# Datasets: daman1209arora /jeebench

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phy
B
1
MCQ
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 : \begin{center} \begin{tabular}{cc} \hline $\lambda(\mu \mathrm{m})$ & $V_{0}($ Volt $)$ \\ \hline 0.3 & 2.0 \\ 0.4 & 1.0 \\ 0.5 & 0.4 \\ \hline \end{tabular} \end{center} Given 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 (A) $6.0 \times 10^{-34}$ (B) $6.4 \times 10^{-34}$ (C) $6.6 \times 10^{-34}$ (D) $6.8 \times 10^{-34}$
phy
D
2
MCQ
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 $h(<l)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $30^{\circ}$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $h / l$ and the frictional force $f$ at the bottom of the stick are $\left(g=10 \mathrm{~ms} \mathrm{~s}^{2}\right)$ (A) $\frac{h}{l}=\frac{\sqrt{3}}{16}, f=\frac{16 \sqrt{3}}{3} \mathrm{~N}$ (B) $\frac{h}{l}=\frac{3}{16}, f=\frac{16 \sqrt{3}}{3} \mathrm{~N}$ (C) $\frac{h}{l}=\frac{3 \sqrt{3}}{16}, f=\frac{8 \sqrt{3}}{3} \mathrm{~N}$ (D) $\frac{h}{l}=\frac{3 \sqrt{3}}{16}, f=\frac{16 \sqrt{3}}{3} \mathrm{~N}$
phy
ABD
6
MCQ(multiple)
Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge $Z e$ are defined by their principal quantum number $n$, where $n \gg 1$. Which of the following statement(s) is(are) true? (A) Relative change in the radii of two consecutive orbitals does not depend on $Z$ (B) Relative change in the radii of two consecutive orbitals varies as $1 / n$ (C) Relative change in the energy of two consecutive orbitals varies as $1 / n^{3}$ (D) Relative change in the angular momenta of two consecutive orbitals varies as $1 / n$
phy
CD
8
MCQ(multiple)
An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true? (A) The temperature distribution over the filament is uniform (B) The resistance over small sections of the filament decreases with time (C) The filament emits more light at higher band of frequencies before it breaks up (D) The filament consumes less electrical power towards the end of the life of the bulb
phy
9
MCQ(multiple)
A plano-convex lens is made of a material of refractive index $n$. When a small object is placed $30 \mathrm{~cm}$ away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of $10 \mathrm{~cm}$ away from the lens. Which of the following statement(s) is(are) true? (A) The refractive index of the lens is 2.5 (B) The radius of curvature of the convex surface is $45 \mathrm{~cm}$ (C) The faint image is erect and real (D) The focal length of the lens is $20 \mathrm{~cm}$
phy
BD
10
MCQ(multiple)
A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material, Boltzmann constant $\left(k_{B}\right)$, the absolute temperature $(T)$, the number per unit volume $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particles. Which of the following expression(s) for $l$ is(are) dimensionally correct? (A) $l=\sqrt{\left(\frac{n q^{2}}{\varepsilon k_{B} T}\right)}$ (B) $l=\sqrt{\left(\frac{\varepsilon k_{B} T}{n q^{2}}\right)}$ (C) $l=\sqrt{\left(\frac{q^{2}}{\varepsilon n^{2 / 3} k_{B} T}\right)}$ (D) $l=\sqrt{\left(\frac{q^{2}}{\varepsilon n^{1 / 3} k_{B} T}\right)}$
phy
ABD
12
MCQ(multiple)
The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation $\vec{r}(t)=\alpha t^{3} \hat{i}+\beta t^{2} \hat{j}$ where $\alpha=10 / 3 \mathrm{~m} \mathrm{~s}^{-3}, \beta=5 \mathrm{~m} \mathrm{~s}^{-2}$ and $m=0.1 \mathrm{~kg}$. At $t=1 \mathrm{~s}$, which of the following statement(s) is(are) true about the particle? (A) The velocity $\vec{v}$ is given by $\vec{v}=(10 \hat{i}+10 \hat{j}) \mathrm{ms}^{-1}$ (B) The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L}=-(5 / 3) \hat{k} \mathrm{~N} \mathrm{~m}$ (C) The force $\vec{F}$ is given by $\vec{F}=(\hat{i}+2 \hat{j}) \mathrm{N}$ (D) The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau}=-(20 / 3) \hat{k} \mathrm{~N} \mathrm{~m}$
phy
9
14
Integer
A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated $(P)$ by the metal. The sensor has a scale that displays $\log _{2}\left(P / P_{0}\right)$, where $P_{0}$ is a constant. When the metal surface is at a temperature of $487^{\circ} \mathrm{C}$, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767{ }^{\circ} \mathrm{C}$ ?
