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JEE-ADVANCED EXAMINATION – MAY 2024

JEE-ADVANCED TEST PAPER 2 WITH SOLUTION

Held on Sunday 26th May 2024, Time: 2:30 PM to 5:30 PM

JEE Advanced Paper 2

Mathematics, Physics, Chemistry

Afternoon Session

3 hours

Paper Overview

Total Questions

Correct

Incorrect

N/A

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Complete Solutions

Q#	Explanation	Question	Solution	Status
1	View ExplanationExplain	Considering only the principal values of the inverse trigonometric functions, the value of \[ \tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) \] is (A) $\frac{7}{24}$ (B) $\frac{-7}{24}$ (C) $\frac{-5}{24}$ (D) $\frac{5}{24}$	Available	Available
2	View ExplanationExplain	Let $S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^{2} \leq 4 x, y^{2} \leq 12-2 x\right.$ and $\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}$ . If the area of the region $S$ is $\alpha \sqrt{2}$ , then $\alpha$ is equal to (A) $\frac{17}{2}$ (B) $\frac{17}{3}$ (C) $\frac{17}{4}$ (D) $\frac{17}{5}$	Available	Available
3	View ExplanationExplain	Let $k \in \mathbb{R}$ . If $\lim _{x \rightarrow 0+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^{6}$ , then the value of $k$ is (A) 1 (B) 2 (C) 3 (D) 4	Available	Available
4	View ExplanationExplain	Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by \[ f(x)=\left\{\begin{array}{cc} x^{2} \sin \left(\frac{\pi}{x^{2}}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right. \] Then which of the following statements is TRUE? (A) $f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$ . (B) $f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$ . (C) The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite. (D) $f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^{2}}, \frac{1}{\pi}\right)$ .	Available	Available
5	View ExplanationExplain	Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that \[ \lim _{x \rightarrow \infty} \frac{\sin \left(x^{2}\right)\left(\log _{e} x\right)^{\alpha} \sin \left(\frac{1}{x^{2}}\right)}{x^{\alpha \beta}\left(\log _{e}(1+x)\right)^{\beta}}=0 . \] Then which of the following is (are) correct? (A) $(-1,3) \in S$ (B) $(-1,1) \in S$ (C) $(1,-1) \in S$ (D) $(1,-2) \in S$	Available	Available
6	View ExplanationExplain	A straight line drawn from the point $P(1,3,2)$ , parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$ , intersects the plane $L_{1}: x-y+3 z=6$ at the point $Q$ . Another straight line which passes through $Q$ and is perpendicular to the plane $L_{1}$ intersects the plane $L_{2}: 2 x-y+z=-4$ at the point $R$ . Then which of the following statements is (are) TRUE? (A) The length of the line segment $P Q$ is $\sqrt{6}$ (B) The coordinates of $R$ are $(1,6,3)$ (C) The centroid of the triangle $P Q R$ is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$ (D) The perimeter of the triangle $P Q R$ is $\sqrt{2}+\sqrt{6}+\sqrt{11}$	Available	Available
7	View ExplanationExplain	Let $A_{1}, B_{1}, C_{1}$ be three points in the $x y$ -plane. Suppose that the lines $A_{1} C_{1}$ and $B_{1} C_{1}$ are tangents to the curve $y^{2}=8 x$ at $A_{1}$ and $B_{1}$ , respectively. If $O=(0,0)$ and $C_{1}=(-4,0)$ , then which of the following statements is (are) TRUE? (A) The length of the line segment $O A_{1}$ is $4 \sqrt{3}$ (B) The length of the line segment $A_{1} B_{1}$ is 16 (C) The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(0,0)$ (D) The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(1,0)$	Available	Available
