subject
stringclasses 3
values | description
stringclasses 16
values | gold
stringlengths 1
6
| index
int64 1
54
| type
stringclasses 4
values | question
stringlengths 107
1.61k
|
|---|---|---|---|---|---|
chem | JEE Adv 2017 Paper 2 | A | 21 | MCQ | The standard state Gibbs free energies of formation of $\mathrm{C}$ (graphite) and $C$ (diamond) at $\mathrm{T}=298 \mathrm{~K}$ are
\[
\begin{gathered}
\Delta_{f} G^{o}[\mathrm{C}(\text { graphite })]=0 \mathrm{~kJ} \mathrm{~mol} \\
\Delta_{f} G^{o}[\mathrm{C}(\text { diamond })]=2.9 \mathrm{~kJ} \mathrm{~mol}^{-1} .
\end{gathered}
\]
The 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
[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 $=10^{5} \mathrm{~Pa}$ ]
[A] 14501 bar
[B] 58001 bar
[C] 1450 bar
[D] 29001 bar |
chem | JEE Adv 2017 Paper 2 | C | 22 | MCQ | Which of the following combination will produce $\mathrm{H}_{2}$ gas?
[A] Fe metal and conc. $\mathrm{HNO}_{3}$
[B] Cu metal and conc. $\mathrm{HNO}_{3}$
[C] $\mathrm{Zn}$ metal and $\mathrm{NaOH}(\mathrm{aq})$
[D] Au metal and $\mathrm{NaCN}(\mathrm{aq})$ in the presence of air |
chem | JEE Adv 2017 Paper 2 | C | 23 | MCQ | 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
[A] $\mathrm{H}_{3} \mathrm{PO}_{3}>\mathrm{H}_{3} \mathrm{PO}_{2}>\mathrm{H}_{3} \mathrm{PO}_{4}>\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{6}$
[B] $\mathrm{H}_{3} \mathrm{PO}_{4}>\mathrm{H}_{3} \mathrm{PO}_{2}>\mathrm{H}_{3} \mathrm{PO}_{3}>\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{6}$
[C] $\mathrm{H}_{3} \mathrm{PO}_{4}>\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{6}>\mathrm{H}_{3} \mathrm{PO}_{3}>\mathrm{H}_{3} \mathrm{PO}_{2}$
[D] $\mathrm{H}_{3} \mathrm{PO}_{2}>\mathrm{H}_{3} \mathrm{PO}_{3}>\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{6}>\mathrm{H}_{3} \mathrm{PO}_{4}$ |
chem | JEE Adv 2017 Paper 2 | AB | 26 | MCQ(multiple) | The correct statement(s) about surface properties is(are)
[A] Adsorption is accompanied by decrease in enthalpy and decrease in entropy of the system
[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
[C] Cloud is an emulsion type of colloid in which liquid is dispersed phase and gas is dispersion medium
[D] Brownian motion of colloidal particles does not depend on the size of the particles but depends on viscosity of the solution |
chem | JEE Adv 2017 Paper 2 | BD | 27 | MCQ(multiple) | 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
[A] With increase in temperature, the value of $K$ for exothermic reaction decreases because the entropy change of the system is positive
[B] With increase in temperature, the value of $K$ for endothermic reaction increases because unfavourable change in entropy of the surroundings decreases
[C] With increase in temperature, the value of $K$ for endothermic reaction increases because the entropy change of the system is negative
[D] With increase in temperature, the value of $K$ for exothermic reaction decreases because favourable change in entropy of the surroundings decreases |
chem | JEE Adv 2017 Paper 2 | AB | 28 | MCQ(multiple) | In a bimolecular reaction, the steric factor $P$ was experimentally determined to be 4.5 . The correct option(s) among the following is(are)
[A] The activation energy of the reaction is unaffected by the value of the steric factor
[B] Experimentally determined value of frequency factor is higher than that predicted by Arrhenius equation
[C] Since $\mathrm{P}=4.5$, the reaction will not proceed unless an effective catalyst is used
[D] The value of frequency factor predicted by Arrhenius equation is higher than that determined experimentally |
chem | JEE Adv 2017 Paper 2 | ABD | 30 | MCQ(multiple) | Among the following, the correct statement(s) is(are)
[A] $\mathrm{Al}\left(\mathrm{CH}_{3}\right)_{3}$ has the three-centre two-electron bonds in its dimeric structure
[B] $\mathrm{BH}_{3}$ has the three-centre two-electron bonds in its dimeric structure