phy
9
15
Integer
The isotope ${ }_{5}^{12} \mathrm{~B}$ having a mass 12.014 u undergoes $\beta$-decay to ${ }_{6}^{12} \mathrm{C}$. ${ }_{6}^{12} \mathrm{C}$ has an excited state of the nucleus $\left({ }_{6}^{12} \mathrm{C}^{*}\right)$ at $4.041 \mathrm{MeV}$ above its ground state. If ${ }_{5}^{12} \mathrm{~B}$ decays to ${ }_{6}^{12} \mathrm{C}^{*}$, what is the maximum kinetic energy of the $\beta$-particle in units of $\mathrm{MeV}$? $\left(1 \mathrm{u}=931.5 \mathrm{MeV} / c^{2}\right.$, where $c$ is the speed of light in vacuum).
phy
6
16
Integer
A hydrogen atom in its ground state is irradiated by light of wavelength 970 A. Taking $h c / e=1.237 \times 10^{-6} \mathrm{eV} \mathrm{m}$ and the ground state energy of hydrogen atom as $-13.6 \mathrm{eV}$, what is the number of lines present in the emission spectrum?
phy
3
17
Integer
Consider two solid spheres $\mathrm{P}$ and $\mathrm{Q}$ each of density $8 \mathrm{gm} \mathrm{cm}^{-3}$ and diameters $1 \mathrm{~cm}$ and $0.5 \mathrm{~cm}$, respectively. Sphere $P$ is dropped into a liquid of density $0.8 \mathrm{gm} \mathrm{cm}^{-3}$ and viscosity $\eta=3$ poiseulles. Sphere $Q$ is dropped into a liquid of density $1.6 \mathrm{gm} \mathrm{cm}^{-3}$ and viscosity $\eta=2$ poiseulles. What is the ratio of the terminal velocities of $P$ and $Q$?
phy
8
18
Integer
Two inductors $L_{1}$ (inductance $1 \mathrm{mH}$, internal resistance $3 \Omega$ ) and $L_{2}$ (inductance $2 \mathrm{mH}$, internal resistance $4 \Omega$ ), and a resistor $R$ (resistance $12 \Omega$ ) are all connected in parallel across a $5 \mathrm{~V}$ battery. The circuit is switched on at time $t=0$. What is the ratio of the maximum to the minimum current $\left(I_{\max } / I_{\min }\right)$ drawn from the battery?
chem
C
20
MCQ
One mole of an ideal gas at $300 \mathrm{~K}$ in thermal contact with surroundings expands isothermally from $1.0 \mathrm{~L}$ to $2.0 \mathrm{~L}$ against a constant pressure of $3.0 \mathrm{~atm}$. In this process, the change in entropy of surroundings $\left(\Delta S_{\text {surr }}\right)$ in $\mathrm{J} \mathrm{K}^{-1}$ is (1 $\mathrm{L} \operatorname{atm}=101.3 \mathrm{~J})$ (A) 5.763 (B) 1.013 (C) -1.013 (D) -5.763
chem
B
21
MCQ
The increasing order of atomic radii of the following Group 13 elements is (A) $\mathrm{Al}<\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}$ (B) $\mathrm{Ga}<\mathrm{Al}<\mathrm{In}<\mathrm{Tl}$ (C) $\mathrm{Al}<\mathrm{In}<\mathrm{Ga}<\mathrm{Tl}$ (D) $\mathrm{Al}<\mathrm{Ga}<\mathrm{Tl}<\mathrm{In}$
chem
B
22
MCQ
Among $[\mathrm{Ni(CO)}_{4}]$, $[\mathrm{NiCl}_{4}]^{2-}$, $[\mathrm{Co(NH_3)_{4}Cl_2}]\mathrm{Cl}$, $\mathrm{Na_3}[\mathrm{CoF_6}]$, $\mathrm{Na_2O_2}$ and $\mathrm{CsO_2}$ number of paramagnetic compounds is (A) 2 (B) 3 (C) 4 (D) 5
chem
A
23
MCQ
On complete hydrogenation, natural rubber produces (A) ethylene-propylene copolymer (B) vulcanised rubber (C) polypropylene (D) polybutylene
chem
BCD
24
MCQ(multiple)
According to the Arrhenius equation, (A) a high activation energy usually implies a fast reaction. (B) rate constant increases with increase in temperature. This is due to a greater number of collisions whose energy exceeds the activation energy. (C) higher the magnitude of activation energy, stronger is the temperature dependence of the rate constant. (D) the pre-exponential factor is a measure of the rate at which collisions occur, irrespective of their energy.