8	View ExplanationExplain	Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{R}$ , and $g: \mathbb{R} \rightarrow(0, \infty)$ be a function such that $g(x+y)=g(x) g(y)$ for all $x, y \in \mathbb{R}$ . If $f\left(\frac{-3}{5}\right)=12$ and $g\left(\frac{-1}{3}\right)=2$ , then the value of $\left(f\left(\frac{1}{4}\right)+g(-2)-8\right) g(0)$ is $\qquad$ .	Available	Available
9	View ExplanationExplain	A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i=1,2,3$ , let $W_{i}, G_{i}$ , and $B_{i}$ denote the events that the ball drawn in the $i^{\text {th }}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P\left(W_{1} \cap G_{2} \cap B_{3}\right)=\frac{2}{5 N}$ and the conditional probability $P\left(B_{3} \mid W_{1} \cap G_{2}\right)=\frac{2}{9}$ , then $N$ equals $\qquad$ .	Available	Available
10	View ExplanationExplain	Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by \[ f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^{2}-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^{2}-x+3\right)} \] Then the number of solutions of $f(x)=0$ in $\mathbb{R}$ is $\qquad$ .	Available	Available
11	View ExplanationExplain	Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$ . If for some real numbers $\alpha, \beta$ , and $\gamma$ , we have \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q}) \] then the value of $\gamma$ is $\qquad$ .	NA	Available
12	View ExplanationExplain	A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha)$ to the parabola $x^{2}=-4 a y$ , where $a>0$ . Let $L$ be the line passing through $(0,-\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$ . Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $A B$ . If $r: s=1: 16$ , then the value of $24 a$ is $\qquad$ .	Available	Available
13	View ExplanationExplain	Let the function $f:[1, \infty) \rightarrow \mathbb{R}$ be defined by \[ f(t)=\left\{\begin{array}{cc} (-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \mathbb{N}, \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2 n-1<t<2 n+1, n \in \mathbb{N} . \end{array}\right. \] Define $g(x)=\int_{1}^{x} f(t) d t, x \in(1, \infty)$ . Let $\alpha$ denote the number of solutions of the equation $g(x)=0$ in the interval $(1,8]$ and $\beta=\lim _{x \rightarrow 1+} \frac{g(x)}{x-1}$ . Then the value of $\alpha+\beta$ is equal to $\qquad$	Available	Available
14	View ExplanationExplain	If $n(X)={ }^{m} C_{6}$ , then the value of $m$ is $\qquad$ . \section*{PARAGRAPH "I"} Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: $\quad R$ has exactly 6 elements.For each $(a, b) \in R$ , we have $\|a-b\| \geq 2$ . Let $Y=\{R \in X$ : The range of $R$ has exactly one element $\}$ and $Z=\{R \in X: R$ is a function from $S$ to $S\}$ . Let $n(A)$ denote the number of elements in a set $A$ .	NA	Available
15	View ExplanationExplain	If the value of $n(Y)+n(Z)$ is $k^{2}$ , then $\|k\|$ is $\qquad$ . \section*{PARAGRAPH "II"} Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]$ be the function defined by $f(x)=\sin ^{2} x$ and let $g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^{2}}$ . (There are two questions based on PARAGRAPH "II", the question given below is one of them)	NA	Available
16	View ExplanationExplain	The value of $2 \int_{0}^{\frac{\pi}{2}} f(x) g(x) d x-\int_{0}^{\frac{\pi}{2}} g(x) d x$ is $\qquad$ . \section*{PARAGRAPH "II"} Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]$ be the function defined by $f(x)=\sin ^{2} x$ and let $g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^{2}}$ . (There are two questions based on PARAGRAPH "II", the question given below is one of them)	NA	Available