[C] $\mathrm{AlCl}_{3}$ has the three-centre two-electron bonds in its dimeric structure
[D] The Lewis acidity of $\mathrm{BCl}_{3}$ is greater than that of $\mathrm{AlCl}_{3}$ |
chem | JEE Adv 2017 Paper 2 | AD | 31 | MCQ(multiple) | The option(s) with only amphoteric oxides is(are)
[A] $\mathrm{Cr}_{2} \mathrm{O}_{3}, \mathrm{BeO}, \mathrm{SnO}, \mathrm{SnO}_{2}$
[B] $\mathrm{Cr}_{2} \mathrm{O}_{3}, \mathrm{CrO}, \mathrm{SnO}, \mathrm{PbO}$
[C] $\mathrm{NO}, \mathrm{B}_{2} \mathrm{O}_{3}, \mathrm{PbO}, \mathrm{SnO}_{2}$
[D] $\mathrm{ZnO}, \mathrm{Al}_{2} \mathrm{O}_{3}, \mathrm{PbO}, \mathrm{PbO}_{2}$ |
math | JEE Adv 2017 Paper 2 | C | 37 | MCQ | 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
[A] $14 x+2 y-15 z=1$
[B] $14 x-2 y+15 z=27$
[C] $\quad 14 x+2 y+15 z=31$
[D] $-14 x+2 y+15 z=3$ |
math | JEE Adv 2017 Paper 2 | D | 38 | MCQ | Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that
\[
\overrightarrow{O P} \cdot \overrightarrow{O Q}+\overrightarrow{O R} \cdot \overrightarrow{O S}=\overrightarrow{O R} \cdot \overrightarrow{O P}+\overrightarrow{O Q} \cdot \overrightarrow{O S}=\overrightarrow{O Q} \cdot \overrightarrow{O R}+\overrightarrow{O P} \cdot \overrightarrow{O S}
\]
Then the triangle $P Q R$ has $S$ as its
[A] centroid
[B] circumcentre
[C] incentre
[D] orthocenter |
math | JEE Adv 2017 Paper 2 | A | 39 | MCQ | If $y=y(x)$ satisfies the differential equation
\[
8 \sqrt{x}(\sqrt{9+\sqrt{x}}) d y=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} d x, \quad x>0
\]
and $y(0)=\sqrt{7}$, then $y(256)=$
[A] 3
[B] 9
[C] 16
[D] 80 |
math | JEE Adv 2017 Paper 2 | D | 40 | MCQ | 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
[A] $f^{\prime}(1) \leq 0$
[B] $0<f^{\prime}(1) \leq \frac{1}{2}$
[C] $\frac{1}{2}<f^{\prime}(1) \leq 1$
[D] $f^{\prime}(1)>1$ |
math | JEE Adv 2017 Paper 2 | B | 41 | MCQ | 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 $5 ?$
[A] 126
[B] 198
[C] 162
[D] 135 |
math | JEE Adv 2017 Paper 2 | D | 42 | MCQ | 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 $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=$
[A] 210
[B] 252
[C] 125
[D] 126 |
math | JEE Adv 2017 Paper 2 | B | 43 | MCQ | 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
[A] $\frac{36}{55}$
[B] $\frac{6}{11}$
[C] $\frac{1}{2}$
[D] $\frac{5}{11}$ |
math | JEE Adv 2017 Paper 2 | AC | 46 | MCQ(multiple) | 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
[A] $f(x)$ is increasing in $(0, \infty)$
[B] $f(x)$ is decreasing in $(0, \infty)$
[C] $\quad f(x)>e^{2 x}$ in $(0, \infty)$
[D] $f^{\prime}(x)<e^{2 x}$ in $(0, \infty)$ |
math | JEE Adv 2017 Paper 2 | AD | 47 | MCQ(multiple) | Let $f(x)=\frac{1-x(1+|1-x|)}{|1-x|} \cos \left(\frac{1}{1-x}\right)$ for $x \neq 1$. Then
[A] $\lim _{x \rightarrow 1^{-}} f(x)=0$
[B] $\lim _{x \rightarrow 1^{-}} f(x)$ does not exist
[C] $\lim _{x \rightarrow 1^{+}} f(x)=0$
[D] $\lim _{x \rightarrow 1^{+}} f(x)$ does not exist |
math | JEE Adv 2017 Paper 2 | BC | 48 | MCQ(multiple) | If $f(x)=\left|\begin{array}{ccc}\cos (2 x) & \cos (2 x) & \sin (2 x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x\end{array}\right|$, then
[A] $f^{\prime}(x)=0$ at exactly three points in $(-\pi, \pi)$
[B] $f^{\prime}(x)=0$ at more than three points in $(-\pi, \pi)$
[C] $f(x)$ attains its maximum at $x=0$
[D] $f(x)$ attains its minimum at $x=0$ |
math | JEE Adv 2017 Paper 2 | BC | 49 | MCQ(multiple) | If the line $x=\alpha$ divides the area of region $R=\left\{(x, y) \in \mathbb{R}^{2}: x^{3} \leq y \leq x, 0 \leq x \leq 1\right\}$ into two equal parts, then
[A] $0<\alpha \leq \frac{1}{2}$
[B] $\frac{1}{2}<\alpha<1$
[C] $\quad 2 \alpha^{4}-4 \alpha^{2}+1=0$
[D] $\alpha^{4}+4 \alpha^{2}-1=0$ |
math | JEE Adv 2017 Paper 2 | BD | 50 | MCQ(multiple) | If $I=\sum_{k=1}^{98} \int_{k}^{k+1} \frac{k+1}{x(x+1)} d x$, then
[A] $I>\log _{e} 99$
[B] $I<\log _{e} 99$
[C] $I<\frac{49}{50}$
[D] $I>\frac{49}{50}$ |
phy | JEE Adv 2018 Paper 1 | BC | 1 | MCQ(multiple) | 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?
