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JEE-ADVANCED EXAMINATION – JUNE 2023

JEE-ADVANCED TEST PAPER 2 WITH SOLUTION

Held on Sunday 04th June 2023, 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	Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and\n $3 \int_{1}^{x} f(t) d t=x f(x)-\frac{x^{3}}{3}, x \in[1, \infty)$ . Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is (A) $\frac{e^{2}+4}{3}$ (B) $\frac{\log _{e} 4+e}{3}$ (C) $\frac{4 e^{2}}{3}$ (D) $\frac{e^{2}-4}{3}$	Available	Available
2	View ExplanationExplain	Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$ , then the probability that the experiment stops with head is (A) $\frac{1}{3}$ (B) $\frac{5}{21}$ (C) $\frac{4}{21}$ (D) $\frac{2}{7}$	Available	Available
3	View ExplanationExplain	For any $y \in \mathbb{R}$ , let $\cot ^{-1}(y) \in(0, \pi)$ and $\tan ^{-1}(y) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ . Then the sum of all the solutions of the equation $\tan ^{-1}\left(\frac{6 y}{9-y^{2}}\right)+\cot ^{-1}\left(\frac{9-y^{2}}{6 y}\right)=\frac{2 \pi}{3}$ for $0<\|y\|<3$ , is equal to (A) $2 \sqrt{3}-3$ (B) $3-2 \sqrt{3}$ (C) $4 \sqrt{3}-6$ (D) $6-4 \sqrt{3}$	Available	Available
4	View ExplanationExplain	Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$ , $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$ , respectively. Then which of the following statements is true? (A) The points $P, Q, R$ and $S$ are NOT coplanar (B) $\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ internally in the ratio $5: 4$ (C) $\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ externally in the ratio $5: 4$ (D) The square of the magnitude of the vector $\vec{b} \times \vec{d}$ is 95	Available	Available
5	View ExplanationExplain	Let $M=\left(a_{i j}\right), i, j \in\{1,2,3\}$ , be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$ , otherwise $a_{i j}=0$ . Then which of the following statements is(are) true? (A) $M$ is invertible (B) There exists a nonzero column matrix $\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\end{array}\right)$ such that $M\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\end{array}\right)=\left(\begin{array}{l}-a_{1} \\ -a_{2} \\ -a_{3}\end{array}\right)$ (C) The set $\left\{X \in \mathbb{R}^{3}: M X=\mathbf{0}\right\} \ eq\{\mathbf{0}\} $, where$ \mathbf{0}=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$ (D) The matrix ( $M-2 I$ ) is invertible, where $I$ is the $3 \times 3$ identity matrix	Available	Available
6	View ExplanationExplain	Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^{2}\left(x-\frac{1}{2}\right)$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following statements is(are) true? (A) The function $f$ is discontinuous exactly at one point in $(0,1)$ (B) There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable (C) The function $f$ is NOT differentiable at more than three points in $(0,1)$ (D) The minimum value of the function $f$ is $-\frac{1}{512}$	Available	Available
7	View ExplanationExplain	Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^{2} f}{d x^{2}}(x)>0$ for all $x \in(-1,1)$ . For $f \in S$ , let $X_{f}$ be the number of points $x \in(-1,1)$ for which $f(x)=x$ . Then which of the following statements is(are) true? (A) There exists a function $f \in S$ such that $X_{f}=0$ (B) For every function $f \in S$ , we have $X_{f} \leq 2$ (C) There exists a function $f \in S$ such that $X_{f}=2$ (D) There does NOT exist any function $f$ in $S$ such that $X_{f}=1$	Available	Available
8	View ExplanationExplain	For $x \in \mathbb{R}$ , let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ . Then the minimum value of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\int_{0}^{x \tan ^{-1} x} \frac{e^{(t-\cos t)}}{1+t^{2023}} d t$ is	Available	Available
9	View ExplanationExplain	For $x \in \mathbb{R}$ , let $y(x)$ be a	Available	Available
10	View ExplanationExplain	Let $X$ be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$ . Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5 . Then the value of $38 p$ is equal to	Available	Available
