "Numeric", "question": "Let $f:[0,2] \\rightarrow \\mathbb{R}$ be the function defined by\n\n\$\nf(x)=(3-\\sin (2 \\pi x)) \\sin \\left(\\pi x-\\frac{\\pi}{4}\\right)-\\sin \\left(3 \\pi x+\\frac{\\pi}{4}\\right)\n\$\n\nIf $\\alpha, \\beta \\in[0,2]$ are such that $\\{x \\in[0,2]: f(x) \\geq 0\\}=[\\alpha, \\beta]$, then what is the value of $\\beta-\\alpha$?", "gold": "1" }, { "description": "JEE Adv 2020 Paper 1", "index": 52, "subject": "math", "type": "Numeric", "question": "In a triangle $P Q R$, let $\\vec{a}=\\overrightarrow{Q R}, \\vec{b}=\\overrightarrow{R P}$ and $\\vec{c}=\\overrightarrow{P Q}$. If\n\n\$\n|\\vec{a}|=3, \\quad|\\vec{b}|=4 \\quad \\text { and } \\quad \\frac{\\vec{a} \\cdot(\\vec{c}-\\vec{b})}{\\vec{c} \\cdot(\\vec{a}-\\vec{b})}=\\frac{|\\vec{a}|}{|\\vec{a}|+|\\vec{b}|},\n\$\n\nthen what is the value of $|\\vec{a} \\times \\vec{b}|^{2}$?", "gold": "108" }, { "description": "JEE Adv 2020 Paper 1", "index": 53, "subject": "math", "type": "Numeric", "question": "For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by\n\n\$\nS=\\left\\{\\left(x^{2}-1\\right)^{2}\\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\\right): a_{0}, a_{1}, a_{2}, a_{3} \\in \\mathbb{R}\\right\\}\n\$\n\nFor a polynomial $f$, let $f^{\\prime}$ and $f^{\\prime \\prime}$ denote its first and second order derivatives, respectively. Then what is the minimum possible value of $\\left(m_{f^{\\prime}}+m_{f^{\\prime \\prime}}\\right)$, where $f \\in S$?", "gold": "3" }, { "description": "JEE Adv 2020 Paper 1", "index": 54, "subject": "math", "type": "Numeric", "question": "Let $e$ denote the base of the natural logarithm. What is the value of the real number $a$ for which the right hand limit\n\n\$\n\\lim _{x \\rightarrow 0^{+}} \\frac{(1-x)^{\\frac{1}{x}}-e^{-1}}{x^{a}}\n\$\n\nis equal to a nonzero real number?", "gold": "1" }, { "description": "JEE Adv 2020 Paper 2", "index": 2, "subject": "phy", "type": "Integer", "question": "A train with cross-sectional area $S_{t}$ is moving with speed $v_{t}$ inside a long tunnel of cross-sectional area $S_{0}\\left(S_{0}=4 S_{t}\\right)$. Assume that almost all the air (density $\\rho$ ) in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $p_{0}$. If the pressure in the region between the sides of the train and the tunnel walls is $p$, then $p_{0}-p=\\frac{7}{2 N} \\rho v_{t}^{2}$. What is the value of $N$?", "gold": "9" }, { "description": "JEE Adv 2020 Paper 2", "index": 4, "subject": "phy", "type": "Integer", "question": "A hot air balloon is carrying some passengers, and a few sandbags of mass $1 \\mathrm{~kg}$ each so that its total mass is $480 \\mathrm{~kg}$. Its effective volume giving the balloon its buoyancy is $V$. The balloon is floating at an equilibrium height of $100 \\mathrm{~m}$. When $N$ number of sandbags are thrown out, the balloon rises to a new equilibrium height close to $150 \\mathrm{~m}$ with its volume $V$ remaining unchanged. If the variation of the density of air with height $h$ from the ground is $\\rho(h)=\\rho_{0} e^{-\\frac{h}{h_{0}}}$, where $\\rho_{0}=1.25 \\mathrm{~kg} \\mathrm{~m}^{-3}$ and $h_{0}=6000 \\mathrm{~m}$, what is the value of $N$?", "gold": "4" }, { "description": "JEE Adv 2020 Paper 2", "index": 10, "subject": "phy", "type": "MCQ(multiple)", "question": "In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\\frac{I}{2}$, the potential difference $V$ is increased to $2 V$, and the separation distance $d$ is reduced to $\\frac{d}{2}$, then\n\n(A) the cut-off wavelength will reduce to half, and the wavelengths of the characteristic $\\mathrm{X}$-rays will remain the same\n\n(B) the cut-off wavelength as well as the wavelengths of the characteristic X-rays will remain the same\n\n(C) the cut-off wavelength will reduce to half, and the intensities of all the $\\mathrm{X}$-rays will decrease\n\n(D) the cut-off wavelength will become two times larger, and the intensity of all the X-rays will decrease", "gold": "AC" }, { "description": "JEE Adv 2020 Paper 2", "index": 11, "subject": "phy", "type": "MCQ(multiple)", "question": "Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \\mathrm{~kg} \\mathrm{~m}^{-3}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then\n\n(A) electric force between the spheres remains unchanged\n\n(B) electric force between the spheres reduces\n\n(C) mass density of the spheres is $840 \\mathrm{~kg} \\mathrm{~m}^{-3}$\n\n(D) the tension in the strings holding the spheres remains unchanged", "gold": "AC" }, { "description": "JEE Adv 2020 Paper 2", "index": 12, "subject": "phy", "type": "MCQ(multiple)", "question": "Starting at time $t=0$ from the origin with speed $1 \\mathrm{~ms}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y=\\frac{x^{2}}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_{x}$ and $a_{y}$, respectively. Then\n\n(A) $a_{x}=1 \\mathrm{~ms}^{-2}$ implies that when the particle is at the origin, $a_{y}=1 \\mathrm{~ms}^{-2}$\n\n(B) $a_{x}=0$ implies $a_{y}=1 \\mathrm{~ms}^{-2}$ at all times\n\n(C) at $t=0$, the particle's velocity points in the $x$-direction\n\n(D) $a_{x}=0$ implies that at $t=1 \\mathrm{~s}$, the angle between the particle's velocity and the $x$ axis is $45^{\\circ}$", "gold": "BCD" }, { "description": "JEE Adv 2020 Paper 2", "index": 15, "subject": "phy", "type": "Numeric", "question": "Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \\pm 0.02 \\mathrm{~V}$. What is the percentage error in the calculation of the energy stored in this combination of capacitors?", "gold": "1.3" }, { "description": "JEE Adv 2020 Paper 2", "index": 16, "subject": "phy", "type": "Numeric", "question": "A cubical solid aluminium (bulk modulus $=-V \\frac{d P}{d V}=70 \\mathrm{GPa}$ block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a $5 \\mathrm{~km}$ deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^{3} \\mathrm{~kg} \\mathrm{~m}^{-3}$ and $10 \\mathrm{~ms}^{-2}$, respectively, what is the change in the edge length of the block in $\\mathrm{mm}$?", "gold": "0.24" }, { "description": "JEE Adv 2020 Paper 2", "index": 18, "subject": "phy", "type": "Numeric", "question": "A container with $1 \\mathrm{~kg}$ of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700 \\mathrm{Wm}^{-2}$ and it is absorbed by the water over an effective area of $0.05 \\mathrm{~m}^{2}$. Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, what will be the difference (in ${ }^{\\circ} \\mathrm{C}$ ) in the temperature of water and the surroundings after a long time? (Ignore effect of the container, and take constant for Newton's law of cooling $=0.001 \\mathrm{~s}^{-1}$, Heat capacity of water $=4200 \\mathrm{~J} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$ )", "gold": "8.33" }, { "description": "JEE Adv 2020 Paper 2", "index": 19, "subject": "chem", "type": "Integer", "question": "The $1^{\\text {st }}, 2^{\\text {nd }}$, and the $3^{\\text {rd }}$ ionization enthalpies, $I_{1}, I_{2}$, and $I_{3}$, of four atoms with atomic numbers $n, n+$ 1, $n+2$, and $n+3$, where $n<10$, are tabulated below. What is the value of $n$ ?\n\n\\begin{center}\n\n\\begin{tabular}{|c|c|c|c|}\n\n\\hline\n\n{$\\begin{array}{c}\\text { Atomic } \\\\\n\n\\text { number }\\end{array}$} & \\multicolumn{3}{|c|}{Ionization Enthalpy $(\\mathrm{kJ} / \\mathrm{mol})$} \\\\\n\n\\cline { 2 - 4 }\n\n & $I_{1}$ & $I_{2}$ & $I_{3}$ \\\\\n\n\\hline\n\n$n$ & 1681 & 3374 & 6050 \\\\\n\n\\hline\n\n$n+1$ & 2081 & 3952 & 6122 \\\\\n\n\\hline\n\n$n+2$ & 496 & 4562 & 6910 \\\\\n\n\\hline\n\n$n+3$ & 738 & 1451 & 7733 \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}", "gold": "9" }, { "description": "JEE Adv 2020 Paper 2", "index": 21, "subject": "chem", "type": "Integer", "question": "In the chemical reaction between stoichiometric quantities of $\\mathrm{KMnO}_{4}$ and $\\mathrm{KI}$ in weakly basic solution, what is the number of moles of $\\mathrm{I}_{2}$ released for 4 moles of $\\mathrm{KMnO}_{4}$ consumed?", "gold": "6" }, { "description": "JEE Adv 2020 Paper 2", "index": 22, "subject": "chem", "type": "Integer", "question": "An acidified solution of potassium chromate was layered with an equal volume of amyl alcohol. When it was shaken after the addition of $1 \\mathrm{~mL}$ of $3 \\% \\mathrm{H}_{2} \\mathrm{O}_{2}$, a blue alcohol layer was obtained. The blue color is due to the formation of a chromium (VI) compound ' $\\mathbf{X}$ '. What is the number of oxygen atoms bonded to chromium through only single bonds in a molecule of $\\mathbf{X}$ ?", "gold": "4" }, { "description": "JEE Adv 2020 Paper 2", "index": 24, "subject": "chem", "type": "Integer", "question": "An organic compound $\\left(\\mathrm{C}_{8} \\mathrm{H}_{10} \\mathrm{O}_{2}\\right)$ rotates plane-polarized light. It produces pink color with neutral $\\mathrm{FeCl}_{3}$ solution. What is the total number of all the possible isomers for this compound?", "gold": "6" }, { "description": "JEE Adv 2020 Paper 2", "index": 27, "subject": "chem", "type": "MCQ(multiple)", "question": "Which among the following statement(s) is(are) true for the extraction of aluminium from bauxite?\n\n(A) Hydrated $\\mathrm{Al}_{2} \\mathrm{O}_{3}$ precipitates, when $\\mathrm{CO}_{2}$ is bubbled through a solution of sodium aluminate.\n\n(B) Addition of $\\mathrm{Na}_{3} \\mathrm{AlF}_{6}$ lowers the melting point of alumina.\n\n(C) $\\mathrm{CO}_{2}$ is evolved at the anode during electrolysis.\n\n(D) The cathode is a steel vessel with a lining of carbon.", "gold": "ABCD" }, { "description": "JEE Adv 2020 Paper 2", "index": 28, "subject": "chem", "type": "MCQ(multiple)", "question": "Choose the correct statement(s) among the following.\n\n(A) $\\mathrm{SnCl}_{2} \\cdot 2 \\mathrm{H}_{2} \\mathrm{O}$ is a reducing agent.\n\n(B) $\\mathrm{SnO}_{2}$ reacts with $\\mathrm{KOH}$ to form $\\mathrm{K}_{2}\\left[\\mathrm{Sn}(\\mathrm{OH})_{6}\\right]$.\n\n(C) A solution of $\\mathrm{PbCl}_{2}$ in $\\mathrm{HCl}$ contains $\\mathrm{Pb}^{2+}$ and $\\mathrm{Cl}^{-}$ions.\n\n(D) The reaction of $\\mathrm{Pb}_{3} \\mathrm{O}_{4}$ with hot dilute nitric acid to give $\\mathrm{PbO}_{2}$ is a redox reaction.", "gold": "AB" }, { "description": "JEE Adv 2020 Paper 2", "index": 32, "subject": "chem", "type": "Numeric", "question": "Liquids $\\mathbf{A}$ and $\\mathbf{B}$ form ideal solution for all compositions of $\\mathbf{A}$ and $\\mathbf{B}$ at $25^{\\circ} \\mathrm{C}$. Two such solutions with 0.25 and 0.50 mole fractions of $\\mathbf{A}$ have the total vapor pressures of 0.3 and 0.4 bar, respectively. What is the vapor pressure of pure liquid $\\mathbf{B}$ in bar?", "gold": "0.2" }, { "description": "JEE Adv 2020 Paper 2", "index": 35, "subject": "chem", "type": "Numeric", "question": "Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum temperature (in K) at which the reduction of cassiterite by coke would take place.\n\nAt $298 \\mathrm{~K}: \\Delta_{f} H^{0}\\left(\\mathrm{SnO}_{2}(s)\\right)=-581.0 \\mathrm{~kJ} \\mathrm{~mol}^{-1}, \\Delta_{f} H^{0}\\left(\\mathrm{CO}_{2}(g)\\right)=-394.0 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$,\n\n$S^{0}\\left(\\mathrm{SnO}_{2}(\\mathrm{~s})\\right)=56.0 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}, S^{0}(\\mathrm{Sn}(\\mathrm{s}))=52.0 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$,\n\n$S^{0}(\\mathrm{C}(s))=6.0 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}, S^{0}\\left(\\mathrm{CO}_{2}(g)\\right)=210.0 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$.\n\nAssume that the enthalpies and the entropies are temperature independent.", "gold": "935" }, { "description": "JEE Adv 2020 Paper 2", "index": 36, "subject": "chem", "type": "Numeric", "question": "An acidified solution of $0.05 \\mathrm{M} \\mathrm{Zn}^{2+}$ is saturated with $0.1 \\mathrm{M} \\mathrm{H}_{2} \\mathrm{~S}$. What is the minimum molar concentration (M) of $\\mathrm{H}^{+}$required to prevent the precipitation of $\\mathrm{ZnS}$ ?\n\nUse $K_{\\mathrm{sp}}(\\mathrm{ZnS})=1.25 \\times 10^{-22}$ and\n\noverall dissociation constant of $\\mathrm{H}_{2} \\mathrm{~S}, K_{\\mathrm{NET}}=K_{1} K_{2}=1 \\times 10^{-21}$.", "gold": "0.2" }, { "description": "JEE Adv 2020 Paper 2", "index": 37, "subject": "math", "type": "Integer", "question": "For a complex number $z$, let $\\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\\sqrt{-1}$. Then what is the minimum possible value of $\\left|z_{1}-z_{2}\\right|^{2}$, where $z_{1}, z_{2} \\in S$ with $\\operatorname{Re}\\left(z_{1}\\right)>0$ and $\\operatorname{Re}\\left(z_{2}\\right)<0$?", "gold": "8" }, { "description": "JEE Adv 2020 Paper 2", "index": 38, "subject": "math", "type": "Integer", "question": "The probability that a missile hits a target successfully is 0.75 . In order to destroy the target completely, at least three successful hits are required. Then what is the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95?", "gold": "6" }, { "description": "JEE Adv 2020 Paper 2", "index": 39, "subject": "math", "type": "Integer", "question": "Let $O$ be the centre of the circle $x^{2}+y^{2}=r^{2}$, where $r>\\frac{\\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x+4 y=5$. If the centre of the circumcircle of the triangle $O P Q$ lies on the line $x+2 y=4$, then what is the value of $r$?", "gold": "2" }, { "description": "JEE Adv 2020 Paper 2", "index": 40, "subject": "math", "type": "Integer", "question": "The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \\times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is -18 , then what is the value of the determinant of $A$?", "gold": "5" }, { "description": "JEE Adv 2020 Paper 2", "index": 41, "subject": "math", "type": "Integer", "question": "Let the functions $f:(-1,1) \\rightarrow \\mathbb{R}$ and $g:(-1,1) \\rightarrow(-1,1)$ be defined by\n\n\$\nf(x)=|2 x-1|+|2 x+1| \\text { and } g(x)=x-[x] .\n\$\n\nwhere $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \\circ g:(-1,1) \\rightarrow \\mathbb{R}$ be the composite function defined by $(f \\circ g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \\circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \\circ g$ is NOT differentiable. Then what is the value of $c+d$?", "gold": "4" }, { "description": "JEE Adv 2020 Paper 2", "index": 42, "subject": "math", "type": "Integer", "question": "What is the value of the limit\n\n\$\n\\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{4 \\sqrt{2}(\\sin 3 x+\\sin x)}{\\left(2 \\sin 2 x \\sin \\frac{3 x}{2}+\\cos \\frac{5 x}{2}\\right)-\\left(\\sqrt{2}+\\sqrt{2} \\cos 2 x+\\cos \\frac{3 x}{2}\\right)}\n\$?", "gold": "8" }, { "description": "JEE Adv 2020 Paper 2", "index": 43, "subject": "math", "type": "MCQ(multiple)", "question": "Let $b$ be a nonzero real number. Suppose $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\\prime}$ of $f$ satisfies the equation\n\n\$\nf^{\\prime}(x)=\\frac{f(x)}{b^{2}+x^{2}}\n\$\n\nfor all $x \\in \\mathbb{R}$, then which of the following statements is/are TRUE?\n\n(A) If $b>0$, then $f$ is an increasing function\n\n(B) If $b<0$, then $f$ is a decreasing function\n\n(C) $f(x) f(-x)=1$ for all $x \\in \\mathbb{R}$\n\n(D) $f(x)-f(-x)=0$ for all $x \\in \\mathbb{R}$", "gold": "AC" }, { "description": "JEE Adv 2020 Paper 2", "index": 44, "subject": "math", "type": "MCQ(multiple)", "question": "Let $a$ and $b$ be positive real numbers such that $a>1$ and $bs .\\end{cases}\n\\]\n\nFor positive integers$m$and$n$, let\n\n\$\ng(m, n)=\\sum_{p=0}^{m+n} \\frac{f(m, n, p)}{\\left(\\begin{array}{c}\n\nn+p \\\\\n\np\n\n\\end{array}\\right)}\n\$\n\nwhere for any nonnegative integer$p$,\n\n\$\nf(m, n, p)=\\sum_{i=0}^{p}\\left(\\begin{array}{c}\n\nm \\\\\n\ni\n\n\\end{array}\\right)\\left(\\begin{array}{c}\n\nn+i \\\\\n\np\n\n\\end{array}\\right)\\left(\\begin{array}{c}\n\np+n \\\\\n\np-i\n\n\\end{array}\\right) .\n\$\n\nThen which of the following statements is/are TRUE?\n\n(A)$g(m, n)=g(n, m)$for all positive integers$m, n$\n\n(B)$g(m, n+1)=g(m+1, n)$for all positive integers$m, n$\n\n(C)$g(2 m, 2 n)=2 g(m, n)$for all positive integers$m, n$\n\n(D)$g(2 m, 2 n)=(g(m, n))^{2}$for all positive integers$m, n$", "gold": "ABD" }, { "description": "JEE Adv 2020 Paper 2", "index": 49, "subject": "math", "type": "Numeric", "question": "An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then what is the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021?", "gold": "495" }, { "description": "JEE Adv 2020 Paper 2", "index": 50, "subject": "math", "type": "Numeric", "question": "In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then what is the number of all possible ways in which this can be done?", "gold": "1080" }, { "description": "JEE Adv 2020 Paper 2", "index": 51, "subject": "math", "type": "Numeric", "question": "Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If$p$is the probability that this perfect square is an odd number, then what is the value of$14 p$?", "gold": "8" }, { "description": "JEE Adv 2020 Paper 2", "index": 52, "subject": "math", "type": "Numeric", "question": "Let the function$f:[0,1] \\rightarrow \\mathbb{R}$be defined by\n\n\$\nf(x)=\\frac{4^{x}}{4^{x}+2}\n\$\n\nThen what is the value of\n\$\nf\\left(\\frac{1}{40}\\right)+f\\left(\\frac{2}{40}\\right)+f\\left(\\frac{3}{40}\\right)+\\cdots+f\\left(\\frac{39}{40}\\right)-f\\left(\\frac{1}{2}\\right)\n\$?", "gold": "19" }, { "description": "JEE Adv 2020 Paper 2", "index": 53, "subject": "math", "type": "Numeric", "question": "Let$f: \\mathbb{R} \\rightarrow \\mathbb{R}$be a differentiable function such that its derivative$f^{\\prime}$is continuous and$f(\\pi)=-6$. If$F:[0, \\pi] \\rightarrow \\mathbb{R}$is defined by$F(x)=\\int_{0}^{x} f(t) d t$, and if\n\n\$\n\\int_{0}^{\\pi}\\left(f^{\\prime}(x)+F(x)\\right) \\cos x d x=2,\n\$\n\nthen what is the value of$f(0)$?", "gold": "4" }, { "description": "JEE Adv 2020 Paper 2", "index": 54, "subject": "math", "type": "Numeric", "question": "Let the function$f:(0, \\pi) \\rightarrow \\mathbb{R}$be defined by\n\n\$\nf(\\theta)=(\\sin \\theta+\\cos \\theta)^{2}+(\\sin \\theta-\\cos \\theta)^{4} .