(A) $v=\sqrt{\frac{k}{2 m}} R$
(B) $v=\sqrt{\frac{k}{m}} R$
(C) $L=\sqrt{m k} R^{2}$
(D) $L=\sqrt{\frac{m k}{2}} R^{2}$ |
phy | JEE Adv 2018 Paper 1 | AC | 2 | MCQ(multiple) | 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?
(A) $|\vec{\tau}|=\frac{1}{3} N m$
(B) The torque $\vec{\tau}$ is in the direction of the unit vector $+\hat{k}$
(C) The velocity of the body at $t=1 s$ is $\vec{v}=\frac{1}{2}(\hat{i}+2 \hat{j}) m s^{-1}$
(D) The magnitude of displacement of the body at $t=1 s$ is $\frac{1}{6} m$ |
phy | JEE Adv 2018 Paper 1 | AC | 3 | MCQ(multiple) | 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?
(A) For a given material of the capillary tube, $h$ decreases with increase in $r$
(B) For a given material of the capillary tube, $h$ is independent of $\sigma$
(C) If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases
(D) $h$ is proportional to contact angle $\theta$ |
phy | JEE Adv 2018 Paper 1 | ABD | 5 | MCQ(multiple) | 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?
(A) If $I_{1}=I_{2}$, then $\vec{B}$ cannot be equal to zero at the origin $(0,0,0)$
(B) If $I_{1}>0$ and $I_{2}<0$, then $\vec{B}$ can be equal to zero at the origin $(0,0,0)$
(C) If $I_{1}<0$ and $I_{2}>0$, then $\vec{B}$ can be equal to zero at the origin $(0,0,0)$
(D) If $I_{1}=I_{2}$, then the $z$-component of the magnetic field at the centre of the loop is $\left(-\frac{\mu_{0} I}{2 R}\right)$ |
phy | JEE Adv 2018 Paper 1 | 2 | 7 | Numeric | 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? |
phy | JEE Adv 2018 Paper 1 | 5 | 8 | Numeric | 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? |
phy | JEE Adv 2018 Paper 1 | 0.75 | 9 | Numeric | 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?
Take $g=10 m s^{-2}$. |
phy | JEE Adv 2018 Paper 1 | 130 | 13 | Numeric | 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? |
chem | JEE Adv 2018 Paper 1 | BC | 19 | MCQ(multiple) | The compound(s) which generate(s) $\mathrm{N}_{2}$ gas upon thermal decomposition below $300^{\circ} \mathrm{C}$ is (are)
(A) $\mathrm{NH}_{4} \mathrm{NO}_{3}$
(B) $\left(\mathrm{NH}_{4}\right)_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}$
(C) $\mathrm{Ba}\left(\mathrm{N}_{3}\right)_{2}$
(D) $\mathrm{Mg}_{3} \mathrm{~N}_{2}$ |
chem | JEE Adv 2018 Paper 1 | BC | 20 | MCQ(multiple) | The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: $\mathrm{Fe}=26, \mathrm{Ni}=28$ )
(A) Total number of valence shell electrons at metal centre in $\mathrm{Fe}(\mathrm{CO})_{5}$ or $\mathrm{Ni}(\mathrm{CO})_{4}$ is 16
(B) These are predominantly low spin in nature
(C) Metal-carbon bond strengthens when the oxidation state of the metal is lowered
(D) The carbonyl C-O bond weakens when the oxidation state of the metal is increased |
chem | JEE Adv 2018 Paper 1 | ABC | 21 | MCQ(multiple) | Based on the compounds of group 15 elements, the correct statement(s) is (are)
(A) $\mathrm{Bi}_{2} \mathrm{O}_{5}$ is more basic than $\mathrm{N}_{2} \mathrm{O}_{5}$
(B) $\mathrm{NF}_{3}$ is more covalent than $\mathrm{BiF}_{3}$
(C) $\mathrm{PH}_{3}$ boils at lower temperature than $\mathrm{NH}_{3}$
(D) The $\mathrm{N}-\mathrm{N}$ single bond is stronger than the $\mathrm{P}-\mathrm{P}$ single bond |
chem | JEE Adv 2018 Paper 1 | 1 | 25 | Numeric | Among the species given below, what is the total number of diamagnetic species?
$\mathrm{H}$ atom, $\mathrm{NO}_{2}$ monomer, $\mathrm{O}_{2}^{-}$(superoxide), dimeric sulphur in vapour phase,
$\mathrm{Mn}_{3} \mathrm{O}_{4},\left(\mathrm{NH}_{4}\right)_{2}\left[\mathrm{FeCl}_{4}\right],\left(\mathrm{NH}_{4}\right)_{2}\left[\mathrm{NiCl}_{4}\right], \mathrm{K}_{2} \mathrm{MnO}_{4}, \mathrm{~K}_{2} \mathrm{CrO}_{4}$ |
chem | JEE Adv 2018 Paper 1 | 2992 | 26 | Numeric | 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?