17	View ExplanationExplain	The value of $\frac{16}{\pi^{3}} \int_{0}^{\frac{\pi}{2}} f(x) g(x) d x$ is $\qquad$ .	Available	Available
18	View ExplanationExplain	A region in the form of an equilateral triangle (in $x-y$ plane) of height $L$ has a uniform magnetic field $\vec{B}$ pointing in the $+z$ -direction. A conducting loop PQR , in the form of an equilateral triangle of the same height $L$ , is placed in the $x-y$ plane with its vertex P at $x=0$ in the orientation shown in the figure. At $t=0$ , the loop starts entering the region of the magnetic field with a uniform velocity $\vec{v}$ along the $+x$ -direction. The plane of the loop and its orientation remain unchanged throughout its motion. ![] Which of the following graph best depicts the variation of the induced emf ( $E$ ) in the loop as a function of the distance ( $x$ ) starting from $x=0$ ?	Diagram Question	Available
19	View ExplanationExplain	A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M(\gg m)$ . The particle is moving in a circular orbit of radius $r_{0}$ with time period $T_{0}$ around the mass $M$ . Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_{\mathrm{c}}(r)=m \alpha / r^{3}$ , where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_{0}$ in the combined gravitational potential due to $M$ and $V_{\mathrm{c}}(r)$ , but with a new time period $T_{1}$ , then $\left(T_{1}^{2}-T_{0}^{2}\right) / T_{1}^{2}$ is given by [ $G$ is the gravitational constant.] (A) $\frac{3 \alpha}{G M r_{0}^{2}}$ (B) $\frac{\alpha}{2 G M r_{0}^{2}}$ (C) $\frac{\alpha}{G M r_{0}^{2}}$ (D) $\frac{2 \alpha}{G M r_{0}^{2}}$	Available	Available
20	View ExplanationExplain	A metal target with atomic number $Z=46$ is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio $r$ of the wavelengths of the $K_{\alpha}$ -line and the cut-off is found to be $r=2$ . If the same electron beam bombards another metal target with $Z=41$ , the value of $r$ will be (A) 2.53 (B) 1.27 (C) 2.24 (D) 1.58	Available	Available
21	View ExplanationExplain	A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $m$ and radius $r$ and it is in a uniform vertical magnetic field $B_{0}$ , as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $g$ , on two conducting supports at P and Q . When a current $I$ is passed through the loop, the loop turns about the line PQ by an angle $\theta$ given by (A) $\tan \theta=\pi r I B_{0} /(m g)$ (B) $\tan \theta=2 \pi r I B_{0} /(m g)$ (C) $\tan \theta=\pi r I B_{0} /(2 m g)$ (D) $\tan \theta=m g /\left(\pi r I B_{0}\right)$	Diagram Question	Available
22	View ExplanationExplain	A small electric dipole $\vec{p}_{0}$ , having a moment of inertia $I$ about its center, is kept at a distance $r$ from the center of a spherical shell of radius $R$ . The surface charge density $\sigma$ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $\theta$ as shown in the figure. While staying at a distance $r$ , the dipole is free to rotate about its center. ![] If released from rest, then which of the following statement(s) is(are) correct? [ $\varepsilon_{0}$ is the permittivity of free space.] (A) The dipole will undergo small oscillations at any finite value of $r$ . (B) The dipole will undergo small oscillations at any finite value of $r>R$ . (C) The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2 \sigma p_{0}}{\epsilon_{0} I}}$ at $r=2 R$ . (D) The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_{0}}{100 \epsilon_{0} I}}$ at $r=10 R$ .	Diagram Question	Available