\n\$\n\nSuppose the function$f$has a local minimum at$\\theta$precisely when$\\theta \\in\\left\\{\\lambda_{1} \\pi, \\ldots, \\lambda_{r} \\pi\\right\\}$, where$0<\\lambda_{1}<\\cdots<\\lambda_{r}<1$. Then what is the value of$\\lambda_{1}+\\cdots+\\lambda_{r}$?", "gold": "0.5" }, { "description": "JEE Adv 2021 Paper 1", "index": 4, "subject": "phy", "type": "MCQ", "question": "A heavy nucleus$Q$of half-life 20 minutes undergoes alpha-decay with probability of$60 \\%$and beta-decay with probability of$40 \\%$. Initially, the number of$Q$nuclei is 1000. The number of alpha-decays of$Q$in the first one hour is\n\n(A) 50\n\n(B) 75\n\n(C) 350\n\n(D) 525", "gold": "D" }, { "description": "JEE Adv 2021 Paper 1", "index": 5, "subject": "phy", "type": "Numeric", "question": "A projectile is thrown from a point$\\mathrm{O}$on the ground at an angle$45^{\\circ}$from the vertical and with a speed$5 \\sqrt{2} \\mathrm{~m} / \\mathrm{s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,$0.5 \\mathrm{~s}$after the splitting. The other part,$t$seconds after the splitting, falls to the ground at a distance$x$meters from the point$\\mathrm{O}$. The acceleration due to gravity$g=10 \\mathrm{~m} / \\mathrm{s}^2$. What is the value of$t$?", "gold": "0.5" }, { "description": "JEE Adv 2021 Paper 1", "index": 6, "subject": "phy", "type": "Numeric", "question": "A projectile is thrown from a point$\\mathrm{O}$on the ground at an angle$45^{\\circ}$from the vertical and with a speed$5 \\sqrt{2} \\mathrm{~m} / \\mathrm{s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,$0.5 \\mathrm{~s}$after the splitting. The other part,$t$seconds after the splitting, falls to the ground at a distance$x$meters from the point$\\mathrm{O}$. The acceleration due to gravity$g=10 \\mathrm{~m} / \\mathrm{s}^2$. What is the value of$x$?", "gold": "7.5" }, { "description": "JEE Adv 2021 Paper 1", "index": 13, "subject": "phy", "type": "MCQ(multiple)", "question": "A particle of mass$M=0.2 \\mathrm{~kg}$is initially at rest in the$x y$-plane at a point$(x=-l$,$y=-h)$, where$l=10 \\mathrm{~m}$and$h=1 \\mathrm{~m}$. The particle is accelerated at time$t=0$with a constant acceleration$a=10 \\mathrm{~m} / \\mathrm{s}^{2}$along the positive$x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by$\\vec{L}$and$\\vec{\\tau}$, respectively.$\\hat{i}, \\hat{j}$and$\\hat{k}$are unit vectors along the positive$x, y$and$z$-directions, respectively. If$\\hat{k}=\\hat{i} \\times \\hat{j}$then which of the following statement(s) is(are) correct?\n\n(A) The particle arrives at the point$(x=l, y=-h)$at time$t=2 \\mathrm{~s}$\n\n(B)$\\vec{\\tau}=2 \\hat{k}$when the particle passes through the point$(x=l, y=-h)$\n\n(C)$\\vec{L}=4 \\hat{k}$when the particle passes through the point$(x=l, y=-h)$\n\n(D)$\\vec{\\tau}=\\hat{k}$when the particle passes through the point$(x=0, y=-h)$", "gold": "ABC" }, { "description": "JEE Adv 2021 Paper 1", "index": 14, "subject": "phy", "type": "MCQ(multiple)", "question": "Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?\n\n(A) The ratio of the longest wavelength to the shortest wavelength in Balmer series is$9 / 5$\n\n(B) There is an overlap between the wavelength ranges of Balmer and Paschen series\n\n(C) The wavelengths of Lyman series are given by$\\left(1+\\frac{1}{m^{2}}\\right) \\lambda_{0}$, where$\\lambda_{0}$is the shortest wavelength of Lyman series and$m$is an integer\n\n(D) The wavelength ranges of Lyman and Balmer series do not overlap", "gold": "AD" }, { "description": "JEE Adv 2021 Paper 1", "index": 17, "subject": "phy", "type": "Integer", "question": "An$\\alpha$-particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are accelerated through a potential$V$and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the$\\alpha$-particle and the sulfur ion move in circular orbits of radii$r_{\\alpha}$and$r_{S}$, respectively. What is the ratio$\\left(r_{S} / r_{\\alpha}\\right)$?", "gold": "4" }, { "description": "JEE Adv 2021 Paper 1", "index": 22, "subject": "chem", "type": "MCQ", "question": "The calculated spin only magnetic moments of$\\left[\\mathrm{Cr}\\left(\\mathrm{NH}_{3}\\right)_{6}\\right]^{3+}$and$\\left[\\mathrm{CuF}_{6}\\right]^{3-}$in$\\mathrm{BM}$, respectively, are\n\n(Atomic numbers of$\\mathrm{Cr}$and$\\mathrm{Cu}$are 24 and 29, respectively)\n\n(A) 3.87 and 2.84\n\n(B) 4.90 and 1.73\n\n(C) 3.87 and 1.73\n\n(D) 4.90 and 2.84", "gold": "A" }, { "description": "JEE Adv 2021 Paper 1", "index": 27, "subject": "chem", "type": "Numeric", "question": "The boiling point of water in a 0.1 molal silver nitrate solution (solution$\\mathbf{A}$) is$\\mathbf{x}^{\\circ} \\mathrm{C}$. To this solution$\\mathbf{A}$, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution$\\mathbf{B}$. The difference in the boiling points of water in the two solutions$\\mathbf{A}$and$\\mathbf{B}$is$\\mathbf{y} \\times 10^{-2}{ }^{\\circ} \\mathrm{C}$.\n(Assume: Densities of the solutions$\\mathbf{A}$and$\\mathbf{B}$are the same as that of water and the soluble salts dissociate completely.\nUse: Molal elevation constant (Ebullioscopic Constant),$K_b=0.5 \\mathrm{~K} \\mathrm{~kg} \\mathrm{~mol}^{-1}$; Boiling point of pure water as$100^{\\circ} \\mathrm{C}$.) What is the value of$\\mathbf{x}$?", "gold": "100.1" }, { "description": "JEE Adv 2021 Paper 1", "index": 28, "subject": "chem", "type": "Numeric", "question": "The boiling point of water in a 0.1 molal silver nitrate solution (solution$\\mathbf{A}$) is$\\mathbf{x}^{\\circ} \\mathrm{C}$. To this solution$\\mathbf{A}$, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution$\\mathbf{B}$. The difference in the boiling points of water in the two solutions$\\mathbf{A}$and$\\mathbf{B}$is$\\mathbf{y} \\times 10^{-2}{ }^{\\circ} \\mathrm{C}$.\n(Assume: Densities of the solutions$\\mathbf{A}$and$\\mathbf{B}$are the same as that of water and the soluble salts dissociate completely.\nUse: Molal elevation constant (Ebullioscopic Constant),$K_b=0.5 \\mathrm{~K} \\mathrm{~kg} \\mathrm{~mol}^{-1}$; Boiling point of pure water as$100^{\\circ} \\mathrm{C}$.) What is the value of$|\\mathbf{y}|$?", "gold": "2.5" }, { "description": "JEE Adv 2021 Paper 1", "index": 31, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) related to colloids is(are)\n\n(A) The process of precipitating colloidal sol by an electrolyte is called peptization.\n\n(B) Colloidal solution freezes at higher temperature than the true solution at the same concentration.\n\n(C) Surfactants form micelle above critical micelle concentration (CMC). CMC depends on temperature.\n\n(D) Micelles are macromolecular colloids.", "gold": "BC" }, { "description": "JEE Adv 2021 Paper 1", "index": 33, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) related to the metal extraction processes is(are)\n\n(A) A mixture of$\\mathrm{PbS}$and$\\mathrm{PbO}$undergoes self-reduction to produce$\\mathrm{Pb}$and$\\mathrm{SO}_{2}$.\n\n(B) In the extraction process of copper from copper pyrites, silica is added to produce copper silicate.\n\n(C) Partial oxidation of sulphide ore of copper by roasting, followed by self-reduction produces blister copper.\n\n(D) In cyanide process, zinc powder is utilized to precipitate gold from$\\mathrm{Na}\\left[\\mathrm{Au}(\\mathrm{CN})_{2}\\right]$.", "gold": "ACD" }, { "description": "JEE Adv 2021 Paper 1", "index": 35, "subject": "chem", "type": "Integer", "question": "What is the maximum number of possible isomers (including stereoisomers) which may be formed on mono-bromination of 1-methylcyclohex-1-ene using$\\mathrm{Br}_{2}$and UV light?", "gold": "13" }, { "description": "JEE Adv 2021 Paper 1", "index": 37, "subject": "math", "type": "MCQ", "question": "Consider a triangle$\\Delta$whose two sides lie on the$x$-axis and the line$x+y+1=0$. If the orthocenter of$\\Delta$is$(1,1)$, then the equation of the circle passing through the vertices of the triangle$\\Delta$is\n\n(A)$x^{2}+y^{2}-3 x+y=0$\n\n(B)$x^{2}+y^{2}+x+3 y=0$\n\n(C)$x^{2}+y^{2}+2 y-1=0$\n\n(D)$x^{2}+y^{2}+x+y=0$", "gold": "B" }, { "description": "JEE Adv 2021 Paper 1", "index": 38, "subject": "math", "type": "MCQ", "question": "The area of the region\n\n\$\n\\left\\{(x, y): 0 \\leq x \\leq \\frac{9}{4}, \\quad 0 \\leq y \\leq 1, \\quad x \\geq 3 y, \\quad x+y \\geq 2\\right\\}\n\$\n\nis\n\n(A)$\\frac{11}{32}$\n\n(B)$\\frac{35}{96}$\n\n(C)$\\frac{37}{96}$\n\n(D)$\\frac{13}{32}$", "gold": "A" }, { "description": "JEE Adv 2021 Paper 1", "index": 39, "subject": "math", "type": "MCQ", "question": "Consider three sets$E_{1}=\\{1,2,3\\}, F_{1}=\\{1,3,4\\}$and$G_{1}=\\{2,3,4,5\\}$. Two elements are chosen at random, without replacement, from the set$E_{1}$, and let$S_{1}$denote the set of these chosen elements. Let$E_{2}=E_{1}-S_{1}$and$F_{2}=F_{1} \\cup S_{1}$. Now two elements are chosen at random, without replacement, from the set$F_{2}$and let$S_{2}$denote the set of these chosen elements.\n\nLet$G_{2}=G_{1} \\cup S_{2}$. Finally, two elements are chosen at random, without replacement, from the set$G_{2}$and let$S_{3}$denote the set of these chosen elements.\n\nLet$E_{3}=E_{2} \\cup S_{3}$. Given that$E_{1}=E_{3}$, let$p$be the conditional probability of the event$S_{1}=\\{1,2\\}$. Then the value of$p$is\n\n(A)$\\frac{1}{5}$\n\n(B)$\\frac{3}{5}$\n\n(C)$\\frac{1}{2}$\n\n(D)$\\frac{2}{5}$", "gold": "A" }, { "description": "JEE Adv 2021 Paper 1", "index": 40, "subject": "math", "type": "MCQ", "question": "Let$\\theta_{1}, \\theta_{2}, \\ldots, \\theta_{10}$be positive valued angles (in radian) such that$\\theta_{1}+\\theta_{2}+\\cdots+\\theta_{10}=2 \\pi$. Define the complex numbers$z_{1}=e^{i \\theta_{1}}, z_{k}=z_{k-1} e^{i \\theta_{k}}$for$k=2,3, \\ldots, 10$, where$i=\\sqrt{-1}$. Consider the statements$P$and$Qgiven below:\n\n\\n\\begin{aligned}\n\n& P:\\left|z_{2}-z_{1}\\right|+\\left|z_{3}-z_{2}\\right|+\\cdots+\\left|z_{10}-z_{9}\\right|+\\left|z_{1}-z_{10}\\right| \\leq 2 \\pi \\\\\n\n& Q:\\left|z_{2}^{2}-z_{1}^{2}\\right|+\\left|z_{3}^{2}-z_{2}^{2}\\right|+\\cdots+\\left|z_{10}^{2}-z_{9}^{2}\\right|+\\left|z_{1}^{2}-z_{10}^{2}\\right| \\leq 4 \\pi\n\n\\end{aligned}\n\\n\nThen,\n\n(A)P$is TRUE and$Q$is FALSE\n\n(B)$Q$is TRUE and$P$is FALSE\n\n(C) both$P$and$Q$are TRUE\n\n(D) both$P$and$Q$are FALSE", "gold": "C" }, { "description": "JEE Adv 2021 Paper 1", "index": 41, "subject": "math", "type": "Numeric", "question": "Three numbers are chosen at random, one after another with replacement, from the set$S=\\{1,2,3, \\ldots, 100\\}$. Let$p_1$be the probability that the maximum of chosen numbers is at least 81 and$p_2$be the probability that the minimum of chosen numbers is at most 40 . What is the value of$\\frac{625}{4} p_{1}$?", "gold": "76.25" }, { "description": "JEE Adv 2021 Paper 1", "index": 42, "subject": "math", "type": "Numeric", "question": "Three numbers are chosen at random, one after another with replacement, from the set$S=\\{1,2,3, \\ldots, 100\\}$. Let$p_1$be the probability that the maximum of chosen numbers is at least 81 and$p_2$be the probability that the minimum of chosen numbers is at most 40 . What is the value of$\\frac{125}{4} p_{2}$?", "gold": "24.5" }, { "description": "JEE Adv 2021 Paper 1", "index": 43, "subject": "math", "type": "Numeric", "question": "Let$\\alpha, \\beta$and$\\gamma$be real numbers such that the system of linear equations\n\$\\begin{gathered}\nx+2 y+3 z=\\alpha \\\\\n4 x+5 y+6 z=\\beta \\\\\n7 x+8 y+9 z=\\gamma-1\n\\end{gathered}\$\nis consistent. Let$|M|$represent the determinant of the matrix\n\$M=\\left[\\begin{array}{ccc}\n\\alpha & 2 & \\gamma \\\\\n\\beta & 1 & 0 \\\\\n-1 & 0 & 1\n\\end{array}\\right]\$\nLet$P$be the plane containing all those$(\\alpha, \\beta, \\gamma)$for which the above system of linear equations is consistent, and$D$be the square of the distance of the point$(0,1,0)$from the plane$P$. What is the value of$|\\mathrm{M}|$?", "gold": "1.00" }, { "description": "JEE Adv 2021 Paper 1", "index": 44, "subject": "math", "type": "Numeric", "question": "Let$\\alpha, \\beta$and$\\gamma$be real numbers such that the system of linear equations\n\$\\begin{gathered}\nx+2 y+3 z=\\alpha \\\\\n4 x+5 y+6 z=\\beta \\\\\n7 x+8 y+9 z=\\gamma-1\n\\end{gathered}\$\nis consistent. Let$|M|$represent the determinant of the matrix\n\$M=\\left[\\begin{array}{ccc}\n\\alpha & 2 & \\gamma \\\\\n\\beta & 1 & 0 \\\\\n-1 & 0 & 1\n\\end{array}\\right]\$\nLet$P$be the plane containing all those$(\\alpha, \\beta, \\gamma)$for which the above system of linear equations is consistent, and$D$be the square of the distance of the point$(0,1,0)$from the plane$P$. What is the value of$D$?", "gold": "1.5" }, { "description": "JEE Adv 2021 Paper 1", "index": 45, "subject": "math", "type": "Numeric", "question": "Consider the lines$\\mathrm{L}_1$and$\\mathrm{L}_2$defined by\n$\\mathrm{L}_1: \\mathrm{x} \\sqrt{2}+\\mathrm{y}-1=0$and$\\mathrm{L}_2: \\mathrm{x} \\sqrt{2}-\\mathrm{y}+1=0$\nFor a fixed constant$\\lambda$, let$\\mathrm{C}$be the locus of a point$\\mathrm{P}$such that the product of the distance of$\\mathrm{P}$from$\\mathrm{L}_1$and the distance of$\\mathrm{P}$from$\\mathrm{L}_2$is$\\lambda^2$. The line$\\mathrm{y}=2 \\mathrm{x}+1$meets$\\mathrm{C}$at two points$\\mathrm{R}$and$\\mathrm{S}$, where the distance between$\\mathrm{R}$and$\\mathrm{S}$is$\\sqrt{270}$.\n\nLet the perpendicular bisector of RS meet$\\mathrm{C}$at two distinct points$\\mathrm{R}^{\\prime}$and$\\mathrm{S}^{\\prime}$. Let$\\mathrm{D}$be the square of the distance between$\\mathrm{R}^{\\prime}$and S'. What is the value of$\\lambda^{2}$?", "gold": "9.00" }, { "description": "JEE Adv 2021 Paper 1", "index": 46, "subject": "math", "type": "Numeric", "question": "Consider the lines$\\mathrm{L}_1$and$\\mathrm{L}_2$defined by\n$\\mathrm{L}_1: \\mathrm{x} \\sqrt{2}+\\mathrm{y}-1=0$and$\\mathrm{L}_2: \\mathrm{x} \\sqrt{2}-\\mathrm{y}+1=0$\nFor a fixed constant$\\lambda$, let$\\mathrm{C}$be the locus of a point$\\mathrm{P}$such that the product of the distance of$\\mathrm{P}$from$\\mathrm{L}_1$and the distance of$\\mathrm{P}$from$\\mathrm{L}_2$is$\\lambda^2$. The line$\\mathrm{y}=2 \\mathrm{x}+1$meets$\\mathrm{C}$at two points$\\mathrm{R}$and$\\mathrm{S}$, where the distance between$\\mathrm{R}$and$\\mathrm{S}$is$\\sqrt{270}$.\n\nLet the perpendicular bisector of RS meet$\\mathrm{C}$at two distinct points$\\mathrm{R}^{\\prime}$and$\\mathrm{S}^{\\prime}$. Let$\\mathrm{D}$be the square of the distance between$\\mathrm{R}^{\\prime}$and S'. What is the value of$D$?", "gold": "77.14" }, { "description": "JEE Adv 2021 Paper 1", "index": 47, "subject": "math", "type": "MCQ(multiple)", "question": "For any$3 \\times 3$matrix$M$, let$|M|$denote the determinant of$M$. Let\n\n\$\nE=\\left[\\begin{array}{ccc}\n\n1 & 2 & 3 \\\\\n\n2 & 3 & 4 \\\\\n\n8 & 13 & 18\n\n\\end{array}\\right], P=\\left[\\begin{array}{ccc}\n\n1 & 0 & 0 \\\\\n\n0 & 0 & 1 \\\\\n\n0 & 1 & 0\n\n\\end{array}\\right] \\text { and } F=\\left[\\begin{array}{ccc}\n\n1 & 3 & 2 \\\\\n\n8 & 18 & 13 \\\\\n\n2 & 4 & 3\n\n\\end{array}\\right]\n\$\n\nIf$Q$is a nonsingular matrix of order$3 \\times 3$, then which of the following statements is (are) TRUE?\n\n(A)$F=P E P$and$P^{2}=\\left[\\begin{array}{lll}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{array}\\right]$\n\n(B)$\\left|E Q+P F Q^{-1}\\right|=|E Q|+\\left|P F Q^{-1}\\right|$\n\n(C)$\\left|(E F)^{3}\\right|>|E F|^{2}$\n\n(D) Sum of the diagonal entries of$P^{-1} E P+F$is equal to the sum of diagonal entries of$E+P^{-1} F P$", "gold": "ABD" }, { "description": "JEE Adv 2021 Paper 1", "index": 48, "subject": "math", "type": "MCQ(multiple)", "question": "Let$f: \\mathbb{R} \\rightarrow \\mathbb{R}$be defined by\n\n\$\nf(x)=\\frac{x^{2}-3 x-6}{x^{2}+2 x+4}\n\$\n\nThen which of the following statements is (are) TRUE ?\n\n(A)$f$is decreasing in the interval$(-2,-1)$\n\n(B)$f$is increasing in the interval$(1,2)$\n\n(C)$f$is onto\n\n(D) Range of$f$is$\\left[-\\frac{3}{2}, 2\\right]$", "gold": "AB" }, { "description": "JEE Adv 2021 Paper 1", "index": 49, "subject": "math", "type": "MCQ(multiple)", "question": "Let$E, F$and$G$be three events having probabilities\n\n\$\nP(E)=\\frac{1}{8}, P(F)=\\frac{1}{6} \\text { and } P(G)=\\frac{1}{4} \\text {, and let } P(E \\cap F \\cap G)=\\frac{1}{10} \\text {. }\n\$\n\nFor any event$H$, if$H^{c}$denotes its complement, then which of the following statements is (are) TRUE ?\n\n(A)$P\\left(E \\cap F \\cap G^{c}\\right) \\leq \\frac{1}{40}$\n\n(B)$P\\left(E^{c} \\cap F \\cap G\\right) \\leq \\frac{1}{15}$\n\n(C)$P(E \\cup F \\cup G) \\leq \\frac{13}{24}$\n\n(D)$P\\left(E^{c} \\cap F^{c} \\cap G^{c}\\right) \\leq \\frac{5}{12}$", "gold": "ABC" }, { "description": "JEE Adv 2021 Paper 1", "index": 50, "subject": "math", "type": "MCQ(multiple)", "question": "For any$3 \\times 3$matrix$M$, let$|M|$denote the determinant of$M$. Let$I$be the$3 \\times 3$identity matrix. Let$E$and$F$be two$3 \\times 3$matrices such that$(I-E F)$is invertible. If$G=(I-E F)^{-1}$, then which of the following statements is (are) TRUE ?\n\n(A)$|F E|=|I-F E||F G E|$\n\n(B)$(I-F E)(I+F G E)=I$\n\n(C)$E F G=G E F$\n\n(D)$(I-F E)(I-F G E)=I$", "gold": "ABC" }, { "description": "JEE Adv 2021 Paper 1", "index": 51, "subject": "math", "type": "MCQ(multiple)", "question": "For any positive integer$n$, let$S_{n}:(0, \\infty) \\rightarrow \\mathbb{R}$be defined by\n\n\$\nS_{n}(x)=\\sum_{k=1}^{n} \\cot ^{-1}\\left(\\frac{1+k(k+1) x^{2}}{x}\\right)\n\$\n\nwhere for any$x \\in \\mathbb{R}, \\cot ^{-1}(x) \\in(0, \\pi)$and$\\tan ^{-1}(x) \\in\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$. Then which of the following statements is (are) TRUE ?\n\n(A)$S_{10}(x)=\\frac{\\pi}{2}-\\tan ^{-1}\\left(\\frac{1+11 x^{2}}{10 x}\\right)$, for all$x>0$\n\n(B)$\\lim _{n \\rightarrow \\infty} \\cot \\left(S_{n}(x)\\right)=x$, for all$x>0$\n\n(C) The equation$S_{3}(x)=\\frac{\\pi}{4}$has a root in$(0, \\infty)$\n\n(D)$\\tan \\left(S_{n}(x)\\right) \\leq \\frac{1}{2}$, for all$n \\geq 1$and$x>0$", "gold": "AB" }, { "description": "JEE Adv 2021 Paper 1", "index": 52, "subject": "math", "type": "MCQ(multiple)", "question": "For any complex number$w=c+i d$, let$\\arg (\\mathrm{w}) \\in(-\\pi, \\pi]$, where$i=\\sqrt{-1}$. Let$\\alpha$and$\\beta$be real numbers such that for all complex numbers$z=x+i y$satisfying$\\arg \\left(\\frac{z+\\alpha}{z+\\beta}\\right)=\\frac{\\pi}{4}$, the ordered pair$(x, y)$lies on the circle\n\n\$\nx^{2}+y^{2}+5 x-3 y+4=0\n\$\n\nThen which of the following statements is (are) TRUE ?