chem
BD
25
MCQ(multiple)
A plot of the number of neutrons $(N)$ against the number of protons $(P)$ of stable nuclei exhibits upward deviation from linearity for atomic number, $Z>20$. For an unstable nucleus having $N / P$ ratio less than 1 , the possible mode(s) of decay is(are) (A) $\beta^{-}$-decay $(\beta$ emission) (B) orbital or $K$-electron capture (C) neutron emission (D) $\beta^{+}$-decay (positron emission)
chem
ACD
26
MCQ(multiple)
The crystalline form of borax has (A) tetranuclear $\left[\mathrm{B}_{4} \mathrm{O}_{5}(\mathrm{OH})_{4}\right]^{2-}$ unit (B) all boron atoms in the same plane (C) equal number of $s p^{2}$ and $s p^{3}$ hybridized boron atoms (D) one terminal hydroxide per boron atom
chem
BC
27
MCQ(multiple)
The compound(s) with TWO lone pairs of electrons on the central atom is(are) (A) $\mathrm{BrF}_{5}$ (B) $\mathrm{ClF}_{3}$ (C) $\mathrm{XeF}_{4}$ (D) $\mathrm{SF}_{4}$
chem
A
28
MCQ(multiple)
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) (A) $\mathrm{CuCl}_{2}$ (B) $\mathrm{BaCl}_{2}$ (C) $\mathrm{Pb}\left(\mathrm{OOCCH}_{3}\right)_{2}$ (D) $\mathrm{Na}_{2}\left[\mathrm{Fe}(\mathrm{CN})_{5} \mathrm{NO}\right]$
chem
9
32
Integer
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, $\left(\frac{M W_{\text {solute }}}{M W_{\text {solvent }}}\right)$?
chem
4
33
Integer
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 $x$?
chem
6
34
Integer
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 $\mathbf{X}$?
math
C
37
MCQ
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 (A) $2(\sec \theta-\tan \theta)$ (B) $2 \sec \theta$ (C) $-2 \tan \theta$ (D) 0
math
A
38
MCQ
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 (A) 380 (B) 320 (C) 260 (D) 95
math
C
39
MCQ
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 (A) $-\frac{7 \pi}{9}$ (B) $-\frac{2 \pi}{9}$ (C) 0 (D) $\frac{5 \pi}{9}$
math
C
40
MCQ
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 $P$ (computer turns out to be defective given that it is produced in plant $T_{1}$ ) $=10 P\left(\right.$ computer turns out to be defective given that it is produced in plant $\left.T_{2}\right)$, where $P(E)$ 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 (A) $\frac{36}{73}$ (B) $\frac{47}{79}$ (C) $\frac{78}{93}$ (D) $\frac{75}{83}$
math
C
41
MCQ
The least value of $\alpha \in \mathbb{R}$ for which $4 \alpha x^{2}+\frac{1}{x} \geq 1$, for all $x>0$, is (A) $\frac{1}{64}$ (B) $\frac{1}{32}$ (C) $\frac{1}{27}$ (D) $\frac{1}{25}$
math
BCD
42
MCQ(multiple)
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 (A) the acute angle between $O Q$ and $O S$ is $\frac{\pi}{3}$ (B) the equation of the plane containing the triangle $O Q S$ is $x-y=0$ (C) the length of the perpendicular from $P$ to the plane containing the triangle $O Q S$ is $\frac{3}{\sqrt{2}}$ (D) the perpendicular distance from $O$ to the straight line containing $R S$ is $\sqrt{\frac{15}{2}}$
math
A
43
MCQ(multiple)
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 (A) $\lim _{x \rightarrow 0+} f^{\prime}\left(\frac{1}{x}\right)=1$ (B) $\lim _{x \rightarrow 0+} x f\left(\frac{1}{x}\right)=2$ (C) $\lim _{x \rightarrow 0+} x^{2} f^{\prime}(x)=0$ (D) $|f(x)| \leq 2$ for all $x \in(0,2)$
math
BC
44
MCQ(multiple)
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 (A) $\alpha=0, k=8$ (B) $4 \alpha-k+8=0$ (C) $\operatorname{det}(P \operatorname{adj}(Q))=2^{9}$ (D) $\operatorname{det}(Q \operatorname{adj}(P))=2^{13}$
math
ACD
45
MCQ(multiple)
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 (A) area of the triangle $X Y Z$ is $6 \sqrt{6}$ (B) the radius of circumcircle of the triangle $X Y Z$ is $\frac{35}{6} \sqrt{6}$ (C) $\sin \frac{X}{2} \sin \frac{Y}{2} \sin \frac{Z}{2}=\frac{4}{35}$ (D) $\sin ^{2}\left(\frac{X+Y}{2}\right)=\frac{3}{5}$
math
46
MCQ(multiple)
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 (A) intersects $y=x+2$ exactly at one point (B) intersects $y=x+2$ exactly at two points (C) intersects $y=(x+2)^{2}$ (D) does NO'T intersect $y=(x+3)^{2}$