11	View ExplanationExplain	Let $A_{1}, A_{2}, A_{3}, \ldots, A_{8}$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P A_{i}$ denote the distance between the points $P$ and $A_{i}$ for $i=1,2, \ldots, 8$ . If $P$ varies over the circle, then the maximum value of the product $P A_{1} \cdot P A_{2} \cdots \cdots P A_{8}$ , is	Available	Available
12	View ExplanationExplain	Let $R=\left\{\left(\begin{array}{lll}a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0\end{array}\right): a, b, c, d \in\{0,3,5,7,11,13,17,19\}\right\}$ . Then the number of invertible matrices in $R$ is	Available	Available
13	View ExplanationExplain	Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point $A=(4,1)$ , where $1<r<3$ . Two distinct common tangents $P Q$ and $S T$ of $C_{1}$ and $C_{2}$ are drawn. The tangent $P Q$ touches $C_{1}$ at $P$ and $C_{2}$ at $Q$ . The tangent $S T$ touches $C_{1}$ at $S$ and $C_{2}$ at $T$ . Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$ -axis at a point $B$ . If $A B=\sqrt{5}$ , then the value of $r^{2}$ is	Available	Available
14	View ExplanationExplain	Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1 . Let $a$ be the area of the triangle $A B C$ . Then the value of $(64 a)^{2}$ is	Available	Available
15	View ExplanationExplain	Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1 . Let $a$ be the area of the triangle $A B C$ . Then the inradius of the triangle $A B C$ is	Available	Available
16	View ExplanationExplain	Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \ldots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{i}$ and $A_{j}$ are friends if they are adjacent along a row or along a column. Assume that each point $A_{i}$ has an equal chance of being chosen. Let $p_{i}$ be the probability that a randomly chosen point has $i$ many friends, $i=0,1,2,3,4$ . Let $X$ be a random variable such that for $i=0,1,2,3,4$ , the probability $P(X=i)=p_{i}$ . Then the value of $7 E(X)$ is	Available	Available
17	View ExplanationExplain	Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \ldots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{i}$ and $A_{j}$ are friends if they are adjacent along a row or along a column. Assume that each point $A_{i}$ has an equal chance of being chosen. Two distinct points are chosen randomly out of the points $A_{1}, A_{2}, \ldots, A_{49}$ . Let $p$ be the probability that they are friends. Then the value of $7 p$ is	Available	Available
18	View ExplanationExplain	An electric dipole is formed by two charges $+q$ and $-q$ located in $x y$ -plane at $(0,2) \mathrm{mm}$ and $(0,-2) \mathrm{mm}$ , respectively, as shown in the figure. The electric potential at point $\mathrm{P}(100,100) \mathrm{mm}$ due to the dipole is $V_{0}$ . The charges $+q$ and $-q$ are then moved to the points $(-1,2) \mathrm{mm}$ and $(1,-2) \mathrm{mm}$ , respectively. What is the value of electric potential at P due to the new dipole?\n (A) $V_{0} / 4$ (B) $V_{0} / 2$ (C) $V_{0} / \sqrt{2}$ (D) $3 V_{0} / 4$	Diagram Question	Available
19	View ExplanationExplain	Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$ , Planck's constant $h$ and the speed of light $c$ , as $Y=c^{\alpha} h^{\beta} G^{\gamma}$ . Which of the following is the correct option? (A) $\alpha=7, \beta=-1, \gamma=-2$ (B) $\alpha=-7, \beta=-1, \gamma=-2$ (C) $\alpha=7, \beta=-1, \gamma=2$ (D) $\alpha=-7, \beta=1, \gamma=-2$	Available	Available
20	View ExplanationExplain	A particle of mass $m$ is moving in the $x y$ -plane such that its velocity at a point ( $x, y$ ) is given as $\overrightarrow{\mathrm{v}}=\alpha(y \hat{x}+2 x \hat{y})$ , where $\alpha$ is a non-zero constant. What is the force $\vec{F}$ acting on the particle? (A) $\vec{F}=2 m \alpha^{2}(x \hat{x}+y \hat{y})$ (B) $\vec{F}=m \alpha^{2}(y \hat{x}+2 x \hat{y})$ (C) $\vec{F}=2 m \alpha^{2}(y \hat{x}+x \hat{y})$ (D) $\vec{F}=m \alpha^{2}(x \hat{x}+2 y \hat{y})$	Available	Available
21	View ExplanationExplain	An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$ . The internal energy of one mole of the gas is $U_{n}$ and the speed of sound in the gas is $\mathrm{v}_{n}$ . At a fixed temperature and pressure, which of the following is the correct option? (A) $\mathrm{v}_{3}<\mathrm{v}_{6}$ and $U_{3}>U_{6}$ (B) $\mathrm{v}_{5}>\mathrm{v}_{3}$ and $U_{3}>U_{5}$ (C) $\mathrm{v}_{5}>\mathrm{v}_{7}$ and $U_{5}<U_{7}$ (D)	Available	Available