(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{Cl}=35.5, \mathrm{Ca}=40, \mathrm{Ni}=59$ ) |
chem | JEE Adv 2018 Paper 1 | 3 | 27 | Numeric | 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.
(i) Remove all the anions (X) except the central one
(ii) Replace all the face centered cations (M) by anions (X)
(iii) Remove all the corner cations (M)
(iv) Replace the central anion (X) with cation (M)
What is the value of $\left(\frac{\text { number of anions }}{\text { number of cations }}\right)$ in $\mathbf{Z}$? |
chem | JEE Adv 2018 Paper 1 | 10 | 28 | Numeric | For the electrochemical cell,
\[
\operatorname{Mg}(\mathrm{s})\left|\mathrm{Mg}^{2+}(\mathrm{aq}, 1 \mathrm{M}) \| \mathrm{Cu}^{2+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{Cu}(\mathrm{s})
\]
the 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 $\boldsymbol{x}$?
(given, $\frac{F}{R}=11500 \mathrm{~K} \mathrm{~V}^{-1}$, where $F$ is the Faraday constant and $R$ is the gas constant, $\ln (10)=2.30)$ |
chem | JEE Adv 2018 Paper 1 | 19 | 30 | Numeric | 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 $\mathrm{T}$ ) |
chem | JEE Adv 2018 Paper 1 | 4.47 | 31 | Numeric | 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 $\left.\mathbf{H B}\left(K_{a}\right)=1 \times 10^{-8}\right)$ |
math | JEE Adv 2018 Paper 1 | ABD | 37 | MCQ(multiple) | 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?
\end{itemize}
(A) $\arg (-1-i)=\frac{\pi}{4}$, where $i=\sqrt{-1}$
(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 $i=\sqrt{-1}$
(C) For any two non-zero complex numbers $z_{1}$ and $z_{2}$,
\[
\arg \left(\frac{z_{1}}{z_{2}}\right)-\arg \left(z_{1}\right)+\arg \left(z_{2}\right)
\]
is an integer multiple of $2 \pi$
(D) For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$, the locus of the point $z$ satisfying the condition
\[
\arg \left(\frac{\left(z-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z-z_{3}\right)\left(z_{2}-z_{1}\right)}\right)=\pi
\]
lies on a straight line |
math | JEE Adv 2018 Paper 1 | BCD | 38 | MCQ(multiple) | 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?
(A) $\angle Q P R=45^{\circ}$
(B) The area of the triangle $P Q R$ is $25 \sqrt{3}$ and $\angle Q R P=120^{\circ}$
(C) The radius of the incircle of the triangle $P Q R$ is $10 \sqrt{3}-15$
(D) The area of the circumcircle of the triangle $P Q R$ is $100 \pi$ |
math | JEE Adv 2018 Paper 1 | CD | 39 | MCQ(multiple) | 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?
(A) The line of intersection of $P_{1}$ and $P_{2}$ has direction ratios $1,2,-1$
(B) The line
\[
\frac{3 x-4}{9}=\frac{1-3 y}{9}=\frac{z}{3}
\]
is perpendicular to the line of intersection of $P_{1}$ and $P_{2}$
(C) The acute angle between $P_{1}$ and $P_{2}$ is $60^{\circ}$
(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 $\frac{2}{\sqrt{3}}$ |
math | JEE Adv 2018 Paper 1 | ABD | 40 | MCQ(multiple) | 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?
(A) There exist $r, s \in \mathbb{R}$, where $r<s$, such that $f$ is one-one on the open interval $(r, s)$
(B) There exists $x_{0} \in(-4,0)$ such that $\left|f^{\prime}\left(x_{0}\right)\right| \leq 1$
(C) $\lim _{x \rightarrow \infty} f(x)=1$
(D) There exists $\alpha \in(-4,4)$ such that $f(\alpha)+f^{\prime \prime}(\alpha)=0$ and $f^{\prime}(\alpha) \neq 0$ |
math | JEE Adv 2018 Paper 1 | BC | 41 | MCQ(multiple) | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two non-constant differentiable functions. If
\[
f^{\prime}(x)=\left(e^{(f(x)-g(x))}\right) g^{\prime}(x) \text { for all } x \in \mathbb{R}
\]
and $f(1)=g(2)=1$, then which of the following statement(s) is (are) TRUE?