\n\n(A)$\\alpha=-1$\n\n(B)$\\alpha \\beta=4$\n\n(C)$\\alpha \\beta=-4$\n\n(D)$\\beta=4$", "gold": "BD" }, { "description": "JEE Adv 2021 Paper 1", "index": 53, "subject": "math", "type": "Integer", "question": "For$x \\in \\mathbb{R}$, what is the number of real roots of the equation\n\n\$\n3 x^{2}-4\\left|x^{2}-1\\right|+x-1=0\n\$?", "gold": "4" }, { "description": "JEE Adv 2021 Paper 1", "index": 54, "subject": "math", "type": "Integer", "question": "In a triangle$A B C$, let$A B=\\sqrt{23}, B C=3$and$C A=4$. Then what is the value of\n\n\$\n\\frac{\\cot A+\\cot C}{\\cot B}\n\$?", "gold": "2" }, { "description": "JEE Adv 2021 Paper 2", "index": 2, "subject": "phy", "type": "MCQ(multiple)", "question": "A source, approaching with speed$u$towards the open end of a stationary pipe of length$L$, is emitting a sound of frequency$f_{s}$. The farther end of the pipe is closed. The speed of sound in air is$v$and$f_{0}$is the fundamental frequency of the pipe. For which of the following combination(s) of$u$and$f_{s}$, will the sound reaching the pipe lead to a resonance?\n\n(A)$u=0.8 v$and$f_{s}=f_{0}$\n\n(B)$u=0.8 v$and$f_{s}=2 f_{0}$\n\n(C)$u=0.8 v$and$f_{s}=0.5 f_{0}$\n\n(D)$u=0.5 v$and$f_{s}=1.5 f_{0}$", "gold": "AD" }, { "description": "JEE Adv 2021 Paper 2", "index": 4, "subject": "phy", "type": "MCQ(multiple)", "question": "A physical quantity$\\vec{S}$is defined as$\\vec{S}=(\\vec{E} \\times \\vec{B}) / \\mu_{0}$, where$\\vec{E}$is electric field,$\\vec{B}$is magnetic field and$\\mu_{0}$is the permeability of free space. The dimensions of$\\vec{S}$are the same as the dimensions of which of the following quantity(ies)?\n\n(A)$\\frac{\\text { Energy }}{\\text { Charge } \\times \\text { Current }}$\n\n(B)$\\frac{\\text { Force }}{\\text { Length } \\times \\text { Time }}$\n\n(C)$\\frac{\\text { Energy }}{\\text { Volume }}$\n\n(D)$\\frac{\\text { Power }}{\\text { Area }}$", "gold": "BD" }, { "description": "JEE Adv 2021 Paper 2", "index": 5, "subject": "phy", "type": "MCQ(multiple)", "question": "A heavy nucleus$N$, at rest, undergoes fission$N \\rightarrow P+Q$, where$P$and$Q$are two lighter nuclei. Let$\\delta=M_{N}-M_{P}-M_{Q}$, where$M_{P}, M_{Q}$and$M_{N}$are the masses of$P$,$Q$and$N$, respectively.$E_{P}$and$E_{Q}$are the kinetic energies of$P$and$Q$, respectively. The speeds of$P$and$Q$are$v_{P}$and$v_{Q}$, respectively. If$c$is the speed of light, which of the following statement(s) is(are) correct?\n\n(A)$E_{P}+E_{Q}=c^{2} \\delta$\n\n(B)$E_{P}=\\left(\\frac{M_{P}}{M_{P}+M_{Q}}\\right) c^{2} \\delta$\n\n(C)$\\frac{v_{P}}{v_{Q}}=\\frac{M_{Q}}{M_{P}}$\n\n(D) The magnitude of momentum for$P$as well as$Q$is$c \\sqrt{2 \\mu \\delta}$, where$\\mu=\\frac{M_{P} M_{Q}}{\\left(M_{P}+M_{Q}\\right)}$", "gold": "ACD" }, { "description": "JEE Adv 2021 Paper 2", "index": 9, "subject": "phy", "type": "Numeric", "question": "A pendulum consists of a bob of mass$m=0.1 \\mathrm{~kg}$and a massless inextensible string of length$L=1.0 \\mathrm{~m}$. It is suspended from a fixed point at height$H=0.9 \\mathrm{~m}$above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse$P=0.2 \\mathrm{~kg}-\\mathrm{m} / \\mathrm{s}$is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is$J \\mathrm{~kg}-\\mathrm{m}^2 / \\mathrm{s}$. The kinetic energy of the pendulum just after the liftoff is$K$Joules. What is the value of$J$?", "gold": "0.18" }, { "description": "JEE Adv 2021 Paper 2", "index": 10, "subject": "phy", "type": "Numeric", "question": "A pendulum consists of a bob of mass$m=0.1 \\mathrm{~kg}$and a massless inextensible string of length$L=1.0 \\mathrm{~m}$. It is suspended from a fixed point at height$H=0.9 \\mathrm{~m}$above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse$P=0.2 \\mathrm{~kg}-\\mathrm{m} / \\mathrm{s}$is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is$J \\mathrm{~kg}-\\mathrm{m}^2 / \\mathrm{s}$. The kinetic energy of the pendulum just after the liftoff is$K$Joules. What is the value of$K$?", "gold": "0.16" }, { "description": "JEE Adv 2021 Paper 2", "index": 11, "subject": "phy", "type": "Numeric", "question": "In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance$\\mathrm{C} \\mu F$across a$200 \\mathrm{~V}, 50 \\mathrm{~Hz}$supply. The power consumed by the lamp is$500 \\mathrm{~W}$while the voltage drop across it is$100 \\mathrm{~V}$. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is$\\varphi$. Assume,$\\pi \\sqrt{3} \\approx 5$. What is the value of$C$?", "gold": "100.00" }, { "description": "JEE Adv 2021 Paper 2", "index": 12, "subject": "phy", "type": "Numeric", "question": "In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance$\\mathrm{C} \\mu F$across a$200 \\mathrm{~V}, 50 \\mathrm{~Hz}$supply. The power consumed by the lamp is$500 \\mathrm{~W}$while the voltage drop across it is$100 \\mathrm{~V}$. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is$\\varphi$. Assume,$\\pi \\sqrt{3} \\approx 5$. What is the value of$\\varphi$?", "gold": "60" }, { "description": "JEE Adv 2021 Paper 2", "index": 21, "subject": "chem", "type": "MCQ(multiple)", "question": "For the following reaction\n\n$2 \\mathbf{X}+\\mathbf{Y} \\stackrel{k}{\\rightarrow} \\mathbf{P}$\n\nthe rate of reaction is$\\frac{d[\\mathbf{P}]}{d t}=k[\\mathbf{X}]$. Two moles of$\\mathbf{X}$are mixed with one mole of$\\mathbf{Y}$to make 1.0 L of solution. At$50 \\mathrm{~s}, 0.5$mole of$\\mathbf{Y}$is left in the reaction mixture. The correct statement(s) about the reaction is(are)\n\n$($Use$: \\ln 2=0.693)$\n\n(A) The rate constant,$k$, of the reaction is$13.86 \\times 10^{-4} \\mathrm{~s}^{-1}$.\n\n(B) Half-life of$\\mathbf{X}$is$50 \\mathrm{~s}$.\n\n(C) At$50 \\mathrm{~s},-\\frac{d[\\mathbf{X}]}{d t}=13.86 \\times 10^{-3} \\mathrm{~mol} \\mathrm{~L}^{-1} \\mathrm{~s}^{-1}$.\n\n(D) At$100 \\mathrm{~s},-\\frac{d[\\mathrm{Y}]}{d t}=3.46 \\times 10^{-3} \\mathrm{~mol} \\mathrm{~L}^{-1} \\mathrm{~s}^{-1}$.", "gold": "BCD" }, { "description": "JEE Adv 2021 Paper 2", "index": 22, "subject": "chem", "type": "MCQ(multiple)", "question": "Some standard electrode potentials at$298 \\mathrm{~K}$are given below:\n\n$\\begin{array}{ll}\\mathrm{Pb}^{2+} / \\mathrm{Pb} & -0.13 \\mathrm{~V} \\\\ \\mathrm{Ni}^{2+} / \\mathrm{Ni} & -0.24 \\mathrm{~V} \\\\ \\mathrm{Cd}^{2+} / \\mathrm{Cd} & -0.40 \\mathrm{~V} \\\\ \\mathrm{Fe}^{2+} / \\mathrm{Fe} & -0.44 \\mathrm{~V}\\end{array}$\n\nTo a solution containing$0.001 \\mathrm{M}$of$\\mathbf{X}^{2+}$and$0.1 \\mathrm{M}$of$\\mathbf{Y}^{2+}$, the metal rods$\\mathbf{X}$and$\\mathbf{Y}$are inserted (at$298 \\mathrm{~K}$) and connected by a conducting wire. This resulted in dissolution of$\\mathbf{X}$. The correct combination(s) of$\\mathbf{X}$and$\\mathbf{Y}$, respectively, is(are)\n\n(Given: Gas constant,$\\mathrm{R}=8.314 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$, Faraday constant,$\\mathrm{F}=96500 \\mathrm{C} \\mathrm{mol}^{-1}$)\n\n(A)$\\mathrm{Cd}$and$\\mathrm{Ni}$\n\n(B)$\\mathrm{Cd}$and$\\mathrm{Fe}$\n\n(C)$\\mathrm{Ni}$and$\\mathrm{Pb}$\n\n(D)$\\mathrm{Ni}$and$\\mathrm{Fe}$", "gold": "ABC" }, { "description": "JEE Adv 2021 Paper 2", "index": 23, "subject": "chem", "type": "MCQ(multiple)", "question": "The pair(s) of complexes wherein both exhibit tetrahedral geometry is(are)\n\n(Note:$\\mathrm{py}=$pyridine\n\nGiven: Atomic numbers of Fe, Co, Ni and$\\mathrm{Cu}$are 26, 27, 28 and 29, respectively)\n\n(A)$\\left[\\mathrm{FeCl}_{4}\\right]^{-}$and$\\left[\\mathrm{Fe}(\\mathrm{CO})_{4}\\right]^{2-}$\n\n(B)$\\left[\\mathrm{Co}(\\mathrm{CO})_{4}\\right]^{-}$and$\\left[\\mathrm{CoCl}_{4}\\right]^{2-}$\n\n(C)$\\left[\\mathrm{Ni}(\\mathrm{CO})_{4}\\right]$and$\\left[\\mathrm{Ni}(\\mathrm{CN})_{4}\\right]^{2-}$\n\n(D)$\\left[\\mathrm{Cu}(\\mathrm{py})_{4}\\right]^{+}$and$\\left[\\mathrm{Cu}(\\mathrm{CN})_{4}\\right]^{3-}$", "gold": "ABD" }, { "description": "JEE Adv 2021 Paper 2", "index": 24, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) related to oxoacids of phosphorous is(are) (A) Upon heating,$\\mathrm{H}_{3} \\mathrm{PO}_{3}$undergoes disproportionation reaction to produce$\\mathrm{H}_{3} \\mathrm{PO}_{4}$and$\\mathrm{PH}_{3}$.\n\n(B) While$\\mathrm{H}_{3} \\mathrm{PO}_{3}$can act as reducing agent,$\\mathrm{H}_{3} \\mathrm{PO}_{4}$cannot.\n\n(C)$\\mathrm{H}_{3} \\mathrm{PO}_{3}$is a monobasic acid.\n\n(D) The$\\mathrm{H}$atom of$\\mathrm{P}-\\mathrm{H}$bond in$\\mathrm{H}_{3} \\mathrm{PO}_{3}$is not ionizable in water.", "gold": "ABD" }, { "description": "JEE Adv 2021 Paper 2", "index": 25, "subject": "chem", "type": "Numeric", "question": "At$298 \\mathrm{~K}$, the limiting molar conductivity of a weak monobasic acid is$4 \\times 10^2 \\mathrm{~S} \\mathrm{~cm}^2 \\mathrm{~mol}^{-1}$. At$298 \\mathrm{~K}$, for an aqueous solution of the acid the degree of dissociation is$\\alpha$and the molar conductivity is$\\mathbf{y} \\times 10^2 \\mathrm{~S} \\mathrm{~cm}^2 \\mathrm{~mol}^{-1}$. At$298 \\mathrm{~K}$, upon 20 times dilution with water, the molar conductivity of the solution becomes$3 \\mathbf{y} \\times 10^2 \\mathrm{~S} \\mathrm{~cm}^2 \\mathrm{~mol}^{-1}$. What is the value of$\\alpha$?", "gold": "0.22" }, { "description": "JEE Adv 2021 Paper 2", "index": 26, "subject": "chem", "type": "Numeric", "question": "At$298 \\mathrm{~K}$, the limiting molar conductivity of a weak monobasic acid is$4 \\times 10^2 \\mathrm{~S} \\mathrm{~cm}^2 \\mathrm{~mol}^{-1}$. At$298 \\mathrm{~K}$, for an aqueous solution of the acid the degree of dissociation is$\\alpha$and the molar conductivity is$\\mathbf{y} \\times 10^2 \\mathrm{~S} \\mathrm{~cm}^2 \\mathrm{~mol}^{-1}$. At$298 \\mathrm{~K}$, upon 20 times dilution with water, the molar conductivity of the solution becomes$3 \\mathbf{y} \\times 10^2 \\mathrm{~S} \\mathrm{~cm}^2 \\mathrm{~mol}^{-1}$. What is the value of$\\mathbf{y}$?", "gold": "0.86" }, { "description": "JEE Adv 2021 Paper 2", "index": 27, "subject": "chem", "type": "Numeric", "question": "Reaction of$\\mathbf{x} \\mathrm{g}$of$\\mathrm{Sn}$with$\\mathrm{HCl}$quantitatively produced a salt. Entire amount of the salt reacted with$\\mathbf{y} g$of nitrobenzene in the presence of required amount of$\\mathrm{HCl}$to produce$1.29 \\mathrm{~g}$of an organic salt (quantitatively).\n(Use Molar masses (in$\\mathrm{g} \\mathrm{mol}^{-1}$) of$\\mathrm{H}, \\mathrm{C}, \\mathrm{N}, \\mathrm{O}, \\mathrm{Cl}$and Sn as 1, 12, 14, 16, 35 and 119 , respectively). What is the value of$\\mathbf{x}$?", "gold": "3.57" }, { "description": "JEE Adv 2021 Paper 2", "index": 28, "subject": "chem", "type": "Numeric", "question": "Reaction of$\\mathbf{x} \\mathrm{g}$of$\\mathrm{Sn}$with$\\mathrm{HCl}$quantitatively produced a salt. Entire amount of the salt reacted with$\\mathbf{y} g$of nitrobenzene in the presence of required amount of$\\mathrm{HCl}$to produce$1.29 \\mathrm{~g}$of an organic salt (quantitatively).\n(Use Molar masses (in$\\mathrm{g} \\mathrm{mol}^{-1}$) of$\\mathrm{H}, \\mathrm{C}, \\mathrm{N}, \\mathrm{O}, \\mathrm{Cl}$and Sn as 1, 12, 14, 16, 35 and 119 , respectively). What is the value of$\\mathbf{y}$?", "gold": "1.23" }, { "description": "JEE Adv 2021 Paper 2", "index": 29, "subject": "chem", "type": "Numeric", "question": "A sample$(5.6 \\mathrm{~g})$containing iron is completely dissolved in cold dilute$\\mathrm{HCl}$to prepare a$250 \\mathrm{~mL}$of solution. Titration of$25.0 \\mathrm{~mL}$of this solution requires$12.5 \\mathrm{~mL}$of$0.03 \\mathrm{M} \\mathrm{KMnO}_4$solution to reach the end point. Number of moles of$\\mathrm{Fe}^{2+}$present in$250 \\mathrm{~mL}$solution is$\\mathbf{x} \\times 10^{-2}$(consider complete dissolution of$\\mathrm{FeCl}_2$). The amount of iron present in the sample is$\\mathbf{y} \\%$by weight.\n(Assume:$\\mathrm{KMnO}_4$reacts only with$\\mathrm{Fe}^{2+}$in the solution\nUse: Molar mass of iron as$56 \\mathrm{~g} \\mathrm{~mol}^{-1}$). What is the value of$\\mathbf{x}$?", "gold": "1.87" }, { "description": "JEE Adv 2021 Paper 2", "index": 30, "subject": "chem", "type": "Numeric", "question": "A sample$(5.6 \\mathrm{~g})$containing iron is completely dissolved in cold dilute$\\mathrm{HCl}$to prepare a$250 \\mathrm{~mL}$of solution. Titration of$25.0 \\mathrm{~mL}$of this solution requires$12.5 \\mathrm{~mL}$of$0.03 \\mathrm{M} \\mathrm{KMnO}_4$solution to reach the end point. Number of moles of$\\mathrm{Fe}^{2+}$present in$250 \\mathrm{~mL}$solution is$\\mathbf{x} \\times 10^{-2}$(consider complete dissolution of$\\mathrm{FeCl}_2$). The amount of iron present in the sample is$\\mathbf{y} \\%$by weight.\n(Assume:$\\mathrm{KMnO}_4$reacts only with$\\mathrm{Fe}^{2+}$in the solution\nUse: Molar mass of iron as$56 \\mathrm{~g} \\mathrm{~mol}^{-1}$). What is the value of$\\mathbf{y}$?", "gold": "18.75" }, { "description": "JEE Adv 2021 Paper 2", "index": 31, "subject": "chem", "type": "MCQ", "question": "Correct match of the$\\mathbf{C}-\\mathbf{H}$bonds (shown in bold) in Column$\\mathbf{J}$with their BDE in Column$\\mathbf{K}$is\n\n\\begin{center}\n\n\\begin{tabular}{|c|c|}\n\n\\hline\n\n$\\begin{array}{l}\\text { Column J } \\\\ \\text { Molecule }\\end{array}$&$\\begin{array}{c}\\text { Column K } \\\\ \\text { BDE }\\left(\\text { kcal mol }^{-1}\\right)\\end{array}$\\\\\n\n\\hline\n\n(P)$\\mathbf{H}-\\mathrm{CH}\\left(\\mathrm{CH}_{3}\\right)_{2}$& (i) 132 \\\\\n\n\\hline\n\n(Q)$\\mathbf{H}-\\mathrm{CH}_{2} \\mathrm{Ph}$& (ii) 110 \\\\\n\n\\hline\n\n(R)$\\mathbf{H}-\\mathrm{CH}=\\mathrm{CH}_{2}$& (iii) 95 \\\\\n\n\\hline\n\n(S)$\\mathrm{H}-\\mathrm{C} \\equiv \\mathrm{CH}$& (iv) 88 \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}\n\n(A)$P$- iii,$Q$- iv, R - ii, S - i\n\n(B)$P-i, Q-i i, R-$iii, S - iv\n\n(C) P - iii, Q - ii, R - i, S - iv\n\n(D)$P-i i, Q-i, R-i v, S-i i i$", "gold": "A" }, { "description": "JEE Adv 2021 Paper 2", "index": 33, "subject": "chem", "type": "MCQ", "question": "The reaction of$\\mathrm{K}_3\\left[\\mathrm{Fe}(\\mathrm{CN})_6\\right]$with freshly prepared$\\mathrm{FeSO}_4$solution produces a dark blue precipitate called Turnbull's blue. Reaction of$\\mathrm{K}_4\\left[\\mathrm{Fe}(\\mathrm{CN})_6\\right]$with the$\\mathrm{FeSO}_4$solution in complete absence of air produces a white precipitate$\\mathbf{X}$, which turns blue in air. Mixing the$\\mathrm{FeSO}_4$solution with$\\mathrm{NaNO}_3$, followed by a slow addition of concentrated$\\mathrm{H}_2 \\mathrm{SO}_4$through the side of the test tube produces a brown ring. Precipitate$\\mathbf{X}$is\n\n(A)$\\mathrm{Fe}_{4}\\left[\\mathrm{Fe}(\\mathrm{CN})_{6}\\right]_{3}$\n\n(B)$\\mathrm{Fe}\\left[\\mathrm{Fe}(\\mathrm{CN})_{6}\\right]$\n\n(C)$\\mathrm{K}_{2} \\mathrm{Fe}\\left[\\mathrm{Fe}(\\mathrm{CN})_{6}\\right]$\n\n(D)$\\mathrm{KFe}\\left[\\mathrm{Fe}(\\mathrm{CN})_{6}\\right]$", "gold": "C" }, { "description": "JEE Adv 2021 Paper 2", "index": 34, "subject": "chem", "type": "MCQ", "question": "The reaction of$\\mathrm{K}_3\\left[\\mathrm{Fe}(\\mathrm{CN})_6\\right]$with freshly prepared$\\mathrm{FeSO}_4$solution produces a dark blue precipitate called Turnbull's blue. Reaction of$\\mathrm{K}_4\\left[\\mathrm{Fe}(\\mathrm{CN})_6\\right]$with the$\\mathrm{FeSO}_4$solution in complete absence of air produces a white precipitate$\\mathbf{X}$, which turns blue in air. Mixing the$\\mathrm{FeSO}_4$solution with$\\mathrm{NaNO}_3$, followed by a slow addition of concentrated$\\mathrm{H}_2 \\mathrm{SO}_4$through the side of the test tube produces a brown ring. Among the following, the brown ring is due to the formation of\n\n(A)$\\left[\\mathrm{Fe}(\\mathrm{NO})_{2}\\left(\\mathrm{SO}_{4}\\right)_{2}\\right]^{2-}$\n\n(B)$\\left[\\mathrm{Fe}(\\mathrm{NO})_{2}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)_{4}\\right]^{3+}$\n\n(C)$\\left[\\mathrm{Fe}(\\mathrm{NO})_{4}\\left(\\mathrm{SO}_{4}\\right)_{2}\\right]$\n\n(D)$\\left[\\mathrm{Fe}(\\mathrm{NO})\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)_{5}\\right]^{2+}$", "gold": "D" }, { "description": "JEE Adv 2021 Paper 2", "index": 36, "subject": "chem", "type": "Integer", "question": "Consider a helium (He) atom that absorbs a photon of wavelength$330 \\mathrm{~nm}$. What is the change in the velocity (in$\\mathrm{cm} \\mathrm{s}^{-1}$) of He atom after the photon absorption?\n\n(Assume: Momentum is conserved when photon is absorbed.\n\nUse: Planck constant$=6.6 \\times 10^{-34} \\mathrm{~J} \\mathrm{~s}$, Avogadro number$=6 \\times 10^{23} \\mathrm{~mol}^{-1}$, Molar mass of$\\mathrm{He}=4 \\mathrm{~g} \\mathrm{~mol}^{-1}$)", "gold": "30" }, { "description": "JEE Adv 2021 Paper 2", "index": 37, "subject": "math", "type": "MCQ(multiple)", "question": "Let\n\n\$\n\\begin{gathered}\n\nS_{1}=\\{(i, j, k): i, j, k \\in\\{1,2, \\ldots, 10\\}\\}, \\\\\n\nS_{2}=\\{(i, j): 1 \\leq i\\frac{p}{r} and \\cos R>\\frac{p}{q}", "gold": "AB" }, { "description": "JEE Adv 2021 Paper 2", "index": 39, "subject": "math", "type": "MCQ(multiple)", "question": "Let f:\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right] \\rightarrow \\mathbb{R} be a continuous function such that\n\n\\[\nf(0)=1 \\text { and } \\int_{0}^{\\frac{\\pi}{3}} f(t) d t=0\n\$\n\nThen which of the following statements is (are) TRUE ?\n\n(A) The equation$f(x)-3 \\cos 3 x=0$has at least one solution in$\\left(0, \\frac{\\pi}{3}\\right)$\n\n(B) The equation$f(x)-3 \\sin 3 x=-\\frac{6}{\\pi}$has at least one solution in$\\left(0, \\frac{\\pi}{3}\\right)$\n\n(C)$\\lim _{x \\rightarrow 0} \\frac{x \\int_{0}^{x} f(t) d t}{1-e^{x^{2}}}=-1$\n\n(D)$\\lim _{x \\rightarrow 0} \\frac{\\sin x \\int_{0}^{x} f(t) d t}{x^{2}}=-1$", "gold": "ABC" }, { "description": "JEE Adv 2021 Paper 2", "index": 40, "subject": "math", "type": "MCQ(multiple)", "question": "For any real numbers$\\alpha$and$\\beta$, let$y_{\\alpha, \\beta}(x), x \\in \\mathbb{R}$, be the solution of the differential equation\n\n\$\n\\frac{d y}{d x}+\\alpha y=x e^{\\beta x}, \\quad y(1)=1\n\$\n\nLet$S=\\left\\{y_{\\alpha, \\beta}(x): \\alpha, \\beta \\in \\mathbb{R}\\right\\}$. Then which of the following functions belong(s) to the set$S$?