math
BC
47
MCQ(multiple)
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 (A) $\quad g^{\prime}(2)=\frac{1}{15}$ (B) $h^{\prime}(1)=666$ (C) $h(0)=16$ (D) $h(g(3))=36$
math
ABC
48
MCQ(multiple)
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 (A) $Q_{2} Q_{3}=12$ (B) $\quad R_{2} R_{3}=4 \sqrt{6}$ (C) area of the triangle $O R_{2} R_{3}$ is $6 \sqrt{2}$ (D) area of the triangle $P Q_{2} Q_{3}$ is $4 \sqrt{2}$
math
AC
49
MCQ(multiple)
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) (A) $\left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)$ (B) $\left(\frac{1}{4}, \frac{1}{2}\right)$ (C) $\left(\frac{1}{3},-\frac{1}{\sqrt{3}}\right)$ (D) $\left(\frac{1}{4},-\frac{1}{2}\right)$
math
2
50
Integer
What is the total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc}x & x^{2} & 1+x^{3} \\ 2 x & 4 x^{2} & 1+8 x^{3} \\ 3 x & 9 x^{2} & 1+27 x^{3}\end{array}\right|=10$?
math
5
51
Integer
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 $n$?
math
1
52
Integer
What is the total number of distinct $x \in[0,1]$ for which $\int_{0}^{x} \frac{t^{2}}{1+t^{4}} d t=2 x-1$?
math
7
53
Integer
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 $6(\alpha+\beta)$?
math
1
54
Integer
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 $P^{2}=-I$?
phy
C
1
MCQ
The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by $E=\frac{3}{5} \frac{Z(Z-1) e^{2}}{4 \pi \varepsilon_{0} R}$ The 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 $\left(1 \mathrm{fm}=10^{-15} \mathrm{~m}\right)$ (A) $2.85 \mathrm{fm}$ (B) $3.03 \mathrm{fm}$ (C) $3.42 \mathrm{fm}$ (D) $3.80 \mathrm{fm}$
phy
C
2
MCQ
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? (A) 64 (B) 90 (C) 108 (D) 120
phy
C
3
MCQ
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 (A) $112 \mathrm{~J}$ (B) $294 \mathrm{~J}$ (C) $588 \mathrm{~J}$ (D) $813 \mathrm{~J}$
phy
A
4
MCQ
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 (A) $0.78 \mathrm{~mm}$ (B) $0.90 \mathrm{~mm}$ (C) $1.56 \mathrm{~mm}$ (D) $2.34 \mathrm{~mm}$
phy
ABD
9
MCQ(multiple)
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? (A) The error in the measurement of $r$ is $10 \%$ (B) The error in the measurement of $T$ is $3.57 \%$ (C) The error in the measurement of $T$ is $2 \%$ (D) The error in the determined value of $g$ is $11 \%$
phy
BC
10
MCQ(multiple)
Consider two identical galvanometers and two identical resistors with resistance $R$. If the internal resistance of the galvanometers $R_{\mathrm{C}}<R / 2$, which of the following statement(s) about any one of the galvanometers is(are) true? (A) The maximum voltage range is obtained when all the components are connected in series (B) The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer (C) The maximum current range is obtained when all the components are connected in parallel (D) The maximum current range is obtained when the two galvanometers are connected in series and the combination is connected in parallel with both the resistors
phy
ABD
12
MCQ(multiple)
A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_{0}$. Consider two cases: (i) when the block is at $x_{0}$; and (ii) when the block is at $x=x_{0}+A$. In both the cases, a particle with mass $m(<M)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass $m$ is placed on the mass $M$ ? (A) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged (B) The final time period of oscillation in both the cases is same (C) The total energy decreases in both the cases (D) The instantaneous speed at $x_{0}$ of the combined masses decreases in both the cases
chem
D
19
MCQ
For the following electrochemical cell at $298 \mathrm{~K}$, $\mathrm{Pt}(s) \mid \mathrm{H}_{2}(g, 1$ bar $)\left|\mathrm{H}^{+}(aq, 1 \mathrm{M}) \| \mathrm{M}^{4+}(aq), \mathrm{M}^{2+}(aq)\right| \operatorname{Pt}(s)$ $E_{\text {cell }}=0.092 \mathrm{~V}$ when $\frac{\left[\mathrm{M}^{2+}(aq)\right]}{\left[M^{4+}(aq)\right]}=10^{x}$. Given : $E_{\mathrm{M}^{4} / \mathrm{M}^{2+}}^{0}=0.151 \mathrm{~V} ; 2.303 \frac{R T}{F}=0.059 \mathrm{~V}$ The value of $x$ is (A) -2 (B) -1 (C) 1 (D) 2