23	View ExplanationExplain	A table tennis ball has radius $(3 / 2) \times 10^{-2} \mathrm{~m}$ and mass $(22 / 7) \times 10^{-3} \mathrm{~kg}$ . It is slowly pushed down into a swimming pool to a depth of $d=0.7 \mathrm{~m}$ below the water surface and then released from rest. It emerges from the water surface at speed $v$ , without getting wet, and rises up to a height $H$ . Which of the following option(s) is(are) correct? [Given: $\pi=22 / 7, g=10 \mathrm{~m} \mathrm{~s}^{-2}$ , density of water $=1 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ , viscosity of water $=1 \times 10^{-3}$ Pa-s.] (A) The work done in pushing the ball to the depth $d$ is 0.077 J . (B) If we neglect the viscous force in water, then the speed $v=7 \mathrm{~m} / \mathrm{s}$ . (C) If we neglect the viscous force in water, then the height $H=1.4 \mathrm{~m}$ . (D) The ratio of the magnitudes of the net force excluding the viscous force to the maximum viscous force in water is $500 / 9$ .	Available	Available
24	View ExplanationExplain	A positive, singly ionized atom of mass number $A_{\mathrm{M}}$ is accelerated from rest by the voltage 192 V . Thereafter, it enters a rectangular region of width $w$ with magnetic field $\vec{B}_{0}=0.1 \widehat{k}$ Tesla, as shown in the figure. The ion finally hits a detector at the distance $x$ below its starting trajectory. [Given: Mass of neutron/proton $=(5 / 3) \times 10^{-27} \mathrm{~kg}$ , charge of the electron $=1.6 \times 10^{-19} \mathrm{C}$ .] ![] Which of the following option(s) is(are) correct? (A) The value of $x$ for $H^{+}$ ion is 4 cm . (B) The value of $x$ for an ion with $A_{\mathrm{M}}=144$ is 48 cm . (C) For detecting ions with $1 \leq A_{\mathrm{M}} \leq 196$ , the minimum height ( $x_{1}-x_{0}$ ) of the detector is 55 cm . (D) The minimum width $w$ of the region of the magnetic field for detecting ions with $A_{\mathrm{M}}=196$ is 56 cm .	Diagram Question	Available
25	View ExplanationExplain	The dimensions of a cone are measured using a scale with a least count of 2 mm . The diameter of the base and the height are both measured to be 20.0 cm . The maximum percentage error in the determination of the volume is $\qquad$ .	Available	Available
26	View ExplanationExplain	A ball is thrown from the location $\left(x_{0}, y_{0}\right)=(0,0)$ of a horizontal playground with an initial speed $v_{0}$ at an angle $\theta_{0}$ from the $+x$ -direction. The ball is to be hit by a stone, which is thrown at the same time from the location $\left(x_{1}, y_{1}\right)=(L, 0)$ . The stone is thrown at an angle ( $180-\theta_{1}$ ) from the $+x$ -direction with a suitable initial speed. For a fixed $v_{0}$ , when $\left(\theta_{0}, \theta_{1}\right)=\left(45^{\circ}, 45^{\circ}\right)$ , the stone hits the ball after time $T_{1}$ , and when $\left(\theta_{0}, \theta_{1}\right)=\left(60^{\circ}, 30^{\circ}\right)$ , it hits the ball after time $T_{2}$ . In such a case, $\left(T_{1} / T_{2}\right)^{2}$ is $\qquad$ .	Available	Available
27	View ExplanationExplain	A charge is kept at the central point P of a cylindrical region. The two edges subtend a half-angle $\theta$ at P , as shown in the figure. When $\theta=30^{\circ}$ , then the electric flux through the curved surface of the cylinder is $\Phi$ . If $\theta=60^{\circ}$ , then the electric flux through the curved surface becomes $\Phi / \sqrt{n}$ , where the value of $n$ is $\qquad$ .	Diagram Question	Available