\n\n(A)$f(x)=\\frac{x^{2}}{2} e^{-x}+\\left(e-\\frac{1}{2}\\right) e^{-x}$\n\n(B)$f(x)=-\\frac{x^{2}}{2} e^{-x}+\\left(e+\\frac{1}{2}\\right) e^{-x}$\n\n(C)$f(x)=\\frac{e^{x}}{2}\\left(x-\\frac{1}{2}\\right)+\\left(e-\\frac{e^{2}}{4}\\right) e^{-x}$\n\n(D)$f(x)=\\frac{e^{x}}{2}\\left(\\frac{1}{2}-x\\right)+\\left(e+\\frac{e^{2}}{4}\\right) e^{-x}$", "gold": "AC" }, { "description": "JEE Adv 2021 Paper 2", "index": 42, "subject": "math", "type": "MCQ(multiple)", "question": "Let$E$denote the parabola$y^{2}=8 x$. Let$P=(-2,4)$, and let$Q$and$Q^{\\prime}$be two distinct points on$E$such that the lines$P Q$and$P Q^{\\prime}$are tangents to$E$. Let$F$be the focus of$E$. Then which of the following statements is (are) TRUE?\n\n(A) The triangle$P F Q$is a right-angled triangle\n\n(B) The triangle$Q P Q^{\\prime}$is a right-angled triangle\n\n(C) The distance between$P$and$F$is$5 \\sqrt{2}$\n\n(D)$F$lies on the line joining$Q$and$Q^{\\prime}$", "gold": "ABD" }, { "description": "JEE Adv 2021 Paper 2", "index": 43, "subject": "math", "type": "Numeric", "question": "Consider the region$R=\\left\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R}: x \\geq 0\\right.$and$\\left.y^2 \\leq 4-x\\right\\}$. Let$\\mathcal{F}$be the family of all circles that are contained in$R$and have centers on the$x$-axis. Let$C$be the circle that has largest radius among the circles in$\\mathcal{F}$. Let$(\\alpha, \\beta)$be a point where the circle$C$meets the curve$y^2=4-x$. What is the radius of the circle$C$?", "gold": "1.5" }, { "description": "JEE Adv 2021 Paper 2", "index": 44, "subject": "math", "type": "Numeric", "question": "Consider the region$R=\\left\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R}: x \\geq 0\\right.$and$\\left.y^2 \\leq 4-x\\right\\}$. Let$\\mathcal{F}$be the family of all circles that are contained in$R$and have centers on the$x$-axis. Let$C$be the circle that has largest radius among the circles in$\\mathcal{F}$. Let$(\\alpha, \\beta)$be a point where the circle$C$meets the curve$y^2=4-x$. What is the value of$\\alpha$?", "gold": "2.00" }, { "description": "JEE Adv 2021 Paper 2", "index": 45, "subject": "math", "type": "Numeric", "question": "Let$f_1:(0, \\infty) \\rightarrow \\mathbb{R}$and$f_2:(0, \\infty) \\rightarrow \\mathbb{R}$be defined by\n\$f_1(x)=\\int_0^x \\prod_{j=1}^{21}(t-j)^j d t, x>0\$\nand\n\$f_2(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0\$\nwhere, for any positive integer$\\mathrm{n}$and real numbers$\\mathrm{a}_1, \\mathrm{a}_2, \\ldots, \\mathrm{a}_{\\mathrm{n}}, \\prod_{i=1}^{\\mathrm{n}} \\mathrm{a}_i$denotes the product of$\\mathrm{a}_1, \\mathrm{a}_2, \\ldots, \\mathrm{a}_{\\mathrm{n}}$. Let$\\mathrm{m}_i$and$\\mathrm{n}_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function$f_i, i=1,2$, in the interval$(0, \\infty)$. What is the value of$2 m_{1}+3 n_{1}+m_{1} n_{1}$?", "gold": "57" }, { "description": "JEE Adv 2021 Paper 2", "index": 46, "subject": "math", "type": "Numeric", "question": "Let$f_1:(0, \\infty) \\rightarrow \\mathbb{R}$and$f_2:(0, \\infty) \\rightarrow \\mathbb{R}$be defined by\n\$f_1(x)=\\int_0^x \\prod_{j=1}^{21}(t-j)^j d t, x>0\$\nand\n\$f_2(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0\$\nwhere, for any positive integer$\\mathrm{n}$and real numbers$\\mathrm{a}_1, \\mathrm{a}_2, \\ldots, \\mathrm{a}_{\\mathrm{n}}, \\prod_{i=1}^{\\mathrm{n}} \\mathrm{a}_i$denotes the product of$\\mathrm{a}_1, \\mathrm{a}_2, \\ldots, \\mathrm{a}_{\\mathrm{n}}$. Let$\\mathrm{m}_i$and$\\mathrm{n}_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function$f_i, i=1,2$, in the interval$(0, \\infty)$. What is the value of$6 m_{2}+4 n_{2}+8 m_{2} n_{2}$?", "gold": "6" }, { "description": "JEE Adv 2021 Paper 2", "index": 47, "subject": "math", "type": "Numeric", "question": "Let$\\mathrm{g}_i:\\left[\\frac{\\pi}{8}, \\frac{3 \\pi}{8}\\right] \\rightarrow \\mathbb{R}, \\mathrm{i}=1,2$, and$f:\\left[\\frac{\\pi}{8}, \\frac{3 \\pi}{8}\\right] \\rightarrow \\mathbb{R}$be functions such that$\\mathrm{g}_1(\\mathrm{x})=1, \\mathrm{~g}_2(\\mathrm{x})=|4 \\mathrm{x}-\\pi|$and$f(\\mathrm{x})=\\sin ^2 \\mathrm{x}$, for all$\\mathrm{x} \\in\\left[\\frac{\\pi}{8}, \\frac{3 \\pi}{8}\\right]$\nDefine$\\mathrm{S}_i=\\int_{\\frac{\\pi}{8}}^{\\frac{3 \\pi}{8}} f(\\mathrm{x}) \\cdot \\mathrm{g}_i(\\mathrm{x}) \\mathrm{dx}, i=1,2$. What is the value of$\\frac{16 S_{1}}{\\pi}$?", "gold": "2" }, { "description": "JEE Adv 2021 Paper 2", "index": 49, "subject": "math", "type": "MCQ", "question": "Let\n\$\\mathrm{M}=\\left\\{(\\mathrm{x}, \\mathrm{y}) \\in \\mathbb{R} \\times \\mathbb{R}: \\mathrm{x}^2+\\mathrm{y}^2 \\leq \\mathrm{r}^2\\right\\}\$\nwhere$\\mathrm{r}>0$. Consider the geometric progression$a_n=\\frac{1}{2^{n-1}}, n=1,2,3, \\ldots$. Let$S_0=0$and, for$n \\geq 1$, let$S_n$denote the sum of the first$n$terms of this progression. For$n \\geq 1$, let$C_n$denote the circle with center$\\left(S_{n-1}, 0\\right)$and radius$a_n$, and$D_n$denote the circle with center$\\left(S_{n-1}, S_{n-1}\\right)$and radius$a_n$. Consider$M$with$r=\\frac{1025}{513}$. Let$k$be the number of all those circles$C_{n}$that are inside$M$. Let$l$be the maximum possible number of circles among these$k$circles such that no two circles intersect. Then\n\n(A)$k+2 l=22$\n\n(B)$2 k+l=26$\n\n(C)$2 k+3 l=34$\n\n(D)$3 k+2 l=40$", "gold": "D" }, { "description": "JEE Adv 2021 Paper 2", "index": 50, "subject": "math", "type": "MCQ", "question": "Let\n\$\\mathrm{M}=\\left\\{(\\mathrm{x}, \\mathrm{y}) \\in \\mathbb{R} \\times \\mathbb{R}: \\mathrm{x}^2+\\mathrm{y}^2 \\leq \\mathrm{r}^2\\right\\}\$\nwhere$\\mathrm{r}>0$. Consider the geometric progression$a_n=\\frac{1}{2^{n-1}}, n=1,2,3, \\ldots$. Let$S_0=0$and, for$n \\geq 1$, let$S_n$denote the sum of the first$n$terms of this progression. For$n \\geq 1$, let$C_n$denote the circle with center$\\left(S_{n-1}, 0\\right)$and radius$a_n$, and$D_n$denote the circle with center$\\left(S_{n-1}, S_{n-1}\\right)$and radius$a_n$. Consider$M$with$r=\\frac{\\left(2^{199}-1\\right) \\sqrt{2}}{2^{198}}$. The number of all those circles$D_{n}$that are inside$M$is\n\n(A) 198\n\n(B) 199\n\n(C) 200\n\n(D) 201", "gold": "B" }, { "description": "JEE Adv 2021 Paper 2", "index": 51, "subject": "math", "type": "MCQ", "question": "Let$\\psi_1:[0, \\infty) \\rightarrow \\mathbb{R}, \\psi_2:[0, \\infty) \\rightarrow \\mathbb{R}, f:[0, \\infty) \\rightarrow \\mathbb{R}$and$g:[0, \\infty) \\rightarrow \\mathbb{R}be functions such that\n\\\begin{aligned}\n& f(0)=\\mathrm{g}(0)=0, \\\\\n& \\psi_1(\\mathrm{x})=\\mathrm{e}^{-\\mathrm{x}}+\\mathrm{x}, \\quad \\mathrm{x} \\geq 0, \\\\\n& \\psi_2(\\mathrm{x})=\\mathrm{x}^2-2 \\mathrm{x}-2 \\mathrm{e}^{-\\mathrm{x}}+2, \\mathrm{x} \\geq 0, \\\\\n& f(\\mathrm{x})=\\int_{-\\mathrm{x}}^{\\mathrm{x}}\\left(|\\mathrm{t}|-\\mathrm{t}^2\\right) \\mathrm{e}^{-\\mathrm{t}^2} \\mathrm{dt}, \\mathrm{x}>0\n\\end{aligned}\\nand\n\$g(x)=\\int_0^{x^2} \\sqrt{t} e^{-t} d t, x>0\$. Which of the following statements is TRUE?\n\n(A)f(\\sqrt{\\ln 3})+g(\\sqrt{\\ln 3})=\\frac{1}{3}$\n\n(B) For every$x>1$, there exists an$\\alpha \\in(1, x)$such that$\\psi_{1}(x)=1+\\alpha x$\n\n(C) For every$x>0$, there exists a$\\beta \\in(0, x)$such that$\\psi_{2}(x)=2 x\\left(\\psi_{1}(\\beta)-1\\right)$\n\n(D)$f$is an increasing function on the interval$\\left[0, \\frac{3}{2}\\right]$", "gold": "C" }, { "description": "JEE Adv 2022 Paper 1", "index": 1, "subject": "math", "type": "Numeric", "question": "Considering only the principal values of the inverse trigonometric functions, what is the value of\n\n\$\n\\frac{3}{2} \\cos ^{-1} \\sqrt{\\frac{2}{2+\\pi^{2}}}+\\frac{1}{4} \\sin ^{-1} \\frac{2 \\sqrt{2} \\pi}{2+\\pi^{2}}+\\tan ^{-1} \\frac{\\sqrt{2}}{\\pi}\n\$?", "gold": "2.35" }, { "description": "JEE Adv 2022 Paper 1", "index": 2, "subject": "math", "type": "Numeric", "question": "Let$\\alpha$be a positive real number. Let$f: \\mathbb{R} \\rightarrow \\mathbb{R}$and$g:(\\alpha, \\infty) \\rightarrow \\mathbb{R}$be the functions defined by\n\n\$\nf(x)=\\sin \\left(\\frac{\\pi x}{12}\\right) \\quad \\text { and } \\quad g(x)=\\frac{2 \\log _{\\mathrm{e}}(\\sqrt{x}-\\sqrt{\\alpha})}{\\log _{\\mathrm{e}}\\left(e^{\\sqrt{x}}-e^{\\sqrt{\\alpha}}\\right)}\n\$\n\nThen what is the value of$\\lim _{x \\rightarrow \\alpha^{+}} f(g(x))$?", "gold": "0.5" }, { "description": "JEE Adv 2022 Paper 1", "index": 3, "subject": "math", "type": "Numeric", "question": "In a study about a pandemic, data of 900 persons was collected. It was found that\n\n190 persons had symptom of fever,\n\n220 persons had symptom of cough,\n\n220 persons had symptom of breathing problem,\n\n330 persons had symptom of fever or cough or both,\n\n350 persons had symptom of cough or breathing problem or both,\n\n340 persons had symptom of fever or breathing problem or both,\n\n30 persons had all three symptoms (fever, cough and breathing problem).\n\nIf a person is chosen randomly from these 900 persons, then what the probability that the person has at most one symptom?", "gold": "0.8" }, { "description": "JEE Adv 2022 Paper 1", "index": 4, "subject": "math", "type": "Numeric", "question": "Let$z$be a complex number with non-zero imaginary part. If\n\n\$\n\\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}\n\$\n\nis a real number, then the value of$|z|^{2}$is", "gold": "0.5" }, { "description": "JEE Adv 2022 Paper 1", "index": 5, "subject": "math", "type": "Numeric", "question": "Let$\\bar{z}$denote the complex conjugate of a complex number$z$and let$i=\\sqrt{-1}$. In the set of complex numbers,what is the number of distinct roots of the equation\n\n\$\n\\bar{z}-z^{2}=i\\left(\\bar{z}+z^{2}\\right)\n\$?", "gold": "4" }, { "description": "JEE Adv 2022 Paper 1", "index": 6, "subject": "math", "type": "Numeric", "question": "Let$l_{1}, l_{2}, \\ldots, l_{100}$be consecutive terms of an arithmetic progression with common difference$d_{1}$, and let$w_{1}, w_{2}, \\ldots, w_{100}$be consecutive terms of another arithmetic progression with common difference$d_{2}$, where$d_{1} d_{2}=10$. For each$i=1,2, \\ldots, 100$, let$R_{i}$be a rectangle with length$l_{i}$, width$w_{i}$and area$A_{i}$. If$A_{51}-A_{50}=1000$, then what is the value of$A_{100}-A_{90}$?", "gold": "18900" }, { "description": "JEE Adv 2022 Paper 1", "index": 7, "subject": "math", "type": "Numeric", "question": "What is the number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits$0,2,3,4,6,7$?", "gold": "569" }, { "description": "JEE Adv 2022 Paper 1", "index": 8, "subject": "math", "type": "Numeric", "question": "Let$A B C$be the triangle with$A B=1, A C=3$and$\\angle B A C=\\frac{\\pi}{2}$. If a circle of radius$r>0$touches the sides$A B, A C$and also touches internally the circumcircle of the triangle$A B C$, then what is the value of$r$?", "gold": "0.83" }, { "description": "JEE Adv 2022 Paper 1", "index": 9, "subject": "math", "type": "MCQ(multiple)", "question": "Consider the equation\n\n\$\n\\int_{1}^{e} \\frac{\\left(\\log _{\\mathrm{e}} x\\right)^{1 / 2}}{x\\left(a-\\left(\\log _{\\mathrm{e}} x\\right)^{3 / 2}\\right)^{2}} d x=1, \\quad a \\in(-\\infty, 0) \\cup(1, \\infty) .\n\$\n\nWhich of the following statements is/are TRUE?\n\n(A) No$a$satisfies the above equation\n\n(B) An integer$a$satisfies the above equation\n\n(C) An irrational number$a$satisfies the above equation\n\n(D) More than one$a$satisfy the above equation", "gold": "CD" }, { "description": "JEE Adv 2022 Paper 1", "index": 10, "subject": "math", "type": "MCQ(multiple)", "question": "Let$a_{1}, a_{2}, a_{3}, \\ldots$be an arithmetic progression with$a_{1}=7$and common difference 8 . Let$T_{1}, T_{2}, T_{3}, \\ldots$be such that$T_{1}=3$and$T_{n+1}-T_{n}=a_{n}$for$n \\geq 1$. Then, which of the following is/are TRUE ?\n\n(A)$T_{20}=1604$\n\n(B)$\\sum_{k=1}^{20} T_{k}=10510$\n\n(C)$T_{30}=3454$\n\n(D)$\\sum_{k=1}^{30} T_{k}=35610$", "gold": "BC" }, { "description": "JEE Adv 2022 Paper 1", "index": 11, "subject": "math", "type": "MCQ(multiple)", "question": "Let$P_{1}$and$P_{2}be two planes given by\n\n\\n\\begin{aligned}\n\n& P_{1}: 10 x+15 y+12 z-60=0, \\\\\n\n& P_{2}: \\quad-2 x+5 y+4 z-20=0 .\n\n\\end{aligned}\n\\n\nWhich of the following straight lines can be an edge of some tetrahedron whose two faces lie onP_{1}$and$P_{2}$?\n\n(A)$\\frac{x-1}{0}=\\frac{y-1}{0}=\\frac{z-1}{5}$\n\n(B)$\\frac{x-6}{-5}=\\frac{y}{2}=\\frac{z}{3}$\n\n(C)$\\frac{x}{-2}=\\frac{y-4}{5}=\\frac{z}{4}$\n\n(D)$\\frac{x}{1}=\\frac{y-4}{-2}=\\frac{z}{3}$", "gold": "ABD" }, { "description": "JEE Adv 2022 Paper 1", "index": 12, "subject": "math", "type": "MCQ(multiple)", "question": "Let$S$be the reflection of a point$Q$with respect to the plane given by\n\n\$\n\\vec{r}=-(t+p) \\hat{i}+t \\hat{j}+(1+p) \\hat{k}\n\$\n\nwhere$t, p$are real parameters and$\\hat{i}, \\hat{j}, \\hat{k}$are the unit vectors along the three positive coordinate axes. If the position vectors of$Q$and$S$are$10 \\hat{i}+15 \\hat{j}+20 \\hat{k}$and$\\alpha \\hat{i}+\\beta \\hat{j}+\\gamma \\hat{k}$respectively, then which of the following is/are TRUE ?\n\n(A)$3(\\alpha+\\beta)=-101$\n\n(B)$3(\\beta+\\gamma)=-71$\n\n(C)$3(\\gamma+\\alpha)=-86$\n\n(D)$3(\\alpha+\\beta+\\gamma)=-121$", "gold": "ABC" }, { "description": "JEE Adv 2022 Paper 1", "index": 14, "subject": "math", "type": "MCQ(multiple)", "question": "Let$|M|$denote the determinant of a square matrix$M$. Let$g:\\left[0, \\frac{\\pi}{2}\\right] \\rightarrow \\mathbb{R}$be the function defined by\n\nwhere\n\n\$\ng(\\theta)=\\sqrt{f(\\theta)-1}+\\sqrt{f\\left(\\frac{\\pi}{2}-\\theta\\right)-1}\n\$\n\n$f(\\theta)=\\frac{1}{2}\\left|\\begin{array}{ccc}1 & \\sin \\theta & 1 \\\\ -\\sin \\theta & 1 & \\sin \\theta \\\\ -1 & -\\sin \\theta & 1\\end{array}\\right|+\\left|\\begin{array}{ccc}\\sin \\pi & \\cos \\left(\\theta+\\frac{\\pi}{4}\\right) & \\tan \\left(\\theta-\\frac{\\pi}{4}\\right) \\\\ \\sin \\left(\\theta-\\frac{\\pi}{4}\\right) & -\\cos \\frac{\\pi}{2} & \\log _{e}\\left(\\frac{4}{\\pi}\\right) \\\\ \\cot \\left(\\theta+\\frac{\\pi}{4}\\right) & \\log _{e}\\left(\\frac{\\pi}{4}\\right) & \\tan \\pi\\end{array}\\right|$\n\nLet$p(x)$be a quadratic polynomial whose roots are the maximum and minimum values of the function$g(\\theta)$, and$p(2)=2-\\sqrt{2}$. Then, which of the following is/are TRUE?\n\n(A)$p\\left(\\frac{3+\\sqrt{2}}{4}\\right)<0$\n\n(B)$p\\left(\\frac{1+3 \\sqrt{2}}{4}\\right)>0$\n\n(C)$p\\left(\\frac{5 \\sqrt{2}-1}{4}\\right)>0$\n\n(D)$p\\left(\\frac{5-\\sqrt{2}}{4}\\right)<0$", "gold": "AC" }, { "description": "JEE Adv 2022 Paper 1", "index": 15, "subject": "math", "type": "MCQ", "question": "Consider the following lists:\nList-I\n(I)$\\left\\{x \\in\\left[-\\frac{2 \\pi}{3}, \\frac{2 \\pi}{3}\\right]: \\cos x+\\sin x=1\\right\\}$\n(II)$\\left\\{x \\in\\left[-\\frac{5 \\pi}{18}, \\frac{5 \\pi}{18}\\right]: \\sqrt{3} \\tan 3 x=1\\right\\}$\n(III)$\\left\\{x \\in\\left[-\\frac{6 \\pi}{5}, \\frac{6 \\pi}{5}\\right]: 2 \\cos (2 x)=\\sqrt{3}\\right\\}$\n(IV)$\\left\\{x \\in\\left[-\\frac{7 \\pi}{4}, \\frac{7 \\pi}{4}\\right]: \\sin x-\\cos x=1\\right\\}$\n\nList-II\n(P) has two elements\n(Q) has three elements\n(R) has four elements\n(S) has five elements\n(T) has six elements\n\nThe correct option is:\n\n(A) (I)$\\rightarrow$(P); (II)$\\rightarrow$(S); (III)$\\rightarrow$(P); (IV)$\\rightarrow$(S)\n\n(B) (I)$\\rightarrow$(P); (II)$\\rightarrow$(P); (III)$\\rightarrow$(T); (IV)$\\rightarrow$(R)\n\n(C) (I)$\\rightarrow$(Q); (II)$\\rightarrow$(P); (III)$\\rightarrow$(T); (IV)$\\rightarrow$(S)\n\n(D) (I)$\\rightarrow$(Q); (II)$\\rightarrow$(S); (III)$\\rightarrow$(P); (IV)$\\rightarrow$(R)", "gold": "B" }, { "description": "JEE Adv 2022 Paper 1", "index": 16, "subject": "math", "type": "MCQ", "question": "Two players,$P_{1}$and$P_{2}$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let$x$and$y$denote the readings on the die rolled by$P_{1}$and$P_{2}$, respectively. If$x>y$, then$P_{1}$scores 5 points and$P_{2}$scores 0 point. If$x=y$, then each player scores 2 points. If$xY_2\\right)$is\n(III) Probability of$\\left(X_3=Y_3\\right)$is\n(IV) Probability of$\\left(X_3>Y_3\\right)$is\n\nList-II\n(P)$\\frac{3}{8}$\n(Q)$\\frac{11}{16}$\n(R)$\\frac{5}{16}$\n(S)$\\frac{355}{864}$\n(T)$\\frac{77}{432}$\n\nThe correct option is:\n\n(A) (I)$\\rightarrow$(Q); (II)$\\rightarrow$(R); (III)$\\rightarrow$(T); (IV)$\\rightarrow$(S)\n\n(B) (I)$\\rightarrow$(Q); (II)$\\rightarrow$(R); (III)$\\rightarrow$(T); (IV)$\\rightarrow$(T)\n\n(C) (I)$\\rightarrow$(P); (II)$\\rightarrow$(R); (III)$\\rightarrow$(Q); (IV)$\\rightarrow$(S)\n\n(D) (I)$\\rightarrow$(P); (II)$\\rightarrow$(R); (III)$\\rightarrow$(Q); (IV)$\\rightarrow$(T)", "gold": "A" }, { "description": "JEE Adv 2022 Paper 1", "index": 17, "subject": "math", "type": "MCQ", "question": "Let$p, q, r$be nonzero real numbers that are, respectively, the$10^{\\text {th }}, 100^{\\text {th }}$and$1000^{\\text {th }}$terms of a harmonic progression. Consider the system of linear equations\n\n\$\n\\begin{gathered}\n\nx+y+z=1 \\\\\n\n10 x+100 y+1000 z=0 \\\\\n\nq r x+p r y+p q z=0\n\n\\end{gathered}\n\$\n\nList-I\n(I) If$\\frac{q}{r}=10$, then the system of linear equations has\n(II) If$\\frac{p}{r} \\neq 100$, then the system of linear equations has\n(III) If$\\frac{p}{q} \\neq 10$, then the system of linear equations has\n(IV) If$\\frac{p}{q}=10$, then the system of linear equations has\n\nList-II\n(P)$x=0, \\quad y=\\frac{10}{9}, z=-\\frac{1}{9}$as a solution\n(Q)$x=\\frac{10}{9}, y=-\\frac{1}{9}, z=0$as a solution\n(III) If$\\frac{p}{q} \\neq 10$, then the system of linear\n(R) infinitely many solutions\n(IV) If$\\frac{p}{q}=10$, then the system of linear\n(S) no solution\n(T) at least one solution\n\nThe correct option is:\n\n(A) (I)$\\rightarrow$(T); (II)$\\rightarrow$(R); (III)$\\rightarrow$(S); (IV)$\\rightarrow$(T)\n\n(B) (I)$\\rightarrow$(Q); (II)$\\rightarrow$(S); (III)$\\rightarrow$(S); (IV)$\\rightarrow$(R)\n\n(C) (I)$\\rightarrow$(Q); (II)$\\rightarrow$(R); (III)$\\rightarrow$(P); (IV)$\\rightarrow$(R)\n\n(D) (I)$\\rightarrow$(T); (II)$\\rightarrow$(S); (III)$\\rightarrow$(P); (IV)$\\rightarrow$(T)", "gold": "B" }, { "description": "JEE Adv 2022 Paper 1", "index": 18, "subject": "math", "type": "MCQ", "question": "Consider the ellipse\n\n\$\n\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\n\$\n\nLet$H(\\alpha, 0), 0<\\alpha<2$, be a point. A straight line drawn through$H$parallel to the$y$-axis crosses the ellipse and its auxiliary circle at points$E$and$F$respectively, in the first quadrant. The tangent to the ellipse at the point$E$intersects the positive$x$-axis at a point$G$. Suppose the straight line joining$F$and the origin makes an angle$\\phi$with the positive$x$-axis.