chem
A
22
MCQ
The geometries of the ammonia complexes of $\mathrm{Ni}^{2+}, \mathrm{Pt}^{2+}$ and $\mathrm{Zn}^{2+}$, respectively, are (A) octahedral, square planar and tetrahedral (B) square planar, octahedral and tetrahedral (C) tetrahedral, square planar and octahedral (D) octahedral, tetrahedral and square planar
chem
AC
25
MCQ(multiple)
According to Molecular Orbital Theory, (A) $\mathrm{C}_{2}^{2-}$ is expected to be diamagnetic (B) $\mathrm{O}_{2}^{2+}$ is expected to have a longer bond length than $\mathrm{O}_{2}$ (C) $\mathrm{N}_{2}^{+}$and $\mathrm{N}_{2}^{-}$have the same bond order (D) $\mathrm{He}_{2}^{+}$has the same energy as two isolated He atoms
chem
AB
26
MCQ(multiple)
Mixture(s) showing positive deviation from Raoult's law at $35^{\circ} \mathrm{C}$ is(are) (A) carbon tetrachloride + methanol (B) carbon disulphide + acetone (C) benzene + toluene (D) phenol + aniline
chem
BCD
27
MCQ(multiple)
The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are) (A) The number of the nearest neighbours of an atom present in the topmost layer is 12 (B) The efficiency of atom packing is $74 \%$ (C) The number of octahedral and tetrahedral voids per atom are 1 and 2 , respectively (D) The unit cell edge length is $2 \sqrt{2}$ times the radius of the atom
chem
ABC
28
MCQ(multiple)
Extraction of copper from copper pyrite $\left(\mathrm{CuFeS}_{2}\right)$ involves (A) crushing followed by concentration of the ore by froth-flotation (B) removal of iron as slag (C) self-reduction step to produce 'blister copper' following evolution of $\mathrm{SO}_{2}$ (D) refining of 'blister copper' by carbon reduction
chem
BD
29
MCQ(multiple)
The nitrogen containing compound produced in the reaction of $\mathrm{HNO}_{3}$ with $\mathrm{P}_{4} \mathrm{O}_{10}$ (A) can also be prepared by reaction of $\mathrm{P}_{4}$ and $\mathrm{HNO}_{3}$ (B) is diamagnetic (C) contains one $\mathrm{N}-\mathrm{N}$ bond (D) reacts with $\mathrm{Na}$ metal producing a brown gas
chem
BC
30
MCQ(multiple)
For 'invert sugar', the correct statement(s) is(are) (Given: specific rotations of (+)-sucrose, (+)-maltose, L-(-)-glucose and L-(+)-fructose in aqueous solution are $+66^{\circ},+140^{\circ},-52^{\circ}$ and $+92^{\circ}$, respectively) (A) 'invert sugar' is prepared by acid catalyzed hydrolysis of maltose (B) 'invert sugar' is an equimolar mixture of $\mathrm{D}$-(+)-glucose and D-(-)-fructose (C) specific rotation of 'invert sugar' is $-20^{\circ}$ (D) on reaction with $\mathrm{Br}_{2}$ water, 'invert sugar' forms saccharic acid as one of the products
math
B
37
MCQ
Let $P=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{array}\right]$ and $I$ be the identity matrix of order 3. If $Q=\left[q_{i j}\right]$ is a matrix such that $P^{50}-Q=I$, then $\frac{q_{31}+q_{32}}{q_{21}}$ equals (A) 52 (B) 103 (C) 201 (D) 205
math
B
38
MCQ
Let $b_{i}>1$ 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 (A) $s>t$ and $a_{101}>b_{101}$ (B) $s>t$ and $a_{101}<b_{101}$ (C) $s<t$ and $a_{101}>b_{101}$ (D) $s<t$ and $a_{101}<b_{101}$
math
A
40
MCQ
The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{2} \cos x}{1+e^{x}} d x$ is equal to (A) $\frac{\pi^{2}}{4}-2$ (B) $\frac{\pi^{2}}{4}+2$ (C) $\pi^{2}-e^{\frac{\pi}{2}}$ (D) $\pi^{2}+e^{\frac{\pi}{2}}$
math
C
42
MCQ
Let $P$ be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is (A) $x+y-3 z=0$ (B) $3 x+z=0$ (C) $x-4 y+7 z=0$ (D) $2 x-y=0$
math
AB
44
MCQ(multiple)
Let $a, b \in \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=a \cos \left(\left|x^{3}-x\right|\right)+b|x| \sin \left(\left|x^{3}+x\right|\right)$. Then $f$ is (A) differentiable at $x=0$ if $a=0$ and $b=1$ (B) differentiable at $x=1$ if $a=1$ and $b=0$ (C) NOT differentiable at $x=0$ if $a=1$ and $b=0$ (D) NOT differentiable at $x=1$ if $a=1$ and $b=1$