(A) $f(2)<1-\log _{\mathrm{e}} 2$
(B) $f(2)>1-\log _{\mathrm{e}} 2$
(C) $g(1)>1-\log _{\mathrm{e}} 2$
(D) $g(1)<1-\log _{\mathrm{e}} 2$ |
math | JEE Adv 2018 Paper 1 | 8 | 43 | Numeric | What is the value of
\[
\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}
\]? |
math | JEE Adv 2018 Paper 1 | 625 | 44 | Numeric | 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? |
math | JEE Adv 2018 Paper 1 | 3748 | 45 | Numeric | 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 $X \cup Y$? |
math | JEE Adv 2018 Paper 1 | 2 | 46 | Numeric | What is the number of real solutions of the equation
\[
\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i}\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i}-\sum_{i=1}^{\infty}(-x)^{i}\right)
\]
lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is
(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.) |
math | JEE Adv 2018 Paper 1 | 1 | 47 | Numeric | For each positive integer $n$, let
\[
y_{n}=\frac{1}{n}((n+1)(n+2) \cdots(n+n))^{\frac{1}{n}}
\]
For $x \in \mathbb{R}$, 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 $[L]$? |
math | JEE Adv 2018 Paper 1 | 3 | 48 | Numeric | 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 $8 \cos ^{2} \alpha$? |
math | JEE Adv 2018 Paper 1 | 0.5 | 49 | Numeric | Let $a, b, c$ be three non-zero real numbers such that the equation
\[
\sqrt{3} a \cos x+2 b \sin x=c, x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\]
has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{3}$. Then, what is the value of $\frac{b}{a}$? |
math | JEE Adv 2018 Paper 1 | 4 | 50 | Numeric | 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 $n$? |
phy | JEE Adv 2018 Paper 2 | ABD | 1 | MCQ(multiple) | 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?
(A) The force applied on the particle is constant
(B) The speed of the particle is proportional to time
(C) The distance of the particle from the origin increases linearly with time
(D) The force is conservative |
phy | JEE Adv 2018 Paper 2 | ACD | 2 | MCQ(multiple) | 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?
(A) The resistive force of liquid on the plate is inversely proportional to $h$
(B) The resistive force of liquid on the plate is independent of the area of the plate
(C) The tangential (shear) stress on the floor of the tank increases with $u_{0}$
(D) The tangential (shear) stress on the plate varies linearly with the viscosity $\eta$ of the liquid |
phy | JEE Adv 2018 Paper 2 | AC | 5 | MCQ(multiple) | 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?
(A) $N_{\alpha}=5$
(B) $N_{\alpha}=6$
(C) $N_{\beta}=2$
(D) $N_{\beta}=4$ |
phy | JEE Adv 2018 Paper 2 | AC | 6 | MCQ(multiple) | 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?
(A) The speed of sound determined from this experiment is $332 \mathrm{~ms}^{-1}$
(B) The end correction in this experiment is $0.9 \mathrm{~cm}$
(C) The wavelength of the sound wave is $66.4 \mathrm{~cm}$
(D) The resonance at $50.7 \mathrm{~cm}$ corresponds to the fundamental harmonic |
phy | JEE Adv 2018 Paper 2 | 6.3 | 7 | Numeric | 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 $e^{-1}=0.37$ |
phy | JEE Adv 2018 Paper 2 | 30 | 8 | Numeric | 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? |
phy | JEE Adv 2018 Paper 2 | 2 | 9 | Numeric | 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? |
phy | JEE Adv 2018 Paper 2 | 5.56 | 10 | Numeric | 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? |
phy | JEE Adv 2018 Paper 2 | 3 | 11 | Numeric | 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 $\pi=3.2$ |
phy | JEE Adv 2018 Paper 2 | 900 | 12 | Numeric | 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? |
phy | JEE Adv 2018 Paper 2 | 24 | 13 | Numeric | 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 $n$?
Mass of the electron $m_{e}=9 \times 10^{-31} \mathrm{~kg}$ and $1.0 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}$. |
phy | JEE Adv 2018 Paper 2 | 3 | 14 | Numeric | 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 $Z$? |
chem | JEE Adv 2018 Paper 2 | ABD | 19 | MCQ(multiple) | 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)
(A) It has two geometrical isomers
(B) It will have three geometrical isomers if bidentate 'en' is replaced by two cyanide ligands
(C) It is paramagnetic
(D) It absorbs light at longer wavelength as compared to $\left[\mathrm{Co}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{4}\right]^{3+}$ |
chem | JEE Adv 2018 Paper 2 | BD | 20 | MCQ(multiple) | The correct option(s) to distinguish nitrate salts of $\mathrm{Mn}^{2+}$ and $\mathrm{Cu}^{2+}$ taken separately is (are)
(A) $\mathrm{Mn}^{2+}$ shows the characteristic green colour in the flame test
(B) Only $\mathrm{Cu}^{2+}$ shows the formation of precipitate by passing $\mathrm{H}_{2} \mathrm{~S}$ in acidic medium
(C) Only $\mathrm{Mn}^{2+}$ shows the formation of precipitate by passing $\mathrm{H}_{2} \mathrm{~S}$ in faintly basic medium
(D) $\mathrm{Cu}^{2+} / \mathrm{Cu}$ has higher reduction potential than $\mathrm{Mn}^{2+} / \mathrm{Mn}$ (measured under similar conditions) |
chem | JEE Adv 2018 Paper 2 | 6 | 25 | Numeric | What is the total number of compounds having at least one bridging oxo group among the molecules given below?
$\mathrm{N}_{2} \mathrm{O}_{3}, \mathrm{~N}_{2} \mathrm{O}_{5}, \mathrm{P}_{4} \mathrm{O}_{6}, \mathrm{P}_{4} \mathrm{O}_{7}, \mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{5}, \mathrm{H}_{5} \mathrm{P}_{3} \mathrm{O}_{10}, \mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}, \mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{5}$ |
chem | JEE Adv 2018 Paper 2 | 6.47 | 26 | Numeric | 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?