28	View ExplanationExplain	Two equilateral-triangular prisms $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism $\mathrm{P}_{1}$ at an angle of incidence $\theta$ such that the outgoing ray undergoes minimum deviation in prism $P_{2}$ . If the respective refractive indices of $P_{1}$ and $\mathrm{P}_{2}$ are $\sqrt{\frac{3}{2}}$ and $\sqrt{3}$ , then $\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]$ , where the value of $\beta$ is $\qquad$ .	Diagram Question	Available
29	View ExplanationExplain	An infinitely long thin wire, having a uniform charge density per unit length of $5 \mathrm{nC} / \mathrm{m}$ , is passing through a spherical shell of radius 1 m , as shown in the figure. A 10 nC charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points P and R , in Volt, is $\qquad$ . [Given: In SI units $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9}, \ln 2=0.7$ . Ignore the area pierced by the wire.]	Diagram Question	Available
30	View ExplanationExplain	A spherical soap bubble inside an air chamber at pressure $P_{0}=10^{5} \mathrm{~Pa}$ has a certain radius so that the excess pressure inside the bubble is $\Delta P=144 \mathrm{~Pa}$ . Now, the chamber pressure is reduced to $8 P_{0} / 27$ so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure $\Delta P$ in both the cases to be much smaller than the chamber pressure. The new excess pressure $\Delta P$ in Pa is.	Available	Available
31	View ExplanationExplain	The $8^{\text {th }}$ bright fringe above the point O oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer ( $\mu \mathrm{m}$ ), is $\qquad$ . \section*{PARAGRAPH I} In a Young's double slit experiment, each of the two slits $A$ and $B$ , as shown in the figure, are oscillating about their fixed center and with a mean separation of 0.8 mm . The distance between the slits at time $t$ is given by $d=(0.8+0.04 \sin \omega t) \mathrm{mm}$ , where $\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}$ . The distance of the screen from the slits is 1 m and the wavelength of the light used to illuminate the slits is $6000 \AA$ . The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point O . ![]	Diagram Question	Available
32	View ExplanationExplain	The maximum speed in $\mu \mathrm{m} / \mathrm{s}$ at which the $8^{\text {th }}$ bright fringe will move is $\qquad$ . \section*{PARAGRAPH I} In a Young's double slit experiment, each of the two slits $A$ and $B$ , as shown in the figure, are oscillating about their fixed center and with a mean separation of 0.8 mm . The distance between the slits at time $t$ is given by $d=(0.8+0.04 \sin \omega t) \mathrm{mm}$ , where $\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}$ . The distance of the screen from the slits is 1 m and the wavelength of the light used to illuminate the slits is $6000 \AA$ . The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point O . ![].	Diagram Question	Available
33	View ExplanationExplain	If the collision occurs at time $t_{0}=0$ , the value of $v_{\mathrm{cm}} /(a \omega)$ will be $\qquad$ . \section*{PARAGRAPH II} Two particles, 1 and 2, each of mass $m$ , are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $x_{0}$ , are oscillating with amplitude $a$ and angular frequency $\omega$ . Thus, their positions at time $t$ are given by $x_{1}(t)=\left(x_{0}+d\right)+a \sin \omega t$ and $x_{2}(t)=\left(x_{0}-d\right)-a \sin \omega t$ , respectively, where $d>2 a$ . Particle 3 of mass $m$ moves towards this system with speed $u_{0}=a \omega / 2$ , and undergoes instantaneous elastic collision with particle 2 , at time $t_{0}$ . Finally, particles 1 and 2 acquire a center of mass speed $v_{\mathrm{cm}}$ and oscillate with amplitude $b$ and the same angular frequency $\omega$ . ![]	Diagram Question	Available