math
45
MCQ(multiple)
Let $f: \mathbb{R} \rightarrow(0, \infty)$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be twice differentiable functions such that $f^{\prime \prime}$ and $g^{\prime \prime}$ are continuous functions on $\mathbb{R}$. Suppose $f^{\prime}(2)=g(2)=0, f^{\prime \prime}(2) \neq 0$ and $g^{\prime}(2) \neq 0$. If $\lim _{x \rightarrow 2} \frac{f(x) g(x)}{f^{\prime}(x) g^{\prime}(x)}=1$, then (A) $f$ has a local minimum at $x=2$ (B) f has a local maximum at $x=2$ (C) $f^{\prime \prime}(2)>f(2)$ (D) $f(x)-f^{\prime \prime}(x)=0$ for at least one $x \in \mathbb{R}$
math
BC
46
MCQ(multiple)
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 (A) $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$ (B) $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$ (C) $g$ is NOT differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$ (D) $g$ is NOT differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$
math
ACD
47
MCQ(multiple)
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 (A) the circle with radius $\frac{1}{2 a}$ and centre $\left(\frac{1}{2 a}, 0\right)$ for $a>0, b \neq 0$ (B) the circle with radius $-\frac{1}{2 a}$ and centre $\left(-\frac{1}{2 a}, 0\right)$ for $a<0, b \neq 0$ (C) the $x$-axis for $a \neq 0, b=0$ (D) the $y$-axis for $a=0, b \neq 0$
math
ACD
48
MCQ(multiple)
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 (A) $S P=2 \sqrt{5}$ (B) $S Q: Q P=(\sqrt{5}+1): 2$ (C) the $x$-intercept of the normal to the parabola at $P$ is 6 (D) the slope of the tangent to the circle at $Q$ is $\frac{1}{2}$
math
BCD
49
MCQ(multiple)
Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations \begin{aligned} & a x+2 y=\lambda \\ & 3 x-2 y=\mu \end{aligned} Which of the following statement(s) is(are) correct? (A) If $a=-3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$ (B) If $a \neq-3$, then the system has a unique solution for all values of $\lambda$ and $\mu$ (C) If $\lambda+\mu=0$, then the system has infinitely many solutions for $a=-3$ (D) If $\lambda+\mu \neq 0$, then the system has no solution for $\alpha=-3$
math
BC
50
MCQ(multiple)
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 statementís) is(are) correct? (A) There is exactly one choice for such $\vec{v}$ (B) There are infinitely many choices for such $\vec{v}$ (C) If $\hat{u}$ lies in the $x y$-plane then $\left|u_{1}\right|=\left|u_{2}\right|$ (D) If $\hat{u}$ lies in the $x z$-plane then $2\left|u_{1}\right|=\left|u_{3}\right|$
phy
ABD
1
MCQ(multiple)
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? [A] The pressure difference between the leading and trailing faces of the plate is proportional to $u v$ [B] The resistive force experienced by the plate is proportional to $v$ [C] The plate will continue to move with constant non-zero acceleration, at all times [D] At a later time the external force $F$ balances the resistive force
phy
C
4
MCQ(multiple)
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? [A] The amount of energy radiated by the body in 1 second is close to 60 Joules [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 [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 [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
phy
ACD
7
MCQ(multiple)
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? [A] For the angle of incidence $i_{1}=A$, the ray inside the prism is parallel to the base of the prism [B] For this prism, the refractive index $\mu$ and the angle of prism $A$ are related as $A=\frac{1}{2} \cos ^{-1}\left(\frac{\mu}{2}\right)$ [C] At minimum deviation, the incident angle $i_{1}$ and the refracting angle $r_{1}$ at the first refracting surface are related by $r_{1}=\left(i_{1} / 2\right)$ [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 $i_{1}=\sin ^{-1}\left[\sin A \sqrt{4 \cos ^{2} \frac{A}{2}-1}-\cos A\right]$
phy
6
8
Integer
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 $\alpha$?