\n\nList-I\n(I) If$\\phi=\\frac{\\pi}{4}$, then the area of the triangle$F G H$is\n(II) If$\\phi=\\frac{\\pi}{3}$, then the area of the triangle$F G H$is\n(III) If$\\phi=\\frac{\\pi}{6}$, then the area of the triangle$F G H$is\n(IV) If$\\phi=\\frac{\\pi}{12}$, then the area of the triangle$F G H$is\n\nList-II\n(P)$\\frac{(\\sqrt{3}-1)^4}{8}$\n(Q) 1\n(R)$\\frac{3}{4}$\n(S)$\\frac{1}{2 \\sqrt{3}}$\n(T)$\\frac{3 \\sqrt{3}}{2}$\n\nThe correct option is:\n\n(A) (I)$\\rightarrow$(R); (II)$\\rightarrow$(S); (III)$\\rightarrow$(Q); (IV)$\\rightarrow$(P)\n\n(B)$\\quad$(I)$\\rightarrow$(R); (II)$\\rightarrow$(T); (III)$\\rightarrow$(S); (IV)$\\rightarrow$(P)\n\n(C)$\\quad$(I)$\\rightarrow$(Q); (II)$\\rightarrow$(T); (III)$\\rightarrow$(S); (IV)$\\rightarrow$(P)\n\n(D)$\\quad$(I)$\\rightarrow$(Q); (II)$\\rightarrow$(S); (III)$\\rightarrow$(Q); (IV)$\\rightarrow$(P)", "gold": "C" }, { "description": "JEE Adv 2022 Paper 1", "index": 19, "subject": "phy", "type": "Numeric", "question": "Two spherical stars$A$and$B$have densities$\\rho_{A}$and$\\rho_{B}$, respectively.$A$and$B$have the same radius, and their masses$M_{A}$and$M_{B}$are related by$M_{B}=2 M_{A}$. Due to an interaction process, star$A$loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains$\\rho_{A}$. The entire mass lost by$A$is deposited as a thick spherical shell on$B$with the density of the shell being$\\rho_{A}$. If$v_{A}$and$v_{B}$are the escape velocities from$A$and$B$after the interaction process, the ratio$\\frac{v_{B}}{v_{A}}=\\sqrt{\\frac{10 n}{15^{1 / 3}}}$. What is the value of$n$?", "gold": "2.3" }, { "description": "JEE Adv 2022 Paper 1", "index": 20, "subject": "phy", "type": "Numeric", "question": "The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction${ }_{7}{ }_{7} \\mathrm{~N}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{1}^{1} \\mathrm{H}+{ }_{8}^{19} \\mathrm{O}$in a laboratory frame is$n$(in$M e V$). Assume that${ }_{7}^{16} \\mathrm{~N}$is at rest in the laboratory frame. The masses of${ }_{7}^{16} \\mathrm{~N},{ }_{2}^{4} \\mathrm{He},{ }_{1}^{1} \\mathrm{H}$and${ }_{8}^{19} \\mathrm{O}$can be taken to be$16.006 u, 4.003 u, 1.008 u$and 19.003$u$, respectively, where$1 u=930 \\mathrm{MeV}^{-2}$. What is the value of$n$?", "gold": "2.32" }, { "description": "JEE Adv 2022 Paper 1", "index": 23, "subject": "phy", "type": "Numeric", "question": "At time$t=0$, a disk of radius$1 \\mathrm{~m}$starts to roll without slipping on a horizontal plane with an angular acceleration of$\\alpha=\\frac{2}{3} \\mathrm{rads}^{-2}$. A small stone is stuck to the disk. At$t=0$, it is at the contact point of the disk and the plane. Later, at time$t=\\sqrt{\\pi} s$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in$m$) reached by the stone measured from the plane is$\\frac{1}{2}+\\frac{x}{10}$. What is the value of$x$?$\\left[\\right.$Take$\\left.g=10 m s^{-2}.\\right]$", "gold": "0.52" }, { "description": "JEE Adv 2022 Paper 1", "index": 25, "subject": "phy", "type": "Numeric", "question": "Consider an LC circuit, with inductance$L=0.1 \\mathrm{H}$and capacitance$C=10^{-3} \\mathrm{~F}$, kept on a plane. The area of the circuit is$1 \\mathrm{~m}^{2}$. It is placed in a constant magnetic field of strength$B_{0}$which is perpendicular to the plane of the circuit. At time$t=0$, the magnetic field strength starts increasing linearly as$B=B_{0}+\\beta t$with$\\beta=0.04 \\mathrm{Ts}^{-1}$. What is the maximum magnitude of the current in the circuit in$m A$?", "gold": "4" }, { "description": "JEE Adv 2022 Paper 1", "index": 26, "subject": "phy", "type": "Numeric", "question": "A projectile is fired from horizontal ground with speed$v$and projection angle$\\theta$. When the acceleration due to gravity is$g$, the range of the projectile is$d$. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is$g^{\\prime}=\\frac{g}{0.81}$, then the new range is$d^{\\prime}=n d$. What is the value of$n$?", "gold": "0.95" }, { "description": "JEE Adv 2022 Paper 1", "index": 32, "subject": "phy", "type": "MCQ(multiple)", "question": "The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be$E_{b}^{p}$and the binding energy of a neutron be$E_{b}^{n}$in the nucleus.\n\nWhich of the following statement(s) is(are) correct?\n\n(A)$E_{b}^{p}-E_{b}^{n}$is proportional to$Z(Z-1)$where$Z$is the atomic number of the nucleus.\n\n(B)$E_{b}^{p}-E_{b}^{n}$is proportional to$A^{-\\frac{1}{3}}$where$A$is the mass number of the nucleus.\n\n(C)$E_{b}^{p}-E_{b}^{n}$is positive.\n\n(D)$E_{b}^{p}$increases if the nucleus undergoes a beta decay emitting a positron.", "gold": "ABD" }, { "description": "JEE Adv 2022 Paper 1", "index": 35, "subject": "phy", "type": "MCQ", "question": "List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.\nList-I\n(I)$10^{-3} \\mathrm{~kg}$of water at$100^{\\circ} \\mathrm{C}$is converted to steam at the same temperature, at a pressure of$10^5 \\mathrm{~Pa}$. The volume of the system changes from$10^{-6} \\mathrm{~m}^3$to$10^{-3} \\mathrm{~m}^3$in the process. Latent heat of water$=2250 \\mathrm{~kJ} / \\mathrm{kg}$.\n(II) 0.2 moles of a rigid diatomic ideal gas with volume$V$at temperature$500 \\mathrm{~K}$undergoes an isobaric expansion to volume$3 \\mathrm{~V}$. Assume$R=8.0 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$.\n(III) One mole of a monatomic ideal gas is compressed adiabatically from volume$V=\\frac{1}{3} m^3$and pressure$2 \\mathrm{kPa}$to volume$\\frac{V}{8}$.\n(IV) Three moles of a diatomic ideal gas whose molecules can vibrate, is given$9 \\mathrm{~kJ}$of heat and undergoes isobaric expansion.\n\nList-II\n(P)$2 \\mathrm{~kJ}$\n(Q)$7 \\mathrm{~kJ}$\n(R)$4 \\mathrm{~kJ}$\n(S)$5 \\mathrm{~kJ}$\n(T)$3 \\mathrm{~kJ}$\n\nWhich one of the following options is correct?\\\n(A)$\\mathrm{I} \\rightarrow \\mathrm{T}$, II$\\rightarrow$R, III$\\rightarrow \\mathrm{S}$, IV$\\rightarrow$Q,\n\n(B) I$\\rightarrow \\mathrm{S}$, II$\\rightarrow$P, III$\\rightarrow \\mathrm{T}$, IV$\\rightarrow$P,\n\n(C) I$\\rightarrow$P, II$\\rightarrow$R, III$\\rightarrow$T, IV$\\rightarrow$Q.\n\n(D) I$\\rightarrow$Q, II$\\rightarrow$R, III$\\rightarrow \\mathrm{S}$, IV$\\rightarrow \\mathrm{T}$,", "gold": "C" }, { "description": "JEE Adv 2022 Paper 1", "index": 37, "subject": "chem", "type": "Numeric", "question": "2 mol of$\\mathrm{Hg}(g)$is combusted in a fixed volume bomb calorimeter with excess of$\\mathrm{O}_{2}$at$298 \\mathrm{~K}$and 1 atm into$\\mathrm{HgO}(s)$. During the reaction, temperature increases from$298.0 \\mathrm{~K}$to$312.8 \\mathrm{~K}$. If heat capacity of the bomb calorimeter and enthalpy of formation of$\\mathrm{Hg}(g)$are$20.00 \\mathrm{~kJ} \\mathrm{~K}^{-1}$and$61.32 \\mathrm{~kJ}\\mathrm{mol}^{-1}$at$298 \\mathrm{~K}$, respectively, the calculated standard molar enthalpy of formation of$\\mathrm{HgO}(s)$at 298$\\mathrm{K}$is$\\mathrm{X} \\mathrm{kJ} \\mathrm{mol} \\mathrm{m}^{-1}$. What is the value of$|\\mathrm{X}|$?\n\n[Given: Gas constant$\\mathrm{R}=8.3 \\mathrm{~J} \\mathrm{~K}^{-1} \\mathrm{~mol}^{-1}$]", "gold": "90.39" }, { "description": "JEE Adv 2022 Paper 1", "index": 38, "subject": "chem", "type": "Numeric", "question": "What is the reduction potential$\\left(E^{0}\\right.$, in$\\left.\\mathrm{V}\\right)$of$\\mathrm{MnO}_{4}^{-}(\\mathrm{aq}) / \\mathrm{Mn}(\\mathrm{s})$?\n\n[Given:$\\left.E_{\\left(\\mathrm{MnO}_{4}^{-}(\\mathrm{aq}) / \\mathrm{MnO}_{2}(\\mathrm{~s})\\right)}^{0}=1.68 \\mathrm{~V} ; E_{\\left(\\mathrm{MnO}_{2}(\\mathrm{~s}) / \\mathrm{Mn}^{2+}(\\mathrm{aq})\\right)}^{0}=1.21 \\mathrm{~V} ; E_{\\left(\\mathrm{Mn}^{2+}(\\mathrm{aq}) / \\mathrm{Mn}(\\mathrm{s})\\right)}^{0}=-1.03 \\mathrm{~V}\\right]$", "gold": "0.77" }, { "description": "JEE Adv 2022 Paper 1", "index": 39, "subject": "chem", "type": "Numeric", "question": "A solution is prepared by mixing$0.01 \\mathrm{~mol}$each of$\\mathrm{H}_{2} \\mathrm{CO}_{3}, \\mathrm{NaHCO}_{3}, \\mathrm{Na}_{2} \\mathrm{CO}_{3}$, and$\\mathrm{NaOH}$in$100 \\mathrm{~mL}$of water. What is the$p \\mathrm{H}$of the resulting solution?\n\n[Given:$p \\mathrm{~K}_{\\mathrm{a} 1}$and$p \\mathrm{~K}_{\\mathrm{a} 2}$of$\\mathrm{H}_{2} \\mathrm{CO}_{3}$are 6.37 and 10.32, respectively;$\\log 2=0.30$]", "gold": "10.02" }, { "description": "JEE Adv 2022 Paper 1", "index": 40, "subject": "chem", "type": "Numeric", "question": "The treatment of an aqueous solution of$3.74 \\mathrm{~g}$of$\\mathrm{Cu}\\left(\\mathrm{NO}_{3}\\right)_{2}$with excess KI results in a brown solution along with the formation of a precipitate. Passing$\\mathrm{H}_{2} \\mathrm{~S}$through this brown solution gives another precipitate$\\mathbf{X}$. What is the amount of$\\mathbf{X}$(in$\\mathrm{g}$)?\n\n[Given: Atomic mass of$\\mathrm{H}=1, \\mathrm{~N}=14, \\mathrm{O}=16, \\mathrm{~S}=32, \\mathrm{~K}=39, \\mathrm{Cu}=63, \\mathrm{I}=127$]", "gold": "0.32" }, { "description": "JEE Adv 2022 Paper 1", "index": 41, "subject": "chem", "type": "Numeric", "question": "Dissolving$1.24 \\mathrm{~g}$of white phosphorous in boiling NaOH solution in an inert atmosphere gives a gas$\\mathbf{Q}$. What is the amount of$\\mathrm{CuSO}_{4}$(in g) required to completely consume the gas$\\mathbf{Q}$?\n\n[Given: Atomic mass of$\\mathrm{H}=1, \\mathrm{O}=16, \\mathrm{Na}=23, \\mathrm{P}=31, \\mathrm{~S}=32, \\mathrm{Cu}=63$]", "gold": "2.38" }, { "description": "JEE Adv 2022 Paper 1", "index": 45, "subject": "chem", "type": "MCQ(multiple)", "question": "For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap of two$2 p_{z}$orbitals is(are)\n\n(A)$\\sigma$orbital has a total of two nodal planes.\n\n(B)$\\sigma^{*}$orbital has one node in the$x z$-plane containing the molecular axis.\n\n(C)$\\pi$orbital has one node in the plane which is perpendicular to the molecular axis and goes through the center of the molecule.\n\n(D)$\\pi^{*}$orbital has one node in the$x y$-plane containing the molecular axis.", "gold": "AD" }, { "description": "JEE Adv 2022 Paper 1", "index": 46, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct option(s) related to adsorption processes is(are)\n\n(A) Chemisorption results in a unimolecular layer.\n\n(B) The enthalpy change during physisorption is in the range of 100 to$140 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$.\n\n(C) Chemisorption is an endothermic process.\n\n(D) Lowering the temperature favors physisorption processes.", "gold": "AD" }, { "description": "JEE Adv 2022 Paper 1", "index": 47, "subject": "chem", "type": "MCQ(multiple)", "question": "The electrochemical extraction of aluminum from bauxite ore involves\n\n(A) the reaction of$\\mathrm{Al}_{2} \\mathrm{O}_{3}$with coke (C) at a temperature$>2500^{\\circ} \\mathrm{C}$.\n\n(B) the neutralization of aluminate solution by passing$\\mathrm{CO}_{2}$gas to precipitate hydrated alumina$\\left(\\mathrm{Al}_{2} \\mathrm{O}_{3} \\cdot 3 \\mathrm{H}_{2} \\mathrm{O}\\right)$.\n\n(C) the dissolution of$\\mathrm{Al}_{2} \\mathrm{O}_{3}$in hot aqueous$\\mathrm{NaOH}$.\n\n(D) the electrolysis of$\\mathrm{Al}_{2} \\mathrm{O}_{3}$mixed with$\\mathrm{Na}_{3} \\mathrm{AlF}_{6}$to give$\\mathrm{Al}$and$\\mathrm{CO}_{2}$.", "gold": "BCD" }, { "description": "JEE Adv 2022 Paper 1", "index": 48, "subject": "chem", "type": "MCQ(multiple)", "question": "The treatment of galena with$\\mathrm{HNO}_{3}$produces a gas that is\n\n(A) paramagnetic\n\n(B) bent in geometry\n\n(C) an acidic oxide\n\n(D) colorless", "gold": "AD" }, { "description": "JEE Adv 2022 Paper 1", "index": 52, "subject": "chem", "type": "MCQ", "question": "LIST-I contains compounds and LIST-II contains reactions\n\n\nLIST-I\n(I)$\\mathrm{H}_2 \\mathrm{O}_2$\n(II)$\\mathrm{Mg}(\\mathrm{OH})_2$\n(III)$\\mathrm{BaCl}_2$\n(IV)$\\mathrm{CaCO}_3$\n\nLIST-II\n(P)$\\mathrm{Mg}\\left(\\mathrm{HCO}_{3}\\right)_{2}+\\mathrm{Ca}(\\mathrm{OH})_{2} \\rightarrow$\n(Q)$\\mathrm{BaO}_{2}+\\mathrm{H}_{2} \\mathrm{SO}_{4} \\rightarrow$\n(R)$\\mathrm{Ca}(\\mathrm{OH})_{2}+\\mathrm{MgCl}_{2} \\rightarrow$\n(S)$\\mathrm{BaO}_{2}+\\mathrm{HCl} \\rightarrow$\n(T)$\\mathrm{Ca}\\left(\\mathrm{HCO}_{3}\\right)_{2}+\\mathrm{Ca}(\\mathrm{OH})_{2} \\rightarrow$\n\nMatch each compound in LIST-I with its formation reaction(s) in LIST-II, and choose the correct option\\\n(A) I$\\rightarrow$Q; II$\\rightarrow$P; III$\\rightarrow \\mathrm{S}$; IV$\\rightarrow$R\n\n(B) I$\\rightarrow$T; II$\\rightarrow$P; III$\\rightarrow$Q; IV$\\rightarrow \\mathrm{R}$\n\n(C) I$\\rightarrow$T; II$\\rightarrow$R; III$\\rightarrow$Q; IV$\\rightarrow$P\n\n(D) I$\\rightarrow$Q; II$\\rightarrow$R; III$\\rightarrow \\mathrm{S}$; IV$\\rightarrow \\mathrm{P}$", "gold": "D" }, { "description": "JEE Adv 2022 Paper 1", "index": 53, "subject": "chem", "type": "MCQ", "question": "LIST-I contains metal species and LIST-II contains their properties.\n\nLIST-I\n(I)$\\left[\\mathrm{Cr}(\\mathrm{CN})_{6}\\right]^{4-}$\n(II)$\\left[\\mathrm{RuCl}_{6}\\right]^{2-}$\n(III)$\\left[\\mathrm{Cr}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)_{6}\\right]^{2+}$\n(IV)$\\left[\\mathrm{Fe}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)_{6}\\right]^{2+}$\n\nLIST-II\n(P)$t_{2 \\mathrm{~g}}$orbitals contain 4 electrons\n(Q)$\\mu$(spin-only$=4.9 \\mathrm{BM}$\n(R) low spin complex ion\n(S) metal ion in$4+$oxidation state\n(T)$d^{4}$species\n\n[Given: Atomic number of$\\mathrm{Cr}=24, \\mathrm{Ru}=44, \\mathrm{Fe}=26$]\n\nMatch each metal species in LIST-I with their properties in LIST-II, and choose the correct option\n\n(A) I$\\rightarrow$R, T; II$\\rightarrow$P, S; III$\\rightarrow$Q, T; IV$\\rightarrow$P, Q\n\n(B) I$\\rightarrow$R, S; II$\\rightarrow$P, T; III$\\rightarrow$P, Q; IV$\\rightarrow$Q, T\n\n(C) I$\\rightarrow \\mathrm{P}, \\mathrm{R} ; \\mathrm{II} \\rightarrow \\mathrm{R}, \\mathrm{S}$; III$\\rightarrow \\mathrm{R}, \\mathrm{T}$; IV$\\rightarrow \\mathrm{P}, \\mathrm{T}$\n\n(D) I$\\rightarrow$Q, T; II$\\rightarrow \\mathrm{S}, \\mathrm{T}$; III$\\rightarrow \\mathrm{P}, \\mathrm{T}$; IV$\\rightarrow \\mathrm{Q}, \\mathrm{R}$", "gold": "A" }, { "description": "JEE Adv 2022 Paper 1", "index": 54, "subject": "chem", "type": "MCQ", "question": "Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option.\n\nLIST-I\n\n(I) Aniline\n(II)$o$-Cresol\n(III) Cysteine\n(IV) Caprolactam\n\nLIST-II\n\n(P) Sodium fusion extract of the compound on boiling with$\\mathrm{FeSO}_{4}$, followed by acidification with conc.$\\mathrm{H}_{2} \\mathrm{SO}_{4}$, gives Prussian blue color.\n(Q) Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.\n(R) Addition of the compound to a saturated solution of$\\mathrm{NaHCO}_{3}$results in effervescence.\n(S) The compound reacts with bromine water to give a white precipitate.\n(T) Treating the compound with neutral$\\mathrm{FeCl}_{3}$solution produces violet color.\n\n(A)$\\mathrm{I} \\rightarrow \\mathrm{P}, \\mathrm{Q}$; II$\\rightarrow \\mathrm{S}$; III$\\rightarrow \\mathrm{Q}$, R; IV$\\rightarrow \\mathrm{P}$\n\n(B)$\\mathrm{I} \\rightarrow \\mathrm{P}$; II$\\rightarrow \\mathrm{R}, \\mathrm{S}$; III$\\rightarrow \\mathrm{R}$; IV$\\rightarrow \\mathrm{Q}, \\mathrm{S}$\n\n(C)$\\mathrm{I} \\rightarrow \\mathrm{Q}, \\mathrm{S}$; II$\\rightarrow \\mathrm{P}, \\mathrm{T}$; III$\\rightarrow \\mathrm{P}$; IV$\\rightarrow \\mathrm{S}$\n\n(D)$\\mathrm{I} \\rightarrow \\mathrm{P}, \\mathrm{S}$; II$\\rightarrow \\mathrm{T}$; III$\\rightarrow \\mathrm{Q}, \\mathrm{R}$; IV$\\rightarrow \\mathrm{P}$", "gold": "D" }, { "description": "JEE Adv 2022 Paper 2", "index": 1, "subject": "math", "type": "Integer", "question": "Let$\\alpha$and$\\beta$be real numbers such that$-\\frac{\\pi}{4}<\\beta<0<\\alpha<\\frac{\\pi}{4}$. If$\\sin (\\alpha+\\beta)=\\frac{1}{3}$and$\\cos (\\alpha-\\beta)=\\frac{2}{3}$, then what is the greatest integer less than or equal to\n\n\$\n\\left(\\frac{\\sin \\alpha}{\\cos \\beta}+\\frac{\\cos \\beta}{\\sin \\alpha}+\\frac{\\cos \\alpha}{\\sin \\beta}+\\frac{\\sin \\beta}{\\cos \\alpha}\\right)^{2}\n\$?", "gold": "1" }, { "description": "JEE Adv 2022 Paper 2", "index": 2, "subject": "math", "type": "Integer", "question": "If$y(x)$is the solution of the differential equation\n\n\$\nx d y-\\left(y^{2}-4 y\\right) d x=0 \\text { for } x>0, \\quad y(1)=2,\n\$\n\nand the slope of the curve$y=y(x)$is never zero, then what is the value of$10 y(\\sqrt{2})$?", "gold": "8" }, { "description": "JEE Adv 2022 Paper 2", "index": 3, "subject": "math", "type": "Numeric", "question": "What is the greatest integer less than or equal to\n\n\$\n\\int_{1}^{2} \\log _{2}\\left(x^{3}+1\\right) d x+\\int_{1}^{\\log _{2} 9}\\left(2^{x}-1\\right)^{\\frac{1}{3}} d x\n\$?", "gold": "5" }, { "description": "JEE Adv 2022 Paper 2", "index": 4, "subject": "math", "type": "Integer", "question": "What is the product of all positive real values of$x$satisfying the equation\n\n\$\nx^{\\left(16\\left(\\log _{5} x\\right)^{3}-68 \\log _{5} x\\right)}=5^{-16}\n\$?", "gold": "1" }, { "description": "JEE Adv 2022 Paper 2", "index": 5, "subject": "math", "type": "Integer", "question": "If\n\n\$\n\\beta=\\lim _{x \\rightarrow 0} \\frac{e^{x^{3}}-\\left(1-x^{3}\\right)^{\\frac{1}{3}}+\\left(\\left(1-x^{2}\\right)^{\\frac{1}{2}}-1\\right) \\sin x}{x \\sin ^{2} x},\n\$\n\nthen what is the value of$6 \\beta$?", "gold": "5" }, { "description": "JEE Adv 2022 Paper 2", "index": 6, "subject": "math", "type": "Integer", "question": "Let$\\beta$be a real number. Consider the matrix\n\n\$\nA=\\left(\\begin{array}{ccc}\n\n\\beta & 0 & 1 \\\\\n\n2 & 1 & -2 \\\\\n\n3 & 1 & -2\n\n\\end{array}\\right)\n\$\n\nIf$A^{7}-(\\beta-1) A^{6}-\\beta A^{5}$is a singular matrix, then what is the value of$9 \\beta$?", "gold": "3" }, { "description": "JEE Adv 2022 Paper 2", "index": 7, "subject": "math", "type": "Integer", "question": "Consider the hyperbola\n\n\$\n\\frac{x^{2}}{100}-\\frac{y^{2}}{64}=1\n\$\n\nwith foci at$S$and$S_{1}$, where$S$lies on the positive$x$-axis. Let$P$be a point on the hyperbola, in the first quadrant. Let$\\angle S P S_{1}=\\alpha$, with$\\alpha<\\frac{\\pi}{2}$. The straight line passing through the point$S$and having the same slope as that of the tangent at$P$to the hyperbola, intersects the straight line$S_{1} P$at$P_{1}$. Let$\\delta$be the distance of$P$from the straight line$S P_{1}$, and$\\beta=S_{1} P$. Then what is the greatest integer less than or equal to$\\frac{\\beta \\delta}{9} \\sin \\frac{\\alpha}{2}$?", "gold": "7" }, { "description": "JEE Adv 2022 Paper 2", "index": 8, "subject": "math", "type": "Integer", "question": "Consider the functions$f, g: \\mathbb{R} \\rightarrow \\mathbb{R}$defined by\n\n\$\nf(x)=x^{2}+\\frac{5}{12} \\quad \\text { and } \\quad g(x)= \\begin{cases}2\\left(1-\\frac{4|x|}{3}\\right), & |x| \\leq \\frac{3}{4} \\\\ 0, & |x|>\\frac{3}{4}\\end{cases}\n\$\n\nIf$\\alpha$is the area of the region\n\n\$\n\\left\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R}:|x| \\leq \\frac{3}{4}, 0 \\leq y \\leq \\min \\{f(x), g(x)\\}\\right\\}\n\$\n\nthen what is the value of$9 \\alpha$?", "gold": "6" }, { "description": "JEE Adv 2022 Paper 2", "index": 9, "subject": "math", "type": "MCQ(multiple)", "question": "Let$P Q R S$be a quadrilateral in a plane, where$Q R=1, \\angle P Q R=\\angle Q R S=70^{\\circ}, \\angle P Q S=15^{\\circ}$and$\\angle P R S=40^{\\circ}$. If$\\angle R P S=\\theta^{\\circ}, P Q=\\alpha$and$P S=\\beta$, then the interval(s) that contain(s) the value of$4 \\alpha \\beta \\sin \\theta^{\\circ}$is/are\n\n(A)$(0, \\sqrt{2})$\n\n(B)$(1,2)$\n\n(C)$(\\sqrt{2}, 3)$\n\n(D)$(2 \\sqrt{2}, 3 \\sqrt{2})$", "gold": "AB" }, { "description": "JEE Adv 2022 Paper 2", "index": 10, "subject": "math", "type": "MCQ(multiple)", "question": "Let\n\n\$\n\\alpha=\\sum_{k=1}^{\\infty} \\sin ^{2 k}\\left(\\frac{\\pi}{6}\\right)\n\$\n\nLet$g:[0,1] \\rightarrow \\mathbb{R}$be the function defined by\n\n\$\ng(x)=2^{\\alpha x}+2^{\\alpha(1-x)}\n\$\n\nThen, which of the following statements is/are TRUE ?