(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~Pb}=207$ ) |
chem | JEE Adv 2018 Paper 2 | 126 | 27 | Numeric | 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).
Few 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
(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{Mn}=55, \mathrm{Cl}=35.5$ ) |
chem | JEE Adv 2018 Paper 2 | -14.6 | 30 | Numeric | 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:
$2 \mathrm{Cu}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightarrow \mathrm{Cu}_{2} \mathrm{O}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g})$
$p_{\mathrm{H}_{2}}$ 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 $\ln \left(p_{\mathrm{H}_{2}}\right)$?
(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.
At $1250 \mathrm{~K}: 2 \mathrm{Cu}(\mathrm{s})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{Cu}_{2} \mathrm{O}(\mathrm{s}) ; \Delta G^{\theta}=-78,000 \mathrm{~J} \mathrm{~mol}^{-1}$
\[
\mathrm{H}_{2}(\mathrm{~g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) ; \quad \Delta G^{\theta}=-1,78,000 \mathrm{~J} \mathrm{~mol}^{-1} ; G \text { is the Gibbs energy) }
\] |
chem | JEE Adv 2018 Paper 2 | 8500 | 31 | Numeric | Consider the following reversible reaction,
\[
\mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g}) \rightleftharpoons \mathrm{AB}(\mathrm{g})
\]
The 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 $300 \mathrm{~K}$?
(Given; $\ln (2)=0.7, R T=2500 \mathrm{~J} \mathrm{~mol}^{-1}$ at $300 \mathrm{~K}$ and $G$ is the Gibbs energy) |
chem | JEE Adv 2018 Paper 2 | -11.62 | 32 | Numeric | 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 $300 \mathrm{~K}$?
(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.)
|
math | JEE Adv 2018 Paper 2 | D | 37 | MCQ(multiple) | For any positive integer $n$, define $f_{n}:(0, \infty) \rightarrow \mathbb{R}$ as
\[
f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)
\]
(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?
(A) $\sum_{j=1}^{5} \tan ^{2}\left(f_{j}(0)\right)=55$
(B) $\sum_{j=1}^{10}\left(1+f_{j}^{\prime}(0)\right) \sec ^{2}\left(f_{j}(0)\right)=10$
(C) For any fixed positive integer $n, \lim _{x \rightarrow \infty} \tan \left(f_{n}(x)\right)=\frac{1}{n}$
(D) For any fixed positive integer $n$, $\lim _{x \rightarrow \infty} \sec ^{2}\left(f_{n}(x)\right)=1$ |
math | JEE Adv 2018 Paper 2 | ACD | 41 | MCQ(multiple) | 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?
(A) If $L$ has exactly one element, then $|s| \neq|t|$
(B) If $|s|=|t|$, then $L$ has infinitely many elements
(C) The number of elements in $L \cap\{z:|z-1+i|=5\}$ is at most 2
(D) If $L$ has more than one element, then $L$ has infinitely many elements |
math | JEE Adv 2018 Paper 2 | 2 | 43 | Numeric | What is the value of the integral
\[
\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} d x
\]? |
math | JEE Adv 2018 Paper 2 | 4 | 44 | Numeric | 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 $P$? |
math | JEE Adv 2018 Paper 2 | 119 | 45 | Numeric | 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 $\frac{1}{5 !}(\beta-\alpha)$? |
math | JEE Adv 2018 Paper 2 | 0.4 | 46 | Numeric | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation
\[
\frac{d y}{d x}=(2+5 y)(5 y-2)
\]
then what is the value of $\lim _{x \rightarrow-\infty} f(x)$? |
math | JEE Adv 2018 Paper 2 | 2 | 47 | Numeric | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation
\[
f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y) \text { for all } x, y \in \mathbb{R} .
\]
Then, the value of $\log _{e}(f(4))$ is |
math | JEE Adv 2018 Paper 2 | 8 | 48 | Numeric | 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 $P R$? |
math | JEE Adv 2018 Paper 2 | 0.5 | 49 | Numeric | 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 $|(\vec{p} \times \vec{q}) \times(\vec{r} \times \vec{t})|$? |
math | JEE Adv 2018 Paper 2 | 646 | 50 | Numeric | Let
\[
X=\left({ }^{10} C_{1}\right)^{2}+2\left({ }^{10} C_{2}\right)^{2}+3\left({ }^{10} C_{3}\right)^{2}+\cdots+10\left({ }^{10} C_{10}\right)^{2}
\]
where ${ }^{10} C_{r}, r \in\{1,2, \cdots, 10\}$ denote binomial coefficients. Then, what is the value of $\frac{1}{1430} X$? |
phy | JEE Adv 2019 Paper 1 | A | 1 | MCQ | 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
[ $G$ is universal gravitational constant]
(A) $\frac{K}{2 \pi r^{2} m^{2} G}$
(B) $\frac{K}{\pi r^{2} m^{2} G}$
(C) $\frac{3 K}{\pi r^{2} m^{2} G}$
(D) $\frac{K}{6 \pi r^{2} m^{2} G}$ |
phy | JEE Adv 2019 Paper 1 | C | 2 | MCQ | 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?