34	View ExplanationExplain	If the collision occurs at time $t_{0}=\pi /(2 \omega)$ , then the value of $4 b^{2} / a^{2}$ will be $\qquad$ . \section*{PARAGRAPH II} Two particles, 1 and 2, each of mass $m$ , are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $x_{0}$ , are oscillating with amplitude $a$ and angular frequency $\omega$ . Thus, their positions at time $t$ are given by $x_{1}(t)=\left(x_{0}+d\right)+a \sin \omega t$ and $x_{2}(t)=\left(x_{0}-d\right)-a \sin \omega t$ , respectively, where $d>2 a$ . Particle 3 of mass $m$ moves towards this system with speed $u_{0}=a \omega / 2$ , and undergoes instantaneous elastic collision with particle 2 , at time $t_{0}$ . Finally, particles 1 and 2 acquire a center of mass speed $v_{\mathrm{cm}}$ and oscillate with amplitude $b$ and the same angular frequency $\omega$ . ![]	Diagram Question	Available
35	View ExplanationExplain	According to Bohr's model, the highest kinetic energy is associated with the electron in the (A) first orbit of H atom (B) first orbit of $\mathrm{He}^{+}$ (C) second orbit of $\mathrm{He}^{+}$ (D) second orbit of $\mathrm{Li}^{2+}$	Available	Available
36	View ExplanationExplain	In a metal deficient oxide sample, $\mathbf{M}_{\mathbf{X}} \mathbf{Y}_{\mathbf{2}} \mathbf{O}_{\mathbf{4}}$ ( $\mathbf{M}$ and $\mathbf{Y}$ are metals), $\mathbf{M}$ is present in both +2 and +3 oxidation states and $\mathbf{Y}$ is in +3 oxidation state. If the fraction of $\mathbf{M}^{2+}$ ions present in $\mathbf{M}$ is $\frac{1}{3}$ , the value of $\mathbf{X}$ is $\qquad$ . (A) 0.25 (B) 0.33 (C) 0.67 (D) 0.75	Available	Available
37	View ExplanationExplain	In the following reaction sequence, the major product $\mathbf{Q}$ is ![]	Diagram Question	Available
38	View ExplanationExplain	The species formed on fluorination of phosphorus pentachloride in a polar organic solvent are (A) $\left[\mathrm{PF}_{4}\right]^{+}\left[\mathrm{PF}_{6}\right]^{-}$ and $\left[\mathrm{PCl}_{4}\right]^{+}\left[\mathrm{PF}_{6}\right]^{-}$ (B) $\left[\mathrm{PCl}_{4}\right]^{+}\left[\mathrm{PCl}_{4} \mathrm{~F}_{2}\right]^{-}$ and $\left[\mathrm{PCl}_{4}\right]^{+}\left[\mathrm{PF}_{6}\right]^{-}$ (C) $\quad \mathrm{PF}_{3}$ and $\mathrm{PCl}_{3}$ (D) $\mathrm{PF}_{5}$ and $\mathrm{PCl}_{3}$	Available	Available
39	View ExplanationExplain	An aqueous solution of hydrazine $\left(\mathrm{N}_{2} \mathrm{H}_{4}\right)$ is electrochemically oxidized by $\mathrm{O}_{2}$ , thereby releasing chemical energy in the form of electrical energy. One of the products generated from the electrochemical reaction is $\mathrm{N}_{2}(\mathrm{~g})$ . Choose the correct statement(s) about the above process (A) $\quad \mathrm{OH}^{-}$ ions react with $\mathrm{N}_{2} \mathrm{H}_{4}$ at the anode to form $\mathrm{N}_{2}(\mathrm{~g})$ and water, releasing 4 electrons to the anode. (B) At the cathode, $\mathrm{N}_{2} \mathrm{H}_{4}$ breaks to $\mathrm{N}_{2}(\mathrm{~g})$ and nascent hydrogen released at the electrode reacts with oxygen to form water. (C) At the cathode, molecular oxygen gets converted to $\mathrm{OH}^{-}$ . (D) Oxides of nitrogen are major by-products of the electrochemical process.	Available	Available