\n\n(A) The minimum value of$g(x)$is$2^{\\frac{7}{6}}$\n\n(B) The maximum value of$g(x)$is$1+2^{\\frac{1}{3}}$\n\n(C) The function$g(x)$attains its maximum at more than one point\n\n(D) The function$g(x)$attains its minimum at more than one point", "gold": "ABC" }, { "description": "JEE Adv 2022 Paper 2", "index": 11, "subject": "math", "type": "MCQ(multiple)", "question": "Let$\\bar{z}$denote the complex conjugate of a complex number$z$. If$z$is a non-zero complex number for which both real and imaginary parts of\n\n\$\n(\\bar{z})^{2}+\\frac{1}{z^{2}}\n\$\n\nare integers, then which of the following is/are possible value(s) of$|z|$?\n\n(A)$\\left(\\frac{43+3 \\sqrt{205}}{2}\\right)^{\\frac{1}{4}}$\n\n(B)$\\left(\\frac{7+\\sqrt{33}}{4}\\right)^{\\frac{1}{4}}$\n\n(C)$\\left(\\frac{9+\\sqrt{65}}{4}\\right)^{\\frac{1}{4}}$\n\n(D)$\\left(\\frac{7+\\sqrt{13}}{6}\\right)^{\\frac{1}{4}}$", "gold": "A" }, { "description": "JEE Adv 2022 Paper 2", "index": 12, "subject": "math", "type": "MCQ(multiple)", "question": "Let$G$be a circle of radius$R>0$. Let$G_{1}, G_{2}, \\ldots, G_{n}$be$n$circles of equal radius$r>0$. Suppose each of the$n$circles$G_{1}, G_{2}, \\ldots, G_{n}$touches the circle$G$externally. Also, for$i=1,2, \\ldots, n-1$, the circle$G_{i}$touches$G_{i+1}$externally, and$G_{n}$touches$G_{1}$externally. Then, which of the following statements is/are TRUE?\n\n(A) If$n=4$, then$(\\sqrt{2}-1) rR$", "gold": "CD" }, { "description": "JEE Adv 2022 Paper 2", "index": 13, "subject": "math", "type": "MCQ(multiple)", "question": "Let$\\hat{i}, \\hat{j}$and$\\hat{k}$be the unit vectors along the three positive coordinate axes. Let\n\n\$\n\\vec{a}=3 \\hat{i}+\\hat{j}-\\hat{k},\n\$\n\$\\vec{b}=\\hat{i}+b_{2} \\hat{j}+b_{3} \\hat{k}, \\quad b_{2}, b_{3} \\in \\mathbb{R},\n\$\n\$\\vec{c}=c_{1} \\hat{i}+c_{2} \\hat{j}+c_{3} \\hat{k}, \\quad c_{1}, c_{2}, c_{3} \\in \\mathbb{R}\n\$\n\nbe three vectors such that$b_{2} b_{3}>0, \\vec{a} \\cdot \\vec{b}=0$and\n\n\$\n\\left(\\begin{array}{ccc}\n\n0 & -c_{3} & c_{2} \\\\\n\nc_{3} & 0 & -c_{1} \\\\\n\n-c_{2} & c_{1} & 0\n\n\\end{array}\\right)\\left(\\begin{array}{l}\n\n1 \\\\\n\nb_{2} \\\\\n\nb_{3}\n\n\\end{array}\\right)=\\left(\\begin{array}{r}\n\n3-c_{1} \\\\\n\n1-c_{2} \\\\\n\n-1-c_{3}\n\n\\end{array}\\right) .\n\$\n\nThen, which of the following is/are TRUE ?\n\n(A)$\\vec{a} \\cdot \\vec{c}=0$\n\n(B)$\\vec{b} \\cdot \\vec{c}=0$\n\n(C)$|\\vec{b}|>\\sqrt{10}$\n\n(D)$|\\vec{c}| \\leq \\sqrt{11}$", "gold": "BCD" }, { "description": "JEE Adv 2022 Paper 2", "index": 14, "subject": "math", "type": "MCQ(multiple)", "question": "For$x \\in \\mathbb{R}$, let the function$y(x)$be the solution of the differential equation\n\n\$\n\\frac{d y}{d x}+12 y=\\cos \\left(\\frac{\\pi}{12} x\\right), \\quad y(0)=0\n\$\n\nThen, which of the following statements is/are TRUE?\n\n(A)$y(x)$is an increasing function\n\n(B)$y(x)$is a decreasing function\n\n(C) There exists a real number$\\beta$such that the line$y=\\beta$intersects the curve$y=y(x)$at infinitely many points\n\n(D)$y(x)$is a periodic function", "gold": "C" }, { "description": "JEE Adv 2022 Paper 2", "index": 15, "subject": "math", "type": "MCQ", "question": "Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?\n\n(A) 21816\n\n(B) 85536\n\n(C) 12096\n\n(D) 156816", "gold": "A" }, { "description": "JEE Adv 2022 Paper 2", "index": 16, "subject": "math", "type": "MCQ", "question": "If$M=\\left(\\begin{array}{rr}\\frac{5}{2} & \\frac{3}{2} \\\\ -\\frac{3}{2} & -\\frac{1}{2}\\end{array}\\right)$, then which of the following matrices is equal to$M^{2022}$?\n\n(A)$\\left(\\begin{array}{rr}3034 & 3033 \\\\ -3033 & -3032\\end{array}\\right)$\n\n(B)$\\left(\\begin{array}{ll}3034 & -3033 \\\\ 3033 & -3032\\end{array}\\right)$\n\n(C)$\\left(\\begin{array}{rr}3033 & 3032 \\\\ -3032 & -3031\\end{array}\\right)$\n\n(D)$\\left(\\begin{array}{rr}3032 & 3031 \\\\ -3031 & -3030\\end{array}\\right)$", "gold": "A" }, { "description": "JEE Adv 2022 Paper 2", "index": 17, "subject": "math", "type": "MCQ", "question": "Suppose that\n\nBox-I contains 8 red, 3 blue and 5 green balls,\n\nBox-II contains 24 red, 9 blue and 15 green balls,\n\nBox-III contains 1 blue, 12 green and 3 yellow balls,\n\nBox-IV contains 10 green, 16 orange and 6 white balls.\n\nA ball is chosen randomly from Box-I; call this ball$b$. If$b$is red then a ball is chosen randomly from Box-II, if$b$is blue then a ball is chosen randomly from Box-III, and if$b$is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to\n\n(A)$\\frac{15}{256}$\n\n(B)$\\frac{3}{16}$\n\n(C)$\\frac{5}{52}$\n\n(D)$\\frac{1}{8}$", "gold": "C" }, { "description": "JEE Adv 2022 Paper 2", "index": 18, "subject": "math", "type": "MCQ", "question": "For positive integer$n$, define\n\n\$\nf(n)=n+\\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\\cdots+\\frac{25 n-7 n^{2}}{7 n^{2}}\n\$\n\nThen, the value of$\\lim _{n \\rightarrow \\infty} f(n)$is equal to\n\n(A)$3+\\frac{4}{3} \\log _{e} 7$\n\n(B)$4-\\frac{3}{4} \\log _{e}\\left(\\frac{7}{3}\\right)$\n\n(C)$4-\\frac{4}{3} \\log _{e}\\left(\\frac{7}{3}\\right)$\n\n(D)$3+\\frac{3}{4} \\log _{e} 7$", "gold": "B" }, { "description": "JEE Adv 2022 Paper 2", "index": 19, "subject": "phy", "type": "Integer", "question": "A particle of mass$1 \\mathrm{~kg}$is subjected to a force which depends on the position as$\\vec{F}=-k(x \\hat{i}+y \\hat{j}) k g \\mathrm{ks}^{-2}$with$k=1 \\mathrm{~kg} \\mathrm{~s}^{-2}$. At time$t=0$, the particle's position$\\vec{r}=\\left(\\frac{1}{\\sqrt{2}} \\hat{i}+\\sqrt{2} \\hat{j}\\right) m$and its velocity$\\vec{v}=\\left(-\\sqrt{2} \\hat{i}+\\sqrt{2} \\hat{j}+\\frac{2}{\\pi} \\hat{k}\\right) m s^{-1}$. Let$v_{x}$and$v_{y}$denote the$x$and the$y$components of the particle's velocity, respectively. Ignore gravity. When$z=0.5 \\mathrm{~m}$, what is the value of$\\left(x v_{y}-y v_{x}\\right)$in$m^{2} s^{-1}$?", "gold": "3" }, { "description": "JEE Adv 2022 Paper 2", "index": 20, "subject": "phy", "type": "Integer", "question": "In a radioactive decay chain reaction,${ }_{90}^{230} \\mathrm{Th}$nucleus decays into${ }_{84}^{214} \\mathrm{Po}$nucleus. What is the ratio of the number of$\\alpha$to number of$\\beta^{-}$particles emitted in this process?", "gold": "2" }, { "description": "JEE Adv 2022 Paper 2", "index": 22, "subject": "phy", "type": "Integer", "question": "In a particular system of units, a physical quantity can be expressed in terms of the electric charge$e$, electron mass$m_{e}$, Planck's constant$h$, and Coulomb's constant$k=\\frac{1}{4 \\pi \\epsilon_{0}}$, where$\\epsilon_{0}$is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is$[B]=[e]^{\\alpha}\\left[m_{e}\\right]^{\\beta}[h]^{\\gamma}[k]^{\\delta}$. The value of$\\alpha+\\beta+\\gamma+\\delta$is", "gold": "4" }, { "description": "JEE Adv 2022 Paper 2", "index": 25, "subject": "phy", "type": "Integer", "question": "On a frictionless horizontal plane, a bob of mass$m=0.1 \\mathrm{~kg}$is attached to a spring with natural length$l_{0}=0.1 \\mathrm{~m}$. The spring constant is$k_{1}=0.009 \\mathrm{Nm}^{-1}$when the length of the spring$l>l_{0}$and is$k_{2}=0.016 \\mathrm{Nm}^{-1}$when$l0$. In addition to the Coulomb force, the particle experiences a vertical force$\\vec{F}=-c \\hat{k}$with$c>0$. Let$\\beta=\\frac{2 c \\epsilon_{0}}{q \\sigma}$. Which of the following statement(s) is(are) correct?\n\n(A) For$\\beta=\\frac{1}{4}$and$z_{0}=\\frac{25}{7} R$, the particle reaches the origin.\n\n(B) For$\\beta=\\frac{1}{4}$and$z_{0}=\\frac{3}{7} R$, the particle reaches the origin.\n\n(C) For$\\beta=\\frac{1}{4}$and$z_{0}=\\frac{R}{\\sqrt{3}}$, the particle returns back to$z=z_{0}$.\n\n(D) For$\\beta>1$and$z_{0}>0$, the particle always reaches the origin.", "gold": "ACD" }, { "description": "JEE Adv 2022 Paper 2", "index": 34, "subject": "phy", "type": "MCQ", "question": "When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is$6.0 \\mathrm{~V}$. This potential drops to$0.6 \\mathrm{~V}$if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take$\\frac{h c}{e}=1.24 \\times10^{-6} \\mathrm{~J} \\mathrm{mC}^{-1}$.]\n\n(A)$1.72 \\times 10^{-7} \\mathrm{~m}, 1.20 \\mathrm{eV}$\n\n(B)$1.72 \\times 10^{-7} \\mathrm{~m}, 5.60 \\mathrm{eV}$\n\n(C)$3.78 \\times 10^{-7} \\mathrm{~m}, 5.60 \\mathrm{eV}$\n\n(D)$3.78 \\times 10^{-7} \\mathrm{~m}, 1.20 \\mathrm{eV}$", "gold": "A" }, { "description": "JEE Adv 2022 Paper 2", "index": 35, "subject": "phy", "type": "MCQ", "question": "Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is$0.5 \\mathrm{~mm}$. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.\n\n\\begin{center}\n\n\\begin{tabular}{|l|l|l|}\n\n\\hline\n\nMeasurement condition & Main scale reading & Circular scale reading \\\\\n\n\\hline\n\n$\\begin{array}{l}\\text { Two arms of gauge touching } \\\\ \\text { each other without wire }\\end{array}$& 0 division & 4 divisions \\\\\n\n\\hline\n\nAttempt-1: With wire & 4 divisions & 20 divisions \\\\\n\n\\hline\n\nAttempt-2: With wire & 4 divisions & 16 divisions \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}\n\nWhat are the diameter and cross-sectional area of the wire measured using the screw gauge?\n\n(A)$2.22 \\pm 0.02 \\mathrm{~mm}, \\pi(1.23 \\pm 0.02) \\mathrm{mm}^{2}$\n\n(B)$2.22 \\pm 0.01 \\mathrm{~mm}, \\pi(1.23 \\pm 0.01) \\mathrm{mm}^{2}$\n\n(C)$2.14 \\pm 0.02 \\mathrm{~mm}, \\pi(1.14 \\pm 0.02) \\mathrm{mm}^{2}$\n\n(D)$2.14 \\pm 0.01 \\mathrm{~mm}, \\pi(1.14 \\pm 0.01) \\mathrm{mm}^{2}$", "gold": "C" }, { "description": "JEE Adv 2022 Paper 2", "index": 37, "subject": "chem", "type": "Integer", "question": "Concentration of$\\mathrm{H}_{2} \\mathrm{SO}_{4}$and$\\mathrm{Na}_{2} \\mathrm{SO}_{4}$in a solution is$1 \\mathrm{M}$and$1.8 \\times 10^{-2} \\mathrm{M}$, respectively. Molar solubility of$\\mathrm{PbSO}_{4}$in the same solution is$\\mathrm{X} \\times 10^{-\\mathrm{Y}} \\mathrm{M}$(expressed in scientific notation). What is the value of$\\mathrm{Y}$?\n\n[Given: Solubility product of$\\mathrm{PbSO}_{4}\\left(K_{s p}\\right)=1.6 \\times 10^{-8}$. For$\\mathrm{H}_{2} \\mathrm{SO}_{4}, K_{a l}$is very large and$\\left.K_{a 2}=1.2 \\times 10^{-2}\\right]$", "gold": "6" }, { "description": "JEE Adv 2022 Paper 2", "index": 38, "subject": "chem", "type": "Integer", "question": "An aqueous solution is prepared by dissolving$0.1 \\mathrm{~mol}$of an ionic salt in$1.8 \\mathrm{~kg}$of water at$35^{\\circ} \\mathrm{C}$. The salt remains$90 \\%$dissociated in the solution. The vapour pressure of the solution is$59.724 \\mathrm{~mm}$of$\\mathrm{Hg}$. Vapor pressure of water at$35{ }^{\\circ} \\mathrm{C}$is$60.000 \\mathrm{~mm}$of$\\mathrm{Hg}$. What is the number of ions present per formula unit of the ionic salt?", "gold": "5" }, { "description": "JEE Adv 2022 Paper 2", "index": 40, "subject": "chem", "type": "Integer", "question": "The reaction of$\\mathrm{Xe}$and$\\mathrm{O}_{2} \\mathrm{~F}_{2}$gives a Xe compound$\\mathbf{P}$. What is the number of moles of$\\mathrm{HF}$produced by the complete hydrolysis of$1 \\mathrm{~mol}$of$\\mathbf{P}$?", "gold": "4" }, { "description": "JEE Adv 2022 Paper 2", "index": 41, "subject": "chem", "type": "Integer", "question": "Thermal decomposition of$\\mathrm{AgNO}_{3}$produces two paramagnetic gases. What is the total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons?", "gold": "6" }, { "description": "JEE Adv 2022 Paper 2", "index": 45, "subject": "chem", "type": "MCQ(multiple)", "question": "To check the principle of multiple proportions, a series of pure binary compounds$\\left(\\mathrm{P}_{\\mathrm{m}} \\mathrm{Q}_{\\mathrm{n}}\\right)$were analyzed and their composition is tabulated below. The correct option(s) is(are)\n\n\\begin{center}\n\n\\begin{tabular}{|l|l|l|}\n\n\\hline\n\nCompound & Weight \\% of P & Weight \\% of Q \\\\\n\n\\hline\n\n$\\mathbf{1}$& 50 & 50 \\\\\n\n\\hline\n\n$\\mathbf{2}$& 44.4 & 55.6 \\\\\n\n\\hline\n\n$\\mathbf{3}$& 40 & 60 \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}\n\n(A) If empirical formula of compound 3 is$\\mathrm{P}_{3} \\mathrm{Q}_{4}$, then the empirical formula of compound 2 is$\\mathrm{P}_{3} \\mathrm{Q}_{5}$.\n\n(B) If empirical formula of compound 3 is$\\mathrm{P}_{3} \\mathrm{Q}_{2}$and atomic weight of element$\\mathrm{P}$is 20 , then the atomic weight of$\\mathrm{Q}$is 45 .\n\n(C) If empirical formula of compound 2 is$\\mathrm{PQ}$, then the empirical formula of the compound$\\mathbf{1}$is$\\mathrm{P}_{5} \\mathrm{Q}_{4}$.\n\n(D) If atomic weight of$\\mathrm{P}$and$\\mathrm{Q}$are 70 and 35 , respectively, then the empirical formula of compound 1 is$\\mathrm{P}_{2} \\mathrm{Q}$.", "gold": "BC" }, { "description": "JEE Adv 2022 Paper 2", "index": 46, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct option(s) about entropy (S) is(are)\n\n$[\\mathrm{R}=$gas constant,$\\mathrm{F}=$Faraday constant,$\\mathrm{T}=$Temperature$]$\n\n(A) For the reaction,$\\mathrm{M}(s)+2 \\mathrm{H}^{+}(a q) \\rightarrow \\mathrm{H}_{2}(g)+\\mathrm{M}^{2+}(a q)$, if$\\frac{d E_{c e l l}}{d T}=\\frac{R}{F}$, then the entropy change of the reaction is$\\mathrm{R}$(assume that entropy and internal energy changes are temperature independent).\n\n(B) The cell reaction,$\\operatorname{Pt}(s)\\left|\\mathrm{H}_{2}(g, 1 \\mathrm{bar})\\right| \\mathrm{H}^{+}(a q, 0.01 \\mathrm{M}) \\| \\mathrm{H}^{+}(a q, 0.1 \\mathrm{M})\\left|\\mathrm{H}_{2}(g, 1 \\mathrm{bar})\\right| \\operatorname{Pt}(s)$, is an entropy driven process.\n\n(C) For racemization of an optically active compound,$\\Delta \\mathrm{S}>0$.\n\n(D)$\\Delta \\mathrm{S}>0$, for$\\left[\\mathrm{Ni}\\left(\\mathrm{H}_{2} \\mathrm{O}_{6}\\right]^{2+}+3\\right.$en$\\rightarrow\\left[\\mathrm{Ni}(\\mathrm{en})_{3}\\right]^{2+}+6 \\mathrm{H}_{2} \\mathrm{O}$(where en$=$ethylenediamine).", "gold": "BCD" }, { "description": "JEE Adv 2022 Paper 2", "index": 47, "subject": "chem", "type": "MCQ(multiple)", "question": "The compound(s) which react(s) with$\\mathrm{NH}_{3}$to give boron nitride (BN) is(are)\n\n(A)$\\mathrm{B}$\n\n(B)$\\mathrm{B}_{2} \\mathrm{H}_{6}$\n\n(C)$\\mathrm{B}_{2} \\mathrm{O}_{3}$\n\n(D)$\\mathrm{HBF}_{4}$", "gold": "ABC" }, { "description": "JEE Adv 2022 Paper 2", "index": 48, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range$900-1500 \\mathrm{~K}$is(are)\n\n(A) Limestone is used to remove silicate impurity.\n\n(B) Pig iron obtained from blast furnace contains about$4 \\%$carbon.\n\n(C) Coke (C) converts$\\mathrm{CO}_{2}$to$\\mathrm{CO}$.\n\n(D) Exhaust gases consist of$\\mathrm{NO}_{2}$and$\\mathrm{CO}$.", "gold": "ABC" }, { "description": "JEE Adv 2022 Paper 2", "index": 50, "subject": "chem", "type": "MCQ(multiple)", "question": "Among the following, the correct statement(s) about polymers is(are)\n\n(A) The polymerization of chloroprene gives natural rubber.\n\n(B) Teflon is prepared from tetrafluoroethene by heating it with persulphate catalyst at high pressures.\n\n(C) PVC are thermoplastic polymers.\n\n(D) Ethene at 350-570 K temperature and 1000-2000 atm pressure in the presence of a peroxide initiator yields high density polythene.", "gold": "BC" }, { "description": "JEE Adv 2022 Paper 2", "index": 51, "subject": "chem", "type": "MCQ", "question": "Atom$\\mathrm{X}$occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in \\%) of the resultant solid is closest to\n\n(A) 25\n\n(B) 35\n\n(C) 55\n\n(D) 75", "gold": "B" }, { "description": "JEE Adv 2022 Paper 2", "index": 52, "subject": "chem", "type": "MCQ", "question": "The reaction of$\\mathrm{HClO}_{3}$with$\\mathrm{HCl}$gives a paramagnetic gas, which upon reaction with$\\mathrm{O}_{3}$produces\n\n(A)$\\mathrm{Cl}_{2} \\mathrm{O}$\n\n(B)$\\mathrm{ClO}_{2}$\n\n(C)$\\mathrm{Cl}_{2} \\mathrm{O}_{6}$\n\n(D)$\\mathrm{Cl}_{2} \\mathrm{O}_{7}$", "gold": "C" }, { "description": "JEE Adv 2022 Paper 2", "index": 53, "subject": "chem", "type": "MCQ", "question": "The reaction of$\\mathrm{Pb}\\left(\\mathrm{NO}_{3}\\right)_{2}$and$\\mathrm{NaCl}$in water produces a precipitate that dissolves upon the addition of$\\mathrm{HCl}$of appropriate concentration. The dissolution of the precipitate is due to the formation of\n\n(A)$\\mathrm{PbCl}_{2}$\n\n(B)$\\mathrm{PbCl}_{4}$\n\n(C)$\\left[\\mathrm{PbCl}_{4}\\right]^{2-}$\n\n(D)$\\left[\\mathrm{PbCl}_{6}\\right]^{2-}$", "gold": "C" }, { "description": "JEE Adv 2023 Paper 1", "index": 1, "subject": "math", "type": "MCQ(multiple)", "question": "Let$S=(0,1) \\cup(1,2) \\cup(3,4)$and$T=\\{0,1,2,3\\}$. Then which of the following statements is(are) true? \n\n\n\n(A) There are infinitely many functions from$S$to$T$\n\n\n\n(B) There are infinitely many strictly increasing functions from$S$to$T$\n\n\n\n(C) The number of continuous functions from$S$to$T$is at most 120\n\n\n\n(D) Every continuous function from$S$to$T$is differentiable", "gold": "ACD" }, { "description": "JEE Adv 2023 Paper 1", "index": 2, "subject": "math", "type": "MCQ(multiple)", "question": "Let$T_{1}$and$T_{2}$be two distinct common tangents to the ellipse$E: \\frac{x^{2}}{6}+\\frac{y^{2}}{3}=1$and the parabola$P: y^{2}=12 x$. Suppose that the tangent$T_{1}$touches$P$and$E$at the points$A_{1}$and$A_{2}$, respectively and the tangent$T_{2}$touches$P$and$E$at the points$A_{4}$and$A_{3}$, respectively. Then which of the following statements is(are) true?