22	View ExplanationExplain	A monochromatic light wave is incident normally on a glass slab of thickness $d$ , as shown in the figure. The refractive index of the slab increases linearly from $n_{1}$ to $n_{2}$ over the height $h$ . Which of the following statement(s) is(are) true about the light wave emerging out of the slab?\n (A) It will deflect up by an angle $\tan ^{-1}\left[\frac{\left(n_{2}^{2}-n_{1}^{2}\right) d}{2 h}\right]$ . (B) It will deflect up by an angle $\tan ^{-1}\left[\frac{\left(n_{2}-n_{1}\right) d}{h}\right]$ . (C) It will not deflect. (D) The deflection angle depends only on ( $n_{2}-n_{1}$ ) and not on the individual values of $n_{1}$ and $n_{2}$ .	Diagram Question	Available
23	View ExplanationExplain	An annular disk of mass $M$ , inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$ , as shown in the figure. At some time, an impulse $\mathcal{J}_{0} \widehat{x}$ is applied at a height $h$ above the center of the disk. If $h=h_{m}$ then the disk rolls without slipping along the $x$ -axis. Which of the following statement(s) is(are) correct?\n (A) For $\mu \ eq 0 $and$ a \rightarrow 0, h_{m}=b / 2$. (B) For $\mu \ eq 0 $and$ a \rightarrow b, h_{m}=b$. (C) For $h=h_{m}$ , the initial angular velocity does not depend on the inner radius $a$ . (D)	Diagram Question	Available
24	View ExplanationExplain	The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}=30(2 \hat{x}+\hat{y}) \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^{7}}{3} z\right)\right] \mathrm{Vm}^{-1}$ . Which of the following option(s) is(are) correct?\n[Given: The speed of light in vacuum, $c=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$ ] (A) $B_{x}=-2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^{7}}{3} z\right)\right] \mathrm{Wbm}^{-2}$ . (B) $B_{y}=2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^{7}}{3} z\right)\right] \mathrm{Wb} \mathrm{m}^{-2}$ . (C) The wave is polarized in the $x y$ -plane with polarization angle $30^{\circ}$ with respect to the $x$ -axis. (D) The refractive index of the medium is 2 .	Available	Available
25	View ExplanationExplain	A thin circular coin of mass 5 gm and radius $4 / 3 \mathrm{~cm}$ is initially in a horizontal $x y$ -plane. The coin is tossed vertically up ( $+z$ direction) by applying an impulse of $\sqrt{\frac{\pi}{2}} \times 10^{-2} \mathrm{~N}$ -s at a distance $2 / 3 \mathrm{~cm}$ from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is $\qquad$ .\n[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ ]\n	Diagram Question	Available
26	View ExplanationExplain	A rectangular conducting loop of length 4 cm and width 2 cm is in the $x y$ -plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2} \hat{x}+\frac{1}{2} \hat{y}$ with a constant speed v . The wire is carrying a steady current $I=10 \mathrm{~A}$ in the positive $x$ -direction. A current of $10 \mu \mathrm{~A}$ flows through the loop when it is at a distance $d=4 \mathrm{~cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$ , then the value of $v$ is $\qquad$ $\mathrm{m} \mathrm{s}^{-1}$ .\n[Given: The permeability of free space $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~N} \mathrm{~A}^{-2}$ ]\n	Diagram Question	Available
27	View ExplanationExplain	A string of length 1 m and mass $2 \times 10^{-5} \mathrm{~kg}$ is under tension $T$ . When the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz . The value of tension $T$ is $\qquad$ Newton.	Available	Available
28	View ExplanationExplain	An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_{0}$ to $\frac{100}{101} V_{0}$ with the movable piston. Considering air as an ideal gas, the height $(h)$ of the liquid column in the capillary above the liquid level in cm is $\qquad$ .\n[Given: Surface tension of the liquid is $0.075 \mathrm{~N} \mathrm{~m}^{-1}$ , atmospheric pressure is $10^{5} \mathrm{~N} \mathrm{~m}^{-2}$ , acceleration due to gravity ( $g$ ) is $10 \mathrm{~m} \mathrm{~s}^{-2}$ , density of the liquid is $10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ and contact angle of capillary surface with the liquid is zero]\n	Diagram Question	Available