40	View ExplanationExplain	The option(s) with correct sequence of reagents for the conversion of $\mathbf{P}$ to $\mathbf{Q}$ is(are) ![] (A) i) Lindlar's catalyst, $\mathrm{H}_{2}$ ; ii) $\mathrm{SnCl}_{2} / \mathrm{HCl}$ ; iii) $\mathrm{NaBH}_{4}$ ; iv) $\mathrm{H}_{3} \mathrm{O}^{+}$ (B) i) Lindlar's catalyst, $\mathrm{H}_{2}$ ; ii) $\mathrm{H}_{3} \mathrm{O}^{+}$ ; iii) $\mathrm{SnCl}_{2} / \mathrm{HCl}$ ; iv) $\mathrm{NaBH}_{4}$ (C) i) $\mathrm{NaBH}_{4}$ ; ii) $\mathrm{SnCl}_{2} / \mathrm{HCl}$ ; iii) $\mathrm{H}_{3} \mathrm{O}^{+}$ ; iv) Lindlar's catalyst, $\mathrm{H}_{2}$ (D) i) Lindlar's catalyst, $\mathrm{H}_{2}$ ; ii) $\mathrm{NaBH}_{4}$ ; iii) $\mathrm{SnCl}_{2} / \mathrm{HCl}$ ; iv) $\mathrm{H}_{3} \mathrm{O}^{+}$	Diagram Question	Available
41	View ExplanationExplain	The compound(s) having peroxide linkage is(are) (A) $\mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{7}$ (B) $\mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{8}$ (C) $\mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{5}$ (D) $\mathrm{H}_{2} \mathrm{SO}_{5}$	Available	Available
42	View ExplanationExplain	To form a complete monolayer of acetic acid on 1 g of charcoal, 100 mL of 0.5 M acetic acid was used. Some of the acetic acid remained unadsorbed. To neutralize the unadsorbed acetic acid, 40 mL of 1 M NaOH solution was required. If each molecule of acetic acid occupies $\mathbf{P} \times 10^{-23} \mathrm{~m}^{2}$ surface area on charcoal, the value of $\mathbf{P}$ is $\qquad$ . [Use given data: Surface area of charcoal $=1.5 \times 10^{2} \mathrm{~m}^{2} \mathrm{~g}^{-1}$ ; Avogadro's number $\left(\mathrm{N}_{\mathrm{A}}\right)=6.0 \times 10^{23} \mathrm{mol}^{-1}$ ]	Available	Available
43	View ExplanationExplain	Vessel-1 contains $\mathbf{w}_{2} \mathrm{~g}$ of a non-volatile solute $\mathbf{X}$ dissolved in $\mathbf{w}_{1} \mathrm{~g}$ of water. Vessel-2 contains $\mathbf{w}_{2} \mathrm{~g}$ of another non-volatile solute $\mathbf{Y}$ dissolved in $\mathbf{w}_{\mathbf{1}} \mathbf{g}$ of water. Both the vessels are at the same temperature and pressure. The molar mass of $\mathbf{X}$ is $80 \%$ of that of $\mathbf{Y}$ . The van't Hoff factor for $\mathbf{X}$ is 1.2 times of that of $\mathbf{Y}$ for their respective concentrations. The elevation of boiling point for solution in Vessel-1 is $\qquad$ \% of the solution in Vessel-2.	Available	Available
44	View ExplanationExplain	For a double strand DNA, one strand is given below: ![] The amount of energy required to split the double strand DNA into two single strands is $\qquad$ kcal $\mathrm{mol}^{-1}$ . [Given: Average energy per H-bond for A-T base pair $=1.0 \mathrm{kcal} \mathrm{mol}^{-1}$ , G-C base pair $=1.5 \mathrm{kcal} \mathrm{mol}^{-1}$ , and A-U base pair $=1.25 \mathrm{kcal} \mathrm{mol}^{-1}$ . Ignore electrostatic repulsion between the phosphate groups.]	Diagram Question	Available
45	View ExplanationExplain	A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into $\mathrm{Pb}-206$ while the rest of it remains undisintegrated. When the age of the sample is $\mathbf{P} \times 10^{8}$ years, the ratio of mass of $\mathrm{Pb}-206$ to that of U-238 in the sample is found to be 7. The value of $\mathbf{P}$ is $\qquad$ . [Given: Half-life of U-238 is $4.5 \times 10^{9}$ years; $\log _{\mathrm{e}} 2=0.693$ ]	Available	Available
46	View ExplanationExplain	Among $\left[\mathrm{Co}(\mathrm{CN})_{4}\right]^{4-},\left[\mathrm{Co}(\mathrm{CO})_{3}(\mathrm{NO})\right], \mathrm{XeF}_{4},\left[\mathrm{PCl}_{4}\right]^{+},\left[\mathrm{PdCl}_{4}\right]^{2-},\left[\mathrm{ICl}_{4}\right]^{-},\left[\mathrm{Cu}(\mathrm{CN})_{4}\right]^{3-}$ and $\mathrm{P}_{4}$ the total number of species with tetrahedral geometry is $\qquad$ .	Available	Available