\n\n\n\n(A) The area of the quadrilateral$A_{1} A_{2} A_{3} A_{4}$is 35 square units\n\n\n\n(B) The area of the quadrilateral$A_{1} A_{2} A_{3} A_{4}$is 36 square units\n\n\n\n(C) The tangents$T_{1}$and$T_{2}$meet the$x$-axis at the point$(-3,0)$\n\n\n\n(D) The tangents$T_{1}$and$T_{2}$meet the$x$-axis at the point$(-6,0)$", "gold": "AC" }, { "description": "JEE Adv 2023 Paper 1", "index": 3, "subject": "math", "type": "MCQ(multiple)", "question": "Let$f:[0,1] \\rightarrow[0,1]$be the function defined by$f(x)=\\frac{x^{3}}{3}-x^{2}+\\frac{5}{9} x+\\frac{17}{36}$. Consider the square region$S=[0,1] \\times[0,1]$. Let$G=\\{(x, y) \\in S: y>f(x)\\}$be called the green region and$R=\\{(x, y) \\in S: y0$. The normal to the parabola at$P$meets the$x$-axis at a point$Q$. The area of the triangle$P F Q$, where$F$is the focus of the parabola, is 120 . If the slope$m$of the normal and$a$are both positive integers, then the pair$(a, m)$is\n\n(A)$(2,3)$\n\n\n\n(B)$(1,3)$\n\n\n\n(C)$(2,4)$\n\n\n\n(D)$(3,4)$", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 8, "subject": "math", "type": "Integer", "question": "Let$\\tan ^{-1}(x) \\in\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$, for$x \\in \\mathbb{R}$. Then what is the number of real solutions of the equation\n\n\n\n$\\sqrt{1+\\cos (2 x)}=\\sqrt{2} \\tan ^{-1}(\\tan x)$in the set$\\left(-\\frac{3 \\pi}{2},-\\frac{\\pi}{2}\\right) \\cup\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right) \\cup\\left(\\frac{\\pi}{2}, \\frac{3 \\pi}{2}\\right)$?", "gold": "3" }, { "description": "JEE Adv 2023 Paper 1", "index": 9, "subject": "math", "type": "Integer", "question": "Let$n \\geq 2$be a natural number and$f:[0,1] \\rightarrow \\mathbb{R}$be the function defined by\n\n\n\n$$\n\nf(x)= \\begin{cases}n(1-2 n x) & \\text { if } 0 \\leq x \\leq \\frac{1}{2 n} \\\\ 2 n(2 n x-1) & \\text { if } \\frac{1}{2 n} \\leq x \\leq \\frac{3}{4 n} \\\\ 4 n(1-n x) & \\text { if } \\frac{3}{4 n} \\leq x \\leq \\frac{1}{n} \\\\ \\frac{n}{n-1}(n x-1) & \\text { if } \\frac{1}{n} \\leq x \\leq 1\\end{cases}\n\n$$\n\n\n\nIf$n$is such that the area of the region bounded by the curves$x=0, x=1, y=0$and$y=f(x)$is 4 , then what is the maximum value of the function$f$?", "gold": "8" }, { "description": "JEE Adv 2023 Paper 1", "index": 10, "subject": "math", "type": "Integer", "question": "Let$7 \\overbrace{5 \\cdots 5}^{r} 7$denote the$(r+2)$digit number where the first and the last digits are 7 and the remaining$r$digits are 5 . Consider the sum$S=77+757+7557+\\cdots+7 \\overbrace{5 \\cdots 5}^{98}7$. If$S=\\frac{7 \\overbrace{5 \\cdots 5}^{99}7+m}{n}$, where$m$and$n$are natural numbers less than 3000 , then what is the value of$m+n$?", "gold": "1219" }, { "description": "JEE Adv 2023 Paper 1", "index": 11, "subject": "math", "type": "Integer", "question": "Let$A=\\left\\{\\frac{1967+1686 i \\sin \\theta}{7-3 i \\cos \\theta}: \\theta \\in \\mathbb{R}\\right\\}$. If$A$contains exactly one positive integer$n$, then what is the value of$n$?", "gold": "281" }, { "description": "JEE Adv 2023 Paper 1", "index": 12, "subject": "math", "type": "Integer", "question": "Let$P$be the plane$\\sqrt{3} x+2 y+3 z=16$and let$S=\\left\\{\\alpha \\hat{i}+\\beta \\hat{j}+\\gamma \\hat{k}: \\alpha^{2}+\\beta^{2}+\\gamma^{2}=1\\right.$and the distance of$(\\alpha, \\beta, \\gamma)$from the plane$P$is$\\left.\\frac{7}{2}\\right\\}$. Let$\\vec{u}, \\vec{v}$and$\\vec{w}$be three distinct vectors in$S$such that$|\\vec{u}-\\vec{v}|=|\\vec{v}-\\vec{w}|=|\\vec{w}-\\vec{u}|$. Let$V$be the volume of the parallelepiped determined by vectors$\\vec{u}, \\vec{v}$and$\\vec{w}$. Then what is the value of$\\frac{80}{\\sqrt{3}} V$?", "gold": "45" }, { "description": "JEE Adv 2023 Paper 1", "index": 13, "subject": "math", "type": "Integer", "question": "Let$a$and$b$be two nonzero real numbers. If the coefficient of$x^{5}$in the expansion of$\\left(a x^{2}+\\frac{70}{27 b x}\\right)^{4}$is equal to the coefficient of$x^{-5}$in the expansion of$\\left(a x-\\frac{1}{b x^{2}}\\right)^{7}$, then the value of$2 b$is", "gold": "3" }, { "description": "JEE Adv 2023 Paper 1", "index": 14, "subject": "math", "type": "MCQ", "question": "Let$\\alpha, \\beta$and$\\gamma$be real numbers. Consider the following system of linear equations \n\n\n\n$x+2 y+z=7$\n\n\n\n$x+\\alpha z=11$\n\n\n\n$2 x-3 y+\\beta z=\\gamma$\n\n\n\nMatch each entry in List-I to the correct entries in List-II.\n\n\n\n\\textbf{List-I}\n\n\n\n(P) If$\\beta=\\frac{1}{2}(7 \\alpha-3)$and$\\gamma=28$, then the system has\n\n\n\n(Q) If$\\beta=\\frac{1}{2}(7 \\alpha-3)$and$\\gamma \\neq 28$, then the system has\n\n\n\n(R) If$\\beta \\neq \\frac{1}{2}(7 \\alpha-3)$where$\\alpha=1$and$\\gamma \\neq 28$, then the system has\n\n\n\n(S) If$\\beta \\neq \\frac{1}{2}(7 \\alpha-3)$where$\\alpha=1$and$\\gamma = 28$, then the system has\n\n\n\n\n\n\\textbf{List-II}\n\n\n\n(1) a unique solution\n\n\n\n(2) no solution\n\n\n\n(3) infinitely many solutions\n\n\n\n(4)$x=11, y=-2$and$z=0$as a solution\n\n\n\n(5)$x=-15, y=4$and$z=0$as a solution\n\n\n\nThe correct option is:\n\n\n\n(A)$(P) \\rightarrow(3) \\quad(Q) \\rightarrow(2) \\quad(R) \\rightarrow(1) \\quad(S) \\rightarrow(4)$\n\n\n\n(B)$(P) \\rightarrow(3) \\quad(Q) \\rightarrow(2) \\quad(R) \\rightarrow(5) \\quad(S) \\rightarrow(4)$\n\n\n\n(C)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(1) \\quad(R) \\rightarrow(4) \\quad(S) \\rightarrow(5)$\n\n\n\n(D)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(1) \\quad(R) \\rightarrow(1) \\quad$(S)$\\rightarrow$(3)", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 15, "subject": "math", "type": "MCQ", "question": "Consider the given data with frequency distribution\n\n\n\n$$\n\n\\begin{array}{ccccccc}\n\nx_{i} & 3 & 8 & 11 & 10 & 5 & 4 \\\\\n\nf_{i} & 5 & 2 & 3 & 2 & 4 & 4\n\n\\end{array}\n\n$$\n\n\n\nMatch each entry in List-I to the correct entries in List-II.\n\n\n\n\\textbf{List-I}\n\n\n\n(P) The mean of the above data is\n\n\n\n(Q) The median of the above data is\n\n\n\n(R) The mean deviation about the mean of the above data is\n\n\n\n(S) The mean deviation about the median of the above data is\n\n\n\n\\textbf{List-II}\n\n\n\n(1) 2.5\n\n\n\n(2) 5\n\n\n\n(3) 6\n\n\n\n(4) 2.7\n\n\n\n(5) 2.4\n\n\n\nThe correct option is:\n\n\n\n(A)$(P) \\rightarrow(3) \\quad$(Q)$\\rightarrow$(2)$\\quad$(R)$\\rightarrow$(4)$\\quad$(S)$\\rightarrow$(5)\n\n\n\n(B)$(P) \\rightarrow(3) \\quad(Q) \\rightarrow(2) \\quad(R) \\rightarrow(1) \\quad(S) \\rightarrow(5)$\n\n\n\n(C)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(3) \\quad(R) \\rightarrow(4) \\quad(S) \\rightarrow(1)$\n\n\n\n(D)$(P) \\rightarrow$(3)$\\quad$(Q)$\\rightarrow$(3)$\\quad$(R)$\\rightarrow$(5)$\\quad$(S)$\\rightarrow$(5)", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 16, "subject": "math", "type": "MCQ", "question": "Let$\\ell_{1}$and$\\ell_{2}$be the lines$\\vec{r}_{1}=\\lambda(\\hat{i}+\\hat{j}+\\hat{k})$and$\\vec{r}_{2}=(\\hat{j}-\\hat{k})+\\mu(\\hat{i}+\\hat{k})$, respectively. Let$X$be the set of all the planes$H$that contain the line$\\ell_{1}$. For a plane$H$, let$d(H)$denote the smallest possible distance between the points of$\\ell_{2}$and$H$. Let$H_{0}$be a plane in$X$for which$d\\left(H_{0}\\right)$is the maximum value of$d(H)$as$H$varies over all planes in$X$.\n\n\n\nMatch each entry in List-I to the correct entries in List-II.\n\n\n\n\\textbf{List-I}\n\n\n\n(P) The value of$d\\left(H_{0}\\right)$is\n\n\n\n(Q) The distance of the point$(0,1,2)$from$H_{0}$is\n\n\n\n(R) The distance of origin from$H_{0}$is\n\n\n\n(S) The distance of origin from the point of intersection of planes$y=z, x=1$and$H_{0}$is\n\n\n\n\\textbf{List-II}\n\n\n\n(1)$\\sqrt{3}$\n\n\n\n(2)$\\frac{1}{\\sqrt{3}}$\n\n\n\n(3) 0\n\n\n\n(4)$\\sqrt{2}$\n\n\n\n(5)$\\frac{1}{\\sqrt{2}}$\n\n\n\nThe correct option is:\n\n\n\n(A)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(4) \\quad(R) \\rightarrow(5) \\quad(S) \\rightarrow(1)$\n\n\n\n(B)$(P) \\rightarrow(5) \\quad(Q) \\rightarrow(4) \\quad(R) \\rightarrow(3) \\quad(S) \\rightarrow$(1)\n\n\n\n(C)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(1) \\quad(R) \\rightarrow(3) \\quad(S) \\rightarrow$(2)\n\n\n\n(D)$(P) \\rightarrow(5) \\quad(Q) \\rightarrow(1) \\quad(R) \\rightarrow(4) \\quad(S) \\rightarrow(2)$", "gold": "B" }, { "description": "JEE Adv 2023 Paper 1", "index": 17, "subject": "math", "type": "MCQ", "question": "Let$z$be a complex number satisfying$|z|^{3}+2 z^{2}+4 \\bar{z}-8=0$, where$\\bar{z}$denotes the complex conjugate of$z$. Let the imaginary part of$z$be nonzero.\n\n\n\nMatch each entry in List-I to the correct entries in List-II.\n\n\n\n\\textbf{List-I}\n\n\n\n(P)$|z|^{2}$is equal to\n\n\n\n(Q)$|Z-\\bar{Z}|^{2}$is equal to\n\n\n\n(R)$|Z|^{2}+|Z+\\bar{Z}|^{2}$is equal to\n\n\n\n(S)$|z+1|^{2}$is equal to\n\n\n\n\\textbf{List-II}\n\n\n\n(1) 12\n\n\n\n(2) 4\n\n\n\n(3) 8\n\n\n\n(4) 10\n\n\n\n(5) 7\n\n\n\nThe correct option is:\n\n\n\n(A)$(P) \\rightarrow(1) \\quad(Q) \\rightarrow(3) \\quad(R) \\rightarrow(5) \\quad(S) \\rightarrow$(4)\n\n\n\n(B)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(1) \\quad(R) \\rightarrow(3) \\quad(S) \\rightarrow(5)$\n\n\n\n(C)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(4) \\quad(R) \\rightarrow(5) \\quad(S) \\rightarrow(1)$\n\n\n\n(D)$(P) \\rightarrow(2) \\quad(Q) \\rightarrow(3) \\quad(R) \\rightarrow(5) \\quad(S) \\rightarrow(4)$", "gold": "B" }, { "description": "JEE Adv 2023 Paper 1", "index": 21, "subject": "phy", "type": "MCQ", "question": "A bar of mass$M=1.00 \\mathrm{~kg}$and length$L=0.20 \\mathrm{~m}$is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass$m=0.10 \\mathrm{~kg}$is moving on the same horizontal surface with$5.00 \\mathrm{~m} \\mathrm{~s}^{-1}$speed on a path perpendicular to the bar. It hits the bar at a distance$L / 2$from the pivoted end and returns back on the same path with speed$\\mathrm{v}$. After this elastic collision, the bar rotates with an angular velocity$\\omega$. Which of the following statement is correct?\n\n\n\n(A)$\\omega=6.98 \\mathrm{rad} \\mathrm{s}^{-1}$and$\\mathrm{v}=4.30 \\mathrm{~m} \\mathrm{~s}^{-1}$\n\n\n\n(B)$\\omega=3.75 \\mathrm{rad} \\mathrm{s}^{-1}$and$\\mathrm{v}=4.30 \\mathrm{~m} \\mathrm{~s}^{-1}$\n\n\n\n(C)$\\omega=3.75 \\mathrm{rad} \\mathrm{s}^{-1}$and$\\mathrm{v}=10.0 \\mathrm{~m} \\mathrm{~s}^{-1}$\n\n\n\n(D)$\\omega=6.80 \\mathrm{rad} \\mathrm{s}^{-1}$and$\\mathrm{v}=4.10 \\mathrm{~m} \\mathrm{~s}^{-1}$", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 23, "subject": "phy", "type": "MCQ", "question": "One mole of an ideal gas expands adiabatically from an initial state$\\left(T_{\\mathrm{A}}, V_{0}\\right)$to final state$\\left(T_{\\mathrm{f}}, 5 V_{0}\\right)$. Another mole of the same gas expands isothermally from a different initial state$\\left(T_{\\mathrm{B}}, V_{0}\\right)$to the same final state$\\left(T_{\\mathrm{f}}, 5 V_{0}\\right)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is$\\gamma$. What is the ratio$T_{\\mathrm{A}} / T_{\\mathrm{B}}$?\n\n\n\n(A)$5^{\\gamma-1}$\n\n\n\n(B)$5^{1-\\gamma}$\n\n\n\n(C)$5^{\\gamma}$\n\n\n\n(D)$5^{1+\\gamma}$", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 24, "subject": "phy", "type": "MCQ", "question": "Two satellites$\\mathrm{P}$and$\\mathrm{Q}$are moving in different circular orbits around the Earth (radius$R$). The heights of$\\mathrm{P}$and$\\mathrm{Q}$from the Earth surface are$h_{\\mathrm{P}}$and$h_{\\mathrm{Q}}$, respectively, where$h_{\\mathrm{P}}=R / 3$. The accelerations of$\\mathrm{P}$and$\\mathrm{Q}$due to Earth's gravity are$g_{\\mathrm{P}}$and$g_{\\mathrm{Q}}$, respectively. If$g_{\\mathrm{P}} / g_{\\mathrm{Q}}=36 / 25$, what is the value of$h_{\\mathrm{Q}}$?\n\n\n\n(A)$3 R / 5$\n\n\n\n(B)$R / 6$\n\n\n\n(C)$6 R / 5$\n\n\n\n(D)$5 R / 6$", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 25, "subject": "phy", "type": "Integer", "question": "A Hydrogen-like atom has atomic number$Z$. Photons emitted in the electronic transitions from level$n=4$to level$n=3$in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is$1.95 \\mathrm{eV}$. If the photoelectric threshold wavelength for the target metal is$310 \\mathrm{~nm}$, what is the value of$Z$?\n\n\n\n[Given:$h c=1240 \\mathrm{eV}-\\mathrm{nm}$and$R h c=13.6 \\mathrm{eV}$, where$R$is the Rydberg constant,$h$is the Planck's constant and$c$is the speed of light in vacuum]", "gold": "3" }, { "description": "JEE Adv 2023 Paper 1", "index": 27, "subject": "phy", "type": "Integer", "question": "In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is$10 \\pm 0.1 \\mathrm{~cm}$and the distance of its real image from the lens is$20 \\pm 0.2 \\mathrm{~cm}$. The error in the determination of focal length of the lens is$n \\%$. What is the value of$n$?", "gold": "1" }, { "description": "JEE Adv 2023 Paper 1", "index": 28, "subject": "phy", "type": "Integer", "question": "A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas$(\\gamma=5 / 3)$and one mole of an ideal diatomic gas$(\\gamma=7 / 5)$. Here,$\\gamma$is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. What is the change in its internal energy in Joule.", "gold": "121" }, { "description": "JEE Adv 2023 Paper 1", "index": 29, "subject": "phy", "type": "Integer", "question": "A person of height$1.6 \\mathrm{~m}$is walking away from a lamp post of height$4 \\mathrm{~m}$along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is$60 \\mathrm{~cm} \\mathrm{~s}^{-1}$, then what is the speed of the tip of the person's shadow on the ground with respect to the person in$\\mathrm{cm} \\mathrm{s}^{-1}$?", "gold": "40" }, { "description": "JEE Adv 2023 Paper 1", "index": 31, "subject": "phy", "type": "MCQ", "question": "List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.\n\n\n\n\\textbf{List-I}\n\n\n\n(P)${ }_{92}^{238} U \\rightarrow{ }_{91}^{234} \\mathrm{~Pa}$\n\n\n\n(Q)${ }_{82}^{214} \\mathrm{~Pb} \\rightarrow{ }_{82}^{210} \\mathrm{~Pb}$\n\n\n\n(R)${ }_{81}^{210} \\mathrm{Tl} \\rightarrow{ }_{82}^{206} \\mathrm{~Pb}$\n\n\n\n(S)${ }_{91}^{228} \\mathrm{~Pa} \\rightarrow{ }_{88}^{224} \\mathrm{Ra}$\n\n\n\n\\textbf{List-II}\n\n\n\n(1) one$\\alpha$particle and one$\\beta^{+}$particle\n\n\n\n(2) three$\\beta^{-}$particles and one$\\alpha$particle\n\n\n\n(3) two$\\beta^{-}$particles and one$\\alpha$particle\n\n\n\n(4) one$\\alpha$particle and one$\\beta^{-}$particle\n\n\n\n(5) one$\\alpha$particle and two$\\beta^{+}$particles\n\n\n\n\n\n\n\n\n\n(A)$P \\rightarrow 4, Q \\rightarrow 3, R \\rightarrow 2, S \\rightarrow 1$\n\n\n\n(B)$P \\rightarrow 4, Q \\rightarrow 1, R \\rightarrow 2, S \\rightarrow 5$\n\n\n\n(C)$P \\rightarrow 5, Q \\rightarrow 3, R \\rightarrow 1, S \\rightarrow 4$\n\n\n\n(D)$P \\rightarrow 5, Q \\rightarrow 1, R \\rightarrow 3, S \\rightarrow 2$", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 32, "subject": "phy", "type": "MCQ", "question": "Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option.\n\n\n\n[Given: Wien's constant as$2.9 \\times 10^{-3} \\mathrm{~m}-\\mathrm{K}$and$\\frac{h c}{e}=1.24 \\times 10^{-6} \\mathrm{~V}-\\mathrm{m}$]\n\n\n\n\\textbf{List-I}\n\n\n\n(P)$2000 \\mathrm{~K}$\n\n\n\n(Q)$3000 \\mathrm{~K}$\n\n\n\n(R)$5000 \\mathrm{~K}$\n\n\n\n(S)$10000 \\mathrm{~K}$\n\n\n\n\\textbf{List-II}\n\n\n\n(1) The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function$4 \\mathrm{eV}$.\n\n\n\n(2) The radiation at peak wavelength is visible to human eye.\n\n\n\n(3) The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction.\n\n\n\n(4) The power emitted per unit area is$1 / 16$of that emitted by a blackbody at temperature$6000 \\mathrm{~K}$.\n\n\n\n(5) The radiation at peak emission wavelength can be used to image human bones.\n\n\n\n(A)$P \\rightarrow 3, Q \\rightarrow 5, R \\rightarrow 2, S \\rightarrow 3$\n\n\n\n(B)$P \\rightarrow 3, Q \\rightarrow 2, R \\rightarrow 4, S \\rightarrow 1$\n\n\n\n(C)$P \\rightarrow 3, Q \\rightarrow 4, R \\rightarrow 2, S \\rightarrow 1$\n\n\n\n(D)$P \\rightarrow 1, Q \\rightarrow 2, R \\rightarrow 5, S \\rightarrow 3$", "gold": "C" }, { "description": "JEE Adv 2023 Paper 1", "index": 33, "subject": "phy", "type": "MCQ", "question": "A series LCR circuit is connected to a$45 \\sin (\\omega t)$Volt source. The resonant angular frequency of the circuit is$10^{5} \\mathrm{rad} \\mathrm{s}^{-1}$and current amplitude at resonance is$I_{0}$. When the angular frequency of the source is$\\omega=8 \\times 10^{4} \\mathrm{rad} \\mathrm{s}^{-1}$, the current amplitude in the circuit is$0.05 I_{0}$. If$L=50 \\mathrm{mH}$, match each entry in List-I with an appropriate value from List-II and choose the correct option.\n\n\n\n\\textbf{List-I}\n\n\n\n(P)$I_{0}$in$\\mathrm{mA}$\n\n\n\n(Q) The quality factor of the circuit\n\n\n\n(R) The bandwidth of the circuit in$\\mathrm{rad} \\mathrm{s}^{-1}$\n\n\n\n(S) The peak power dissipated at resonance in Watt\n\n\n\n\\textbf{List-II}\n\n\n\n(1) 44.4\n\n\n\n(2) 18\n\n\n\n(3) 400\n\n\n\n(4) 2250\n\n\n\n(5) 500\n\n\n\n(A)$P \\rightarrow 2, Q \\rightarrow 3, R \\rightarrow 5, S \\rightarrow 1$\n\n\n\n(B)$P \\rightarrow 3, Q \\rightarrow 1, R \\rightarrow 4, S \\rightarrow 2$\n\n\n\n(C)$P \\rightarrow 4, Q \\rightarrow 5, R \\rightarrow 3, S \\rightarrow 1$\n\n\n\n(D)$P \\rightarrow 4, Q \\rightarrow 2, R \\rightarrow 1, S \\rightarrow 5$", "gold": "B" }, { "description": "JEE Adv 2023 Paper 1", "index": 35, "subject": "chem", "type": "MCQ(multiple)", "question": "The correct statement(s) related to processes involved in the extraction of metals is(are)\n\n\n\n(A) Roasting of Malachite produces Cuprite.\n\n\n\n(B) Calcination of Calamine produces Zincite.