phy
5
9
Integer
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 $n_{f}$?
phy
6
11
Integer
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).
phy
5
12
Integer
${ }^{131} \mathrm{I}$ 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 $\ln 2 \approx 0.7$ ).
chem
ABC
19
MCQ(multiple)
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) [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 $\mathrm{p}_{1}$ [B] If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic [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 [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 $\mathrm{T}_{1} \neq \mathrm{T}_{2}$
chem
ABD
21
MCQ(multiple)
The correct statement(s) about the oxoacids, $\mathrm{HClO}_{4}$ and $\mathrm{HClO}$, is(are) [A] The central atom in both $\mathrm{HClO}_{4}$ and $\mathrm{HClO}$ is $s p^{3}$ hybridized [B] $\mathrm{HClO}_{4}$ is more acidic than $\mathrm{HClO}$ because of the resonance stabilization of its anion [C] $\mathrm{HClO}_{4}$ is formed in the reaction between $\mathrm{Cl}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$ [D] The conjugate base of $\mathrm{HClO}_{4}$ is weaker base than $\mathrm{H}_{2} \mathrm{O}$
chem
CD
22
MCQ(multiple)
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 [A] the physical state of $\mathrm{X}_{2}$ at room temperature changes from gas to solid down the group [B] decrease in ionization energy down the group [C] decrease in $\pi^{*}-\sigma^{*}$ gap down the group [D] decrease in HOMO-LUMO gap down the group
chem
BCD
23
MCQ(multiple)
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 $\mathbf{Y}$. Among the following options, which statement(s) is(are) correct? [A] Addition of silver nitrate to $\mathbf{Y}$ gives only two equivalents of silver chloride [B] The hybridization of the central metal ion in $\mathbf{Y}$ is $\mathrm{d}^{2} \mathrm{sp}^{3}$ [C] $\mathbf{Z}$ is a tetrahedral complex [D] When $\mathbf{X}$ and $\mathbf{Z}$ are in equilibrium at $0^{\circ} \mathrm{C}$, the colour of the solution is pink
chem
2
26
Integer
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 $N$?
chem
6
27
Integer
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 $Z$?
chem
6
28
Integer
What is the sum of the number of lone pairs of electrons on each central atom in the following species? $\left[\mathrm{TeBr}_{6}\right]^{2-},\left[\mathrm{BrF}_{2}\right]^{+}, \mathrm{SNF}_{3}$, and $\left[\mathrm{XeF}_{3}\right]^{-}$ (Atomic numbers: $\mathrm{N}=7, \mathrm{~F}=9, \mathrm{~S}=16, \mathrm{Br}=35, \mathrm{Te}=52, \mathrm{Xe}=54$ )
chem
5
29
Integer
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: $\mathrm{H}=1, \mathrm{He}=2, \mathrm{Li}=3, \mathrm{Be}=4, \mathrm{~B}=5, \mathrm{C}=6, \mathrm{~N}=7, \mathrm{O}=8, \mathrm{~F}=9$ )
math
ABC
37
MCQ(multiple)
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? [A] $a, 4,1$ [B] $a, 4,2$ [C] $2 a, 8,1$ [D] $2 a, 4,1$
math
C
38
MCQ(multiple)
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 $k$ ? [A] $p=-2, h=2, k=-4$ [B] $p=-1, h=1, k=-3$ [C] $p=2, h=3, k=-4$ [D] $p=5, h=4, k=-3$
math
AB
40
MCQ(multiple)
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 $(0,1)$ ? [A] $x^{9}-f(x)$ [B] $x-\int_{0}^{\frac{\pi}{2}-x} f(t) \cos t d t$ [C] e^{x}-\int_{0}^{x} f(t) \sin t d t$[D] f(x)+\int_{0}^{\frac{\pi}{2}} f(t) \sin t d t$
math
BD
41
MCQ(multiple)
Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries? [A]$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] [B]$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]$[C]$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right] [D]$\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$
math
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 $x ?$ [A]$-1+\sqrt{1-y^{2}}$ [B]$-1-\sqrt{1-y^{2}}$ [C]$1+\sqrt{1+y^{2}}$ [D]$1-\sqrt{1+y^{2}}$
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 [A] $P(Y)=\frac{4}{15}$ [B] $P\left(X^{\prime} \mid Y\right)=\frac{1}{2}$ [C] \quad P(X \cap Y)=\frac{1}{5}$[D]$P(X \cup Y)=\frac{2}{5}$math JEE Adv 2017 Paper 1 2 44 Integer For how many values of$p$, the circle$x^{2}+y^{2}+2 x+4 y-p=0$and the coordinate axes have exactly three common points? math JEE Adv 2017 Paper 1 2 45 Integer Let$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 $g(x)=\int_{x}^{\frac{\pi}{2}}\left[f^{\prime}(t) \operatorname{cosec} t-\cot t \operatorname{cosec} t f(t)\right] d t$ for$x \in\left(0, \frac{\pi}{2}\right]$, then what is the$\lim _{x \rightarrow 0} g(x)$? math JEE Adv 2017 Paper 1 1 46 Integer For a real number$\alpha$, if the system $\left[\begin{array}{ccc} 1 & \alpha & \alpha^{2} \\ \alpha & 1 & \alpha \\ \alpha^{2} & \alpha & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right]$ of linear equations, has infinitely many solutions, then what is the value of$1+\alpha+\alpha^{2}$? math JEE Adv 2017 Paper 1 5 47 Integer 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$\frac{y}{9 x}$? math JEE Adv 2017 Paper 1 6 48 Integer 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? phy JEE Adv 2017 Paper 2 A 1 MCQ 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 [A]$R$[B]$R^{3}$[C]$\frac{1}{R}$[D]$R^{2 / 3}$phy JEE Adv 2017 Paper 2 D 3 MCQ 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$[\mathrm{A}] \quad \lambda_{d} / \lambda[\mathrm{B}] \quad \lambda_{d}^{2} / \lambda^{2}[\mathrm{C}] \lambda_{d}^{3} / \lambda[\mathrm{D}] \lambda_{d}^{3} / \lambda^{2}$phy JEE Adv 2017 Paper 2 B 6 MCQ 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 (Ignore the rotation and revolution of the Earth and the presence of any other planet)$[\mathrm{A}] \quad v_{S}=22 \mathrm{~km} \mathrm{~s}^{-1}[\mathrm{B}] v_{S}=42 \mathrm{~km} \mathrm{~s}^{-1}[\mathrm{C}] \quad v_{S}=62 \mathrm{~km} \mathrm{~s}^{-1}$[D]$v_{S}=72 \mathrm{~km} \mathrm{~s}^{-1}$phy JEE Adv 2017 Paper 2 B 7 MCQ 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 [A]$0.2 \%$[B]$1 \%$[C]$3 \%$[D]$5 \%$phy JEE Adv 2017 Paper 2 AD 9 MCQ(multiple) The instantaneous voltages at three terminals marked$X, Y$and$Zare given by \begin{aligned} & V_{X}=V_{0} \sin \omega t, \\ & V_{Y}=V_{0} \sin \left(\omega t+\frac{2 \pi}{3}\right) \text { and } \\ & V_{Z}=V_{0} \sin \left(\omega t+\frac{4 \pi}{3}\right) . \end{aligned} An ideal voltmeter is configured to read\mathrm{rms}$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 [A]$\quad V_{X Y}^{r m s}=V_{0} \sqrt{\frac{3}{2}}$[B]$\quad V_{Y Z}^{r m s}=V_{0} \sqrt{\frac{1}{2}}$[C]$\quad V_{X Y}^{r m s}=V_{0}$[D] independent of the choice of the two terminals chem JEE Adv 2017 Paper 2 B 20 MCQ For the following cell, $\mathrm{Zn}(s)\left|\mathrm{ZnSO}_{4}(a q) \| \mathrm{CuSO}_{4}(a q)\right| \mathrm{Cu}(s)$ when 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 [$\mathrm{F}$is Faraday constant;$\mathrm{R}$is gas constant;$\mathrm{T}$is temperature;$E^{o}($cell$)=1.1 \mathrm{~V}$] [A]$1.1 \mathrm{~F}$[B]$2.303 \mathrm{RT}-2.2 \mathrm{~F}$[C]$2.303 \mathrm{RT}+1.1 \mathrm{~F}$[D]-2.2 \mathrm{~F}$