(A) The potential at the center of the shell is reduced by $2 \alpha V_{0}$
(B) The magnitude of electric field at the center of the shell is reduced by $\frac{\alpha V_{0}}{2 R}$
(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 $\frac{1-\alpha}{1-2 \alpha}$
(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 $\frac{\alpha V_{0}}{2 R}$ |
phy | JEE Adv 2019 Paper 1 | A | 3 | MCQ | 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
\[
T(t)=T_{0}\left(1+\beta t^{\frac{1}{4}}\right)
\]
where $\beta$ is a constant with appropriate dimension while $T_{0}$ is a constant with dimension of temperature. The heat capacity of the metal is,
(A) $\frac{4 P\left(T(t)-T_{0}\right)^{3}}{\beta^{4} T_{0}^{4}}$
(B) $\frac{4 P\left(T(t)-T_{0}\right)^{4}}{\beta^{4} T_{0}^{5}}$
(C) $\frac{4 P\left(T(t)-T_{0}\right)^{2}}{\beta^{4} T_{0}^{3}}$
(D) $\frac{4 P\left(T(t)-T_{0}\right)}{\beta^{4} T_{0}^{2}}$ |
phy | JEE Adv 2019 Paper 1 | B | 4 | MCQ | 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,
[Given: $\ln 10=2.3]$
(A) 1.15
(B) 9.2
(C) 2.3
(D) 4.6 |
phy | JEE Adv 2019 Paper 1 | ABC | 8 | MCQ(multiple) | 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?
$\left[\epsilon_{0}\right.$ is the permittivity of free space]
(A) If $h>2 R$ and $r>R$ then $\Phi=\mathrm{Q} / \epsilon_{0}$
(B) If $h<8 R / 5$ and $r=3 R / 5$ then $\Phi=0$
(C) If $h>2 R$ and $r=3 R / 5$ then $\Phi=Q / 5 \epsilon_{0}$
(D) If $h>2 R$ and $r=4 R / 5$ then $\Phi=\mathrm{Q} / 5 \epsilon_{0}$ |
phy | JEE Adv 2019 Paper 1 | ABC | 11 | MCQ(multiple) | 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?
(A) The dimension of linear momentum is $L^{-1}$
(B) The dimension of energy is $L^{-2}$
(C) The dimension of force is $L^{-3}$
(D) The dimension of power is $L^{-5}$ |
phy | JEE Adv 2019 Paper 1 | BC | 12 | MCQ(multiple) | 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?
(A) The resistance of the Voltmeter will be $100 \mathrm{k} \Omega$
(B) The resistance of the Ammeter will be $0.02 \Omega$ (round off to $2^{\text {nd }}$ decimal place)
(C) The measured value of $R$ will be $978 \Omega<R<982 \Omega$
(D) If the ideal cell is replaced by a cell having internal resistance of $5 \Omega$ then the measured value of $R$ will be more than $1000 \Omega$ |
phy | JEE Adv 2019 Paper 1 | 1 | 16 | Numeric | A parallel plate capacitor of capacitance $C$ has spacing $d$ between two plates having area $A$. The region between the plates is filled with $N$ dielectric layers, parallel to its plates, each with thickness $\delta=\frac{d}{N}$. The dielectric constant of the $m^{t h}$ layer is $K_{m}=K\left(1+\frac{m}{N}\right)$. For a very large $N\left(>10^{3}\right)$, 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] |
chem | JEE Adv 2019 Paper 1 | C | 19 | MCQ | The green colour produced in the borax bead test of a chromium(III) salt is due to
(A) $\mathrm{Cr}\left(\mathrm{BO}_{2}\right)_{3}$
(B) $\mathrm{Cr}_{2}\left(\mathrm{~B}_{4} \mathrm{O}_{7}\right)_{3}$
(C) $\mathrm{Cr}_{2} \mathrm{O}_{3}$
(D) $\mathrm{CrB}$ |
chem | JEE Adv 2019 Paper 1 | C | 20 | MCQ | Calamine, malachite, magnetite and cryolite, respectively, are
(A) $\mathrm{ZnSO}_{4}, \mathrm{CuCO}_{3}, \mathrm{Fe}_{2} \mathrm{O}_{3}, \mathrm{AlF}_{3}$
(B) $\mathrm{ZnSO}_{4}, \mathrm{Cu}(\mathrm{OH})_{2}, \mathrm{Fe}_{3} \mathrm{O}_{4}, \mathrm{Na}_{3} \mathrm{AlF}_{6}$
(C) $\mathrm{ZnCO}_{3}, \mathrm{CuCO}_{3} \cdot \mathrm{Cu}(\mathrm{OH})_{2}, \mathrm{Fe}_{3} \mathrm{O}_{4}, \mathrm{Na}_{3} \mathrm{AlF}_{6}$
(D) $\mathrm{ZnCO}_{3}, \mathrm{CuCO}_{3}, \mathrm{Fe}_{2} \mathrm{O}_{3}, \mathrm{Na}_{3} \mathrm{AlF}_{6}$ |
chem | JEE Adv 2019 Paper 1 | AB | 23 | MCQ(multiple) | A tin chloride $\mathrm{Q}$ undergoes the following reactions (not balanced)
$\mathrm{Q}+\mathrm{Cl}^{-} \rightarrow \mathrm{X}$
$\mathrm{Q}+\mathrm{Me}_{3} \mathrm{~N} \rightarrow \mathrm{Y}$
$\mathbf{Q}+\mathrm{CuCl}_{2} \rightarrow \mathbf{Z}+\mathrm{CuCl}$
$\mathrm{X}$ is a monoanion having pyramidal geometry. Both $\mathrm{Y}$ and $\mathrm{Z}$ are neutral compounds.