47	View ExplanationExplain	An organic compound $\mathbf{P}$ having molecular formula $\mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}_{3}$ gives ferric chloride test and does not have intramolecular hydrogen bond. The compound $\mathbf{P}$ reacts with 3 equivalents of $\mathrm{NH}_{2} \mathrm{OH}$ to produce oxime $\mathbf{Q}$ . Treatment of $\mathbf{P}$ with excess methyl iodide in the presence of KOH produces compound $\mathbf{R}$ as the major product. Reaction of $\mathbf{R}$ with excess iso-butylmagnesium bromide followed by treatment with $\mathrm{H}_{3} \mathrm{O}^{+}$ gives compound $\mathbf{S}$ as the major product. The total number of methyl ( $-\mathrm{CH}_{3}$ ) group(s) in compound $\mathbf{S}$ is $\qquad$ .	Available	Available
48	View ExplanationExplain	An organic compound $\mathbf{P}$ with molecular formula $\mathrm{C}_{9} \mathrm{H}_{18} \mathrm{O}_{2}$ decolorizes bromine water and also shows positive iodoform test. $\mathbf{P}$ on ozonolysis followed by treatment with $\mathrm{H}_{2} \mathrm{O}_{2}$ gives $\mathbf{Q}$ and $\mathbf{R}$ . While compound $\mathbf{Q}$ shows positive iodoform test, compound $\mathbf{R}$ does not give positive iodoform test. $\mathbf{Q}$ and $\mathbf{R}$ on oxidation with pyridinium chlorochromate (PCC) followed by heating give $\mathbf{S}$ and $\mathbf{T}$ , respectively. Both $\mathbf{S}$ and $\mathbf{T}$ show positive iodoform test. Complete copolymerization of 500 moles of $\mathbf{Q}$ and 500 moles of $\mathbf{R}$ gives one mole of a single acyclic copolymer U. [Given, atomic mass: $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16$ ] Sum of number of oxygen atoms in $\mathbf{S}$ and $\mathbf{T}$ is $\qquad$ .	Available	Available
49	View ExplanationExplain	An organic compound $\mathbf{P}$ with molecular formula $\mathrm{C}_{9} \mathrm{H}_{18} \mathrm{O}_{2}$ decolorizes bromine water and also shows positive iodoform test. $\mathbf{P}$ on ozonolysis followed by treatment with $\mathrm{H}_{2} \mathrm{O}_{2}$ gives $\mathbf{Q}$ and $\mathbf{R}$ . While compound $\mathbf{Q}$ shows positive iodoform test, compound $\mathbf{R}$ does not give positive iodoform test. $\mathbf{Q}$ and $\mathbf{R}$ on oxidation with pyridinium chlorochromate (PCC) followed by heating give $\mathbf{S}$ and $\mathbf{T}$ , respectively. Both $\mathbf{S}$ and $\mathbf{T}$ show positive iodoform test. Complete copolymerization of 500 moles of $\mathbf{Q}$ and 500 moles of $\mathbf{R}$ gives one mole of a single acyclic copolymer U. [Given, atomic mass: $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16$ ] The molecular weight of $\mathbf{U}$ is $\qquad$ .	Available	Available
50	View ExplanationExplain	When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex $\mathbf{P}$ is formed. In a strong acidic medium, the equilibrium shifts completely towards $\mathbf{P}$ . Addition of zinc chloride to $\mathbf{P}$ in a slightly acidic medium results in a sparingly soluble complex Q. The number of moles of potassium iodide required to produce two moles of $\mathbf{P}$ is $\qquad$ .	Available	Available
51	View ExplanationExplain	When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex $\mathbf{P}$ is formed. In a strong acidic medium, the equilibrium shifts completely towards $\mathbf{P}$ . Addition of zinc chloride to $\mathbf{P}$ in a slightly acidic medium results in a sparingly soluble complex Q. The number of zinc ions present in the molecular formula of $\mathbf{Q}$ is $\qquad$ .	Available	Available