\n\n\n\n(C) Copper pyrites is heated with silica in a reverberatory furnace to remove iron.\n\n\n\n(D) Impure silver is treated with aqueous KCN in the presence of oxygen followed by reduction with zinc metal.", "gold": "BCD" }, { "description": "JEE Adv 2023 Paper 1", "index": 39, "subject": "chem", "type": "MCQ", "question": "Plotting$1 / \\Lambda_{\\mathrm{m}}$against$\\mathrm{c} \\Lambda_{\\mathrm{m}}$for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line with y-axis intercept of$\\mathrm{P}$and slope of$\\mathrm{S}$. The ratio$\\mathrm{P} / \\mathrm{S}$is\n\n\n\n$\\left[\\Lambda_{\\mathrm{m}}=\\right.$molar conductivity\n\n\n\n$\\Lambda_{\\mathrm{m}}^{\\mathrm{o}}=$limiting molar conductivity\n\n\n\n$\\mathrm{c}=$molar concentration\n\n\n\n$\\mathrm{K}_{\\mathrm{a}}=$dissociation constant of$\\mathrm{HX}$]\n\n\n\n(A)$\\mathrm{K}_{\\mathrm{a}} \\Lambda_{\\mathrm{m}}^{\\mathrm{o}}$\n\n\n\n(B)$\\mathrm{K}_{\\mathrm{a}} \\Lambda_{\\mathrm{m}}^{\\mathrm{o}} / 2$\n\n\n\n(C)$2 \\mathrm{~K}_{\\mathrm{a}} \\Lambda_{\\mathrm{m}}^{\\mathrm{o}}$\n\n\n\n(D)$1 /\\left(\\mathrm{K}_{\\mathrm{a}} \\Lambda_{\\mathrm{m}}^{\\mathrm{o}}\\right)$", "gold": "A" }, { "description": "JEE Adv 2023 Paper 1", "index": 40, "subject": "chem", "type": "MCQ", "question": "On decreasing the$p \\mathrm{H}$from 7 to 2, the solubility of a sparingly soluble salt (MX) of a weak acid (HX) increased from$10^{-4} \\mathrm{~mol} \\mathrm{~L}^{-1}$to$10^{-3} \\mathrm{~mol} \\mathrm{~L}^{-1}$. The$p\\mathrm{~K}_{\\mathrm{a}}$of$\\mathrm{HX}$is\n\n\n\n(A) 3\n\n\n\n(B) 4\n\n\n\n(C) 5\n\n\n\n(D) 2", "gold": "B" }, { "description": "JEE Adv 2023 Paper 1", "index": 42, "subject": "chem", "type": "Integer", "question": "The stoichiometric reaction of$516 \\mathrm{~g}$of dimethyldichlorosilane with water results in a tetrameric cyclic product$\\mathbf{X}$in$75 \\%$yield. What is the weight (in g) obtained of$\\mathbf{X}$?\n\n\n\n[Use, molar mass$\\left(\\mathrm{g} \\mathrm{mol}^{-1}\\right): \\mathrm{H}=1, \\mathrm{C}=12, \\mathrm{O}=16, \\mathrm{Si}=28, \\mathrm{Cl}=35.5$]", "gold": "222" }, { "description": "JEE Adv 2023 Paper 1", "index": 43, "subject": "chem", "type": "Integer", "question": "A gas has a compressibility factor of 0.5 and a molar volume of$0.4 \\mathrm{dm}^{3} \\mathrm{~mol}^{-1}$at a temperature of$800 \\mathrm{~K}$and pressure$\\mathbf{x}$atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be$\\mathbf{y} \\mathrm{dm}^{3} \\mathrm{~mol}^{-1}$. What is the value of$\\mathbf{x} / \\mathbf{y}$?\n\n\n\n[Use: Gas constant,$\\mathrm{R}=8 \\times 10^{-2} \\mathrm{~L} \\mathrm{~atm} \\mathrm{~K} \\mathrm{Kol}^{-1}$]", "gold": "100" }, { "description": "JEE Adv 2023 Paper 1", "index": 48, "subject": "chem", "type": "MCQ", "question": "Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.\n\n\n\n\\textbf{List-I}\n\n\n\n(P)$\\mathrm{P}_{2} \\mathrm{O}_{3}+3 \\mathrm{H}_{2} \\mathrm{O} \\rightarrow$\n\n\n\n(Q)$\\mathrm{P}_{4}+3 \\mathrm{NaOH}+3 \\mathrm{H}_{2} \\mathrm{O} \\rightarrow$\n\n\n\n(R)$\\mathrm{PCl}_{5}+\\mathrm{CH}_{3} \\mathrm{COOH} \\rightarrow$\n\n\n\n(S)$\\mathrm{H}_{3} \\mathrm{PO}_{2}+2 \\mathrm{H}_{2} \\mathrm{O}+4 \\mathrm{AgNO}_{3} \\rightarrow$\n\n\n\n\\textbf{List-II}\n\n\n\n(1)$\\mathrm{P}(\\mathrm{O})\\left(\\mathrm{OCH}_{3}\\right) \\mathrm{Cl}_{2}$\n\n\n\n(2)$\\mathrm{H}_{3} \\mathrm{PO}_{3}$\n\n\n\n(3)$\\mathrm{PH}_{3}$\n\n\n\n(4)$\\mathrm{POCl}_{3}$\n\n\n\n(5)$\\mathrm{H}_{3} \\mathrm{PO}_{4}$\n\n\n\n(A)$\\mathrm{P} \\rightarrow 2$;$\\mathrm{Q} \\rightarrow 3$;$\\mathrm{R} \\rightarrow 1$;$\\mathrm{S} \\rightarrow 5$\n\n\n\n(B)$\\mathrm{P} \\rightarrow 3$; Q$\\rightarrow 5$; R$\\rightarrow 4$;$\\mathrm{S} \\rightarrow 2$\n\n\n\n(C)$\\mathrm{P} \\rightarrow 5 ; \\mathrm{Q} \\rightarrow 2 ; \\mathrm{R} \\rightarrow 1$; S$\\rightarrow 3$\n\n\n\n(D) P$\\rightarrow 2$; Q$\\rightarrow 3$; R$\\rightarrow 4$;$\\mathrm{S} \\rightarrow 5$", "gold": "D" }, { "description": "JEE Adv 2023 Paper 1", "index": 49, "subject": "chem", "type": "MCQ", "question": "Match the electronic configurations in List-I with appropriate metal complex ions in List-II and choose the correct option.\n\n\n\n[Atomic Number:$\\mathrm{Fe}=26, \\mathrm{Mn}=25$,$\\mathrm{Co}=27$]\n\n\n\n\\textbf{List-I}\n\n\n\n(P)$t_{2 g}^{6} e_{g}^{0}$\n\n\n\n(Q)$t_{2 g}^{3} e_{g}^{2}$\n\n\n\n(R)$\\mathrm{e}^{2} \\mathrm{t}_{2}^{3}$\n\n\n\n(S)$t_{2 g}^{4} e_{g}^{2}$\n\n\n\n\\textbf{List-II}\n\n\n\n(1)$\\left[\\mathrm{Fe}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)_{6}\\right]^{2+}$\n\n\n\n(2)$\\left[\\mathrm{Mn}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)_{6}\\right]^{2+}$\n\n\n\n(3)$\\left[\\mathrm{Co}\\left(\\mathrm{NH}_{3}\\right)_{6}\\right]^{3+}$\n\n\n\n(4)$\\left[\\mathrm{FeCl}_{4}\\right]^{-}$\n\n\n\n(5)$\\left[\\mathrm{CoCl}_{4}\\right]^{2-}$\n\n\n\n\n\n(A)$\\mathrm{P} \\rightarrow 1$; Q$\\rightarrow 4$;$\\mathrm{R} \\rightarrow 2$;$\\mathrm{S} \\rightarrow 3$\n\n\n\n(B)$\\mathrm{P} \\rightarrow 1 ; \\mathrm{Q} \\rightarrow 2 ; \\mathrm{R} \\rightarrow 4 ; \\mathrm{S} \\rightarrow 5$\n\n\n\n(C)$\\mathrm{P} \\rightarrow 3 ; \\mathrm{Q} \\rightarrow 2 ; \\mathrm{R} \\rightarrow 5 ; \\mathrm{S} \\rightarrow 1$\n\n\n\n(D)$\\mathrm{P} \\rightarrow 3 ; \\mathrm{Q} \\rightarrow 2 ; \\mathrm{R} \\rightarrow 4$;$\\mathrm{S} \\rightarrow 1$", "gold": "D" }, { "description": "JEE Adv 2023 Paper 1", "index": 51, "subject": "chem", "type": "MCQ", "question": "The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.\n\n\n\n\\textbf{List-I}\n\n\n\n(P) Etard reaction\n\n\n\n(Q) Gattermann reaction\n\n\n\n(R) Gattermann-Koch reaction\n\n\n\n(S) Rosenmund reduction\n\n\n\n\n\n\n\n\\textbf{List-II}\n\n\n\n(1) Acetophenone$\\stackrel{\\mathrm{Zn}-\\mathrm{Hg}, \\mathrm{HCl}}{\\longrightarrow}$\n\n\n\n(2) Toluene$\\underset{\\text{(ii)}\\mathrm{SOCl}_{2}}{\\stackrel{\\text { (i) } \\mathrm{KMnO}_{4}, \\mathrm{KOH}, \\Delta}{\\longrightarrow}}$\n\n\n\n\n\n(3) Benzene$\\underset{\\text { anhyd. } \\mathrm{AlCl}_{3}}{\\stackrel{\\mathrm{CH}_{3} \\mathrm{Cl}}{\\longrightarrow}}$\n\n\n\n(4) Aniline$\\underset{273-278 \\mathrm{~K}}{\\stackrel{\\mathrm{NaNO}_{2} / \\mathrm{HCl}}{\\longrightarrow}}$\n\n\n\n(5) Phenol$\\stackrel{\\mathrm{Zn}, \\Delta}{\\longrightarrow}$\n\n\n\n\n\n(A)$\\mathrm{P} \\rightarrow 2$;$\\mathrm{Q} \\rightarrow 4 ; \\mathrm{R} \\rightarrow 1 ; \\mathrm{S} \\rightarrow 3$\n\n\n\n(B)$\\mathrm{P} \\rightarrow 1$;$\\mathrm{Q} \\rightarrow 3$;$\\mathrm{R} \\rightarrow 5$;$\\mathrm{S} \\rightarrow 2$\n\n\n\n(C)$\\mathrm{P} \\rightarrow 3$;$\\mathrm{Q} \\rightarrow 2 ; \\mathrm{R} \\rightarrow 1$;$\\mathrm{S} \\rightarrow 4$\n\n\n\n(D)$\\mathrm{P} \\rightarrow 3$;$\\mathrm{Q} \\rightarrow 4$; R$\\rightarrow 5$;$\\mathrm{S} \\rightarrow 2$", "gold": "D" }, { "description": "JEE Adv 2023 Paper 2", "index": 1, "subject": "math", "type": "MCQ", "question": "Let$f:[1, \\infty) \\rightarrow \\mathbb{R}$be a differentiable function such that$f(1)=\\frac{1}{3}$and$3 \\int_{1}^{x} f(t) d t=x f(x)-\\frac{x^{3}}{3}, x \\in[1, \\infty)$. Let$e$denote the base of the natural logarithm. Then the value of$f(e)$is\n\n\n\n(A)$\\frac{e^{2}+4}{3}$\n\n\n\n(B)$\\frac{\\log _{e} 4+e}{3}$\n\n\n\n(C)$\\frac{4 e^{2}}{3}$\n\n\n\n(D)$\\frac{e^{2}-4}{3}$", "gold": "C" }, { "description": "JEE Adv 2023 Paper 2", "index": 2, "subject": "math", "type": "MCQ", "question": "Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is$\\frac{1}{3}$, then the probability that the experiment stops with head is\n\n\n\n(A)$\\frac{1}{3}$\n\n\n\n(B)$\\frac{5}{21}$\n\n\n\n(C)$\\frac{4}{21}$\n\n\n\n(D)$\\frac{2}{7}$", "gold": "B" }, { "description": "JEE Adv 2023 Paper 2", "index": 3, "subject": "math", "type": "MCQ", "question": "For any$y \\in \\mathbb{R}$, let$\\cot ^{-1}(y) \\in(0, \\pi)$and$\\tan ^{-1}(y) \\in\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$. Then the sum of all the solutions of the equation$\\tan ^{-1}\\left(\\frac{6 y}{9-y^{2}}\\right)+\\cot ^{-1}\\left(\\frac{9-y^{2}}{6 y}\\right)=\\frac{2 \\pi}{3}$for$0<|y|<3$, is equal to\n\n\n\n(A)$2 \\sqrt{3}-3$\n\n\n\n(B)$3-2 \\sqrt{3}$\n\n\n\n(C)$4 \\sqrt{3}-6$\n\n\n\n(D)$6-4 \\sqrt{3}$", "gold": "C" }, { "description": "JEE Adv 2023 Paper 2", "index": 4, "subject": "math", "type": "MCQ", "question": "Let the position vectors of the points$P, Q, R$and$S$be$\\vec{a}=\\hat{i}+2 \\hat{j}-5 \\hat{k}, \\vec{b}=3 \\hat{i}+6 \\hat{j}+3 \\hat{k}$,$\\vec{c}=\\frac{17}{5} \\hat{i}+\\frac{16}{5} \\hat{j}+7 \\hat{k}$and$\\vec{d}=2 \\hat{i}+\\hat{j}+\\hat{k}$, respectively. Then which of the following statements is true?\n\n\n\n(A) The points$P, Q, R$and$S$are NOT coplanar\n\n\n\n(B)$\\frac{\\vec{b}+2 \\vec{d}}{3}$is the position vector of a point which divides$P R$internally in the ratio$5: 4$\n\n\n\n(C)$\\frac{\\vec{b}+2 \\vec{d}}{3}$is the position vector of a point which divides$P R$externally in the ratio$5: 4$\n\n\n\n(D) The square of the magnitude of the vector$\\vec{b} \\times \\vec{d}$is 95", "gold": "B" }, { "description": "JEE Adv 2023 Paper 2", "index": 5, "subject": "math", "type": "MCQ(multiple)", "question": "Let$M=\\left(a_{i j}\\right), i, j \\in\\{1,2,3\\}$, be the$3 \\times 3$matrix such that$a_{i j}=1$if$j+1$is divisible by$i$, otherwise$a_{i j}=0$. Then which of the following statements is(are) true?\n\n\n\n(A)$M$is invertible\n\n\n\n(B) There exists a nonzero column matrix$\\left(\\begin{array}{l}a_{1} \\\\ a_{2} \\\\ a_{3}\\end{array}\\right)$such that$M\\left(\\begin{array}{l}a_{1} \\\\ a_{2} \\\\ a_{3}\\end{array}\\right)=\\left(\\begin{array}{c}-a_{1} \\\\ -a_{2} \\\\ -a_{3}\\end{array}\\right)$\n\n\n\n(C) The set$\\left\\{X \\in \\mathbb{R}^{3}: M X=\\mathbf{0}\\right\\} \\neq\\{\\boldsymbol{0}\\}$, where$\\mathbf{0}=\\left(\\begin{array}{l}0 \\\\ 0 \\\\ 0\\end{array}\\right)$\n\n\n\n(D) The matrix$(M-2 I)$is invertible, where$I$is the$3 \\times 3$identity matrix", "gold": "BC" }, { "description": "JEE Adv 2023 Paper 2", "index": 6, "subject": "math", "type": "MCQ(multiple)", "question": "Let$f:(0,1) \\rightarrow \\mathbb{R}$be the function defined as$f(x)=[4 x]\\left(x-\\frac{1}{4}\\right)^{2}\\left(x-\\frac{1}{2}\\right)$, where$[x]$denotes the greatest integer less than or equal to$x$. Then which of the following statements is(are) true?\n\n\n\n(A) The function$f$is discontinuous exactly at one point in$(0,1)$\n\n\n\n(B) There is exactly one point in$(0,1)$at which the function$f$is continuous but NOT differentiable\n\n\n\n(C) The function$f$is NOT differentiable at more than three points in$(0,1)$\n\n\n\n(D) The minimum value of the function$f$is$-\\frac{1}{512}$", "gold": "AB" }, { "description": "JEE Adv 2023 Paper 2", "index": 7, "subject": "math", "type": "MCQ(multiple)", "question": "Let$S$be the set of all twice differentiable functions$f$from$\\mathbb{R}$to$\\mathbb{R}$such that$\\frac{d^{2} f}{d x^{2}}(x)>0$for all$x \\in(-1,1)$. For$f \\in S$, let$X_{f}$be the number of points$x \\in(-1,1)$for which$f(x)=x$. Then which of the following statements is(are) true?\n\n\n\n(A) There exists a function$f \\in S$such that$X_{f}=0$\n\n\n\n(B) For every function$f \\in S$, we have$X_{f} \\leq 2$\n\n\n\n(C) There exists a function$f \\in S$such that$X_{f}=2$\n\n\n\n(D) There does NOT exist any function$f$in$S$such that$X_{f}=1$", "gold": "ABC" }, { "description": "JEE Adv 2023 Paper 2", "index": 8, "subject": "math", "type": "Integer", "question": "For$x \\in \\mathbb{R}$, let$\\tan ^{-1}(x) \\in\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$. Then what is the minimum value of the function$f: \\mathbb{R} \\rightarrow \\mathbb{R}$defined by$f(x)=\\int_{0}^{x \\tan ^{-1} x} \\frac{e^{(t-\\cos t)}}{1+t^{2023}} d t$?", "gold": "0" }, { "description": "JEE Adv 2023 Paper 2", "index": 9, "subject": "math", "type": "Integer", "question": "For$x \\in \\mathbb{R}$, let$y(x)$be a solution of the differential equation$\\left(x^{2}-5\\right) \\frac{d y}{d x}-2 x y=-2 x\\left(x^{2}-5\\right)^{2}$such that$y(2)=7$.\n\n\n\nThen what is the maximum value of the function$y(x)$?", "gold": "16" }, { "description": "JEE Adv 2023 Paper 2", "index": 10, "subject": "math", "type": "Integer", "question": "Let$X$be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in$X$while 02244 and 44422 are not in$X$. Suppose that each element of$X$has an equal chance of being chosen. Let$p$be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5 . Then what is the value of$38 p$equal to?", "gold": "31" }, { "description": "JEE Adv 2023 Paper 2", "index": 11, "subject": "math", "type": "Integer", "question": "Let$A_{1}, A_{2}, A_{3}, \\ldots, A_{8}$be the vertices of a regular octagon that lie on a circle of radius 2. Let$P$be a point on the circle and let$P A_{i}$denote the distance between the points$P$and$A_{i}$for$i=1,2, \\ldots, 8$. If$P$varies over the circle, then what is the maximum value of the product$P A_{1} \\cdot P A_{2} \\cdots P A_{8}?", "gold": "512" }, { "description": "JEE Adv 2023 Paper 2", "index": 12, "subject": "math", "type": "Integer", "question": "$R=\\left\\{\\left(\\begin{array}{lll}a & 3 & b \\\\ c & 2 & d \\\\ 0 & 5 & 0\\end{array}\\right): a, b, c, d \\in\\{0,3,5,7,11,13,17,19\\}\\right\\}$. Then what is the number of invertible matrices in $R$?", "gold": "3780" }, { "description": "JEE Adv 2023 Paper 2", "index": 13, "subject": "math", "type": "Integer", "question": "Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1U_{6}$\n\n\n\n(B) $v_{5}>v_{3}$ and $U_{3}>U_{5}$\n\n\n\n(C) $v_{5}>v_{7}$ and $U_{5}$ Packing efficiency of unit cell of $y>$ Packing efficiency of unit cell of $z$\n\n\n\n(B) $\\mathrm{L}_{\\mathrm{y}}>\\mathrm{L}_{\\mathrm{z}}$\n\n\n\n(C) $\\mathrm{L}_{\\mathrm{x}}>\\mathrm{L}_{\\mathrm{y}}$\n\n\n\n(D) Density of $x>$ Density of $y$", "gold": "ABD" }, { "description": "JEE Adv 2023 Paper 2", "index": 42, "subject": "chem", "type": "Integer", "question": "$\\quad \\mathrm{H}_{2} \\mathrm{~S}$ (5 moles) reacts completely with acidified aqueous potassium permanganate solution. In this reaction, the number of moles of water produced is $\\mathbf{x}$, and the number of moles of electrons involved is $\\mathbf{y}$. The value of $(\\mathbf{x}+\\mathbf{y})$ is", "gold": "18" }, { "description": "JEE Adv 2023 Paper 2", "index": 43, "subject": "chem", "type": "Integer", "question": "Among $\\left[\\mathrm{I}_{3}\\right]^{+},\\left[\\mathrm{SiO}_{4}\\right]^{4-}, \\mathrm{SO}_{2} \\mathrm{Cl}_{2}, \\mathrm{XeF}_{2}, \\mathrm{SF}_{4}, \\mathrm{ClF}_{3}, \\mathrm{Ni}(\\mathrm{CO})_{4}, \\mathrm{XeO}_{2} \\mathrm{~F}_{2},\\left[\\mathrm{PtCl}_{4}\\right]^{2-}, \\mathrm{XeF}_{4}$, and $\\mathrm{SOCl}_{2}$, what is the total number of species having $s p^{3}$ hybridised central atom?", "gold": "5" }, { "description": "JEE Adv 2023 Paper 2", "index": 44, "subject": "chem", "type": "Integer", "question": "Consider the following molecules: $\\mathrm{Br}_{3} \\mathrm{O}_{8}, \\mathrm{~F}_{2} \\mathrm{O}, \\mathrm{H}_{2} \\mathrm{~S}_{4} \\mathrm{O}_{6}, \\mathrm{H}_{2} \\mathrm{~S}_{5} \\mathrm{O}_{6}$, and $\\mathrm{C}_{3} \\mathrm{O}_{2}$.\n\n\n\nCount the number of atoms existing in their zero oxidation state in each molecule. What is their sum?", "gold": "6" }, { "description": "JEE Adv 2023 Paper 2", "index": 46, "subject": "chem", "type": "Integer", "question": "$50 \\mathrm{~mL}$ of 0.2 molal urea solution (density $=1.012 \\mathrm{~g} \\mathrm{~mL}^{-1}$ at $300 \\mathrm{~K}$ ) is mixed with $250 \\mathrm{~mL}$ of a solution containing $0.06 \\mathrm{~g}$ of urea. Both the solutions were prepared in the same solvent. What is the osmotic pressure (in Torr) of the resulting solution at $300 \\mathrm{~K}$?\n\n\n\n[Use: Molar mass of urea $=60 \\mathrm{~g} \\mathrm{~mol}^{-1}$; gas constant, $\\mathrm{R}=62$ L Torr K${ }^{-1} \\mathrm{~mol}^{-1}$;\n\n\n\nAssume, $\\Delta_{\\text {mix }} \\mathrm{H}=0, \\Delta_{\\text {mix }} \\mathrm{V}=0$ ]", "gold": "682" }, { "description": "JEE Adv 2023 Paper 2", "index": 50, "subject": "chem", "type": "Numeric", "question": "A trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\\mathrm{Sn} / \\mathrm{HCl}$ gives a major product, which on treatment with an excess of $\\mathrm{NaNO}_{2} / \\mathrm{HCl}$ at $0{ }^{\\circ} \\mathrm{C}$ provides $\\mathbf{P}$ as the product. $\\mathbf{P}$, upon treatment with excess of $\\mathrm{H}_{2} \\mathrm{O}$ at room temperature, gives the product $\\mathbf{Q}$. Bromination of $\\mathbf{Q}$ in aqueous medium furnishes the product $\\mathbf{R}$. The compound $\\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\\mathbf{S}$.\n\n\n\nThe molar mass difference between compounds $\\mathbf{Q}$ and $\\mathbf{R}$ is $474 \\mathrm{~g} \\mathrm{~mol}^{-1}$ and between compounds $\\mathbf{P}$ and $\\mathbf{S}$ is $172.5 \\mathrm{~g} \\mathrm{~mol}^{-1}$. What is the number of heteroatoms present in one molecule of $\\mathbf{R}$?\n\n\n\n[Use: Molar mass (in g mol${ }^{-1}$ ): $\\mathrm{H}=1, \\mathrm{C}=12, \\mathrm{~N}=14, \\mathrm{O}=16, \\mathrm{Br}=80, \\mathrm{Cl}=35.5$\n\n\n\nAtoms other than $\\mathrm{C}$ and $\\mathrm{H}$ are considered as heteroatoms]", "gold": "9" }, { "description": "JEE Adv 2023 Paper 2", "index": 51, "subject": "chem", "type": "Numeric", "question": "A trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\\mathrm{Sn} / \\mathrm{HCl}$ gives a major product, which on treatment with an excess of $\\mathrm{NaNO}_{2} / \\mathrm{HCl}$ at $0{ }^{\\circ} \\mathrm{C}$ provides $\\mathbf{P}$ as the product. $\\mathbf{P}$, upon treatment with excess of $\\mathrm{H}_{2} \\mathrm{O}$ at room temperature, gives the product $\\mathbf{Q}$. Bromination of $\\mathbf{Q}$ in aqueous medium furnishes the product $\\mathbf{R}$. The compound $\\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\\mathbf{S}$.\n\n\n\nThe molar mass difference between compounds $\\mathbf{Q}$ and $\\mathbf{R}$ is $474 \\mathrm{~g} \\mathrm{~mol}^{-1}$ and between compounds $\\mathbf{P}$ and $\\mathbf{S}$ is $172.5 \\mathrm{~g} \\mathrm{~mol}^{-1}$. What is the total number of carbon atoms and heteroatoms present in one molecule of $\\mathbf{S}$?\n\n\n\n[Use: Molar mass (in $\\mathrm{g} \\mathrm{mol}^{-1}$ ): $\\mathrm{H}=1, \\mathrm{C}=12, \\mathrm{~N}=14, \\mathrm{O}=16, \\mathrm{Br}=80, \\mathrm{Cl}=35.5$\n\n\n\nAtoms other than $\\mathrm{C}$ and $\\mathrm{H}$ are considered as heteroatoms]", "gold": "51" } ]