Choose the correct option(s)
(A) The central atom in $\mathrm{X}$ is $s p^{3}$ hybridized
(B) There is a coordinate bond in $\mathrm{Y}$
(C) The oxidation state of the central atom in $\mathrm{Z}$ is +2
(D) The central atom in $\mathrm{Z}$ has one lone pair of electrons |
chem | JEE Adv 2019 Paper 1 | ABD | 24 | MCQ(multiple) | 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)
(A) In aqueous acidic solution, $\mathrm{Y}$ undergoes disproportionation reaction to give $\mathrm{Z}$ and $\mathrm{MnO}_{2}$
(B) Both $\mathrm{Y}$ and $\mathrm{Z}$ are coloured and have tetrahedral shape
(C) $\mathrm{Y}$ is diamagnetic in nature while $\mathrm{Z}$ is paramagnetic
(D) In both $\mathrm{Y}$ and $\mathrm{Z}, \pi$-bonding occurs between $p$-orbitals of oxygen and $d$-orbitals of manganese |
chem | JEE Adv 2019 Paper 1 | CD | 25 | MCQ(multiple) | Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.
(A) $2 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})$
(B) $2 \mathrm{C}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow \mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{~g})$
(C) $\frac{3}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{O}_{3}(\mathrm{~g})$
(D) $\frac{1}{8} \mathrm{~S}_{8}(\mathrm{~s})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{SO}_{2}(\mathrm{~g})$ |
chem | JEE Adv 2019 Paper 1 | ACD | 26 | MCQ(multiple) | 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?
(A) $u_{rms}$ is doubled when its temperature is increased four times
(B) $\varepsilon_{av}}$ is doubled when its temperature is increased four times
(C) $\varepsilon_{av}$ at a given temperature does not depend on its molecular mass
(D) $u_{rms}$ is inversely proportional to the square root of its molecular mass |
chem | JEE Adv 2019 Paper 1 | BD | 27 | MCQ(multiple) | 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.
(A) $\mathrm{BeCl}_{2}, \mathrm{CO}_{2}, \mathrm{BCl}_{3}, \mathrm{CHCl}_{3}$
(B) $\mathrm{NO}_{2}, \mathrm{NH}_{3}, \mathrm{POCl}_{3}, \mathrm{CH}_{3} \mathrm{Cl}$
(C) $\mathrm{BF}_{3}, \mathrm{O}_{3}, \mathrm{SF}_{6}, \mathrm{XeF}_{6}$
(D) $\mathrm{SO}_{2}, \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{Cl}, \mathrm{H}_{2} \mathrm{Se}, \mathrm{BrF}_{5}$ |
chem | JEE Adv 2019 Paper 1 | ABD | 28 | MCQ(multiple) | In the decay sequence,
${ }_{92}^{238} \mathrm{U} \stackrel{-\mathrm{x}_{1}}{\longrightarrow}{ }_{90}^{234} \mathrm{Th} \stackrel{-\mathrm{x}_{2}}{\longrightarrow}{ }_{91}^{234} \mathrm{~Pa} \stackrel{-\mathrm{x}_{3}}{\longrightarrow}{ }^{234} \mathbf{Z} \stackrel{-\mathrm{x}_{4}}{\longrightarrow}{ }_{90}^{230} \mathrm{Th}$
$\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}$ and $\mathrm{x}_{4}$ are particles/radiation emitted by the respective isotopes. The correct option(s) is $($ are $)$
(A) $x_{1}$ will deflect towards negatively charged plate
(B) $\mathrm{x}_{2}$ is $\beta^{-}$
(C) $x_{3}$ is $\gamma$-ray
(D) $\mathbf{Z}$ is an isotope of uranium |
chem | JEE Adv 2019 Paper 1 | ACD | 29 | MCQ(multiple) | Which of the following statement(s) is(are) true?
(A) Monosaccharides cannot be hydrolysed to give polyhydroxy aldehydes and ketones
(B) Oxidation of glucose with bromine water gives glutamic acid
(C) Hydrolysis of sucrose gives dextrorotatory glucose and laevorotatory fructose
(D) The two six-membered cyclic hemiacetal forms of $\mathrm{D}-(+)$-glucose are called anomers |
chem | JEE Adv 2019 Paper 1 | 4 | 31 | Numeric | 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? |
chem | JEE Adv 2019 Paper 1 | 19 | 32 | Numeric | 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 $\mathrm{Y}$? |
chem | JEE Adv 2019 Paper 1 | 1.02 | 34 | Numeric | 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) |