29	View ExplanationExplain	In a radioactive decay process, the activity is defined as $A=-\frac{d N}{d t}$ , where $N(t)$ is the number of radioactive nuclei at time $t$ . Two radioactive sources, $S_{1}$ and $S_{2}$ have same activity at time $t=0$ . At a later time, the activities of $S_{1}$ and $S_{2}$ are $A_{1}$ and $A_{2}$ , respectively. When $S_{1}$ and $S_{2}$ have just completed their $3^{\text {rd }}$ and $7^{\text {th }}$ half-lives, respectively, the ratio $A_{1} / A_{2}$ is $\qquad$ .	Available	Available
30	View ExplanationExplain	One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the $P-V$ diagrams below. In cycle I, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes $a^{\prime}, b^{\prime}, c^{\prime}$ and $d^{\prime}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is $W_{I}$ and that during cycle II is $W_{I I}$ . The ratio $W_{I} / W_{I I}$ is $\qquad$ .	Diagram Question	Available
31	View ExplanationExplain	$S_{1}$ and $S_{2}$ are two identical sound sources of frequency 656 Hz . The source $S_{1}$ is located at $O$ and $S_{2}$ moves anti-clockwise with a uniform speed $4 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$ on a circular path around $O$ , as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$ . The source $S_{1}$ can move along direction $O P$ .\n[Given: The speed of sound in air is $324 \mathrm{~m} \mathrm{~s}^{-1}$ ]. When only $S_{2}$ is emitting sound and it is at $Q$ , the frequency of sound measured by the detector in Hz is $\qquad$ .	Diagram Question	Available
32	View ExplanationExplain	$S_{1}$ and $S_{2}$ are two identical sound sources of frequency 656 Hz . The source $S_{1}$ is located at $O$ and $S_{2}$ moves anti-clockwise with a uniform speed $4 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$ on a circular path around $O$ , as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$ . The source $S_{1}$ can move along direction $O P$ .\n[Given: The speed of sound in air is $324 \mathrm{~m} \mathrm{~s}^{-1}$ ]. Consider both sources emitting sound. When $S_{2}$ is at $R$ and $S_{1}$ approaches the detector with a speed $4 \mathrm{~m} \mathrm{~s}^{-1}$ , the beat frequency measured by the detector is $\qquad$ Hz.	Diagram Question	Available
33	View ExplanationExplain	A cylindrical furnace has height $(H)$ and diameter $(D)$ both 1 m . It is maintained at temperature 360 K . The air gets heated inside the furnace at constant pressure $P_{a}$ and its temperature becomes $T=360 \mathrm{~K}$ . The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{~m}$ and height $h=9 \mathrm{~m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $\rho_{a}= 1.2 \mathrm{~kg} \mathrm{~m}^{-3}$ , pressure $P_{a}$ and temperature $T_{a}=300 \mathrm{~K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.\n[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $\pi=3.14$ ]. Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is $\qquad$ $\mathrm{gm} \mathrm{s}^{-1}$ .	Diagram Question	Available
34	View ExplanationExplain	A cylindrical furnace has height $(H)$ and diameter $(D)$ both 1 m . It is maintained at temperature 360 K . The air gets heated inside the furnace at constant pressure $P_{a}$ and its temperature becomes $T=360 \mathrm{~K}$ . The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{~m}$ and height $h=9 \mathrm{~m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $\rho_{a}= 1.2 \mathrm{~kg} \mathrm{~m}^{-3}$ , pressure $P_{a}$ and temperature $T_{a}=300 \mathrm{~K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.\n[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $\pi=3.14$ ]. When the chimney is closed using a cap at the top, a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $\Delta P$ is $\qquad$ $\mathrm{N} \mathrm{m}^{-2}$ .	Diagram Question	Available
35	View ExplanationExplain	The correct molecular orbital diagram for $\mathrm{F}_{2}$ molecule in the ground state is\n\n\begin{figure}\n\captionsetup{labelformat=empty}\n\caption{	Diagram Question	Available
36	View ExplanationExplain	Consider the following statements related to colloids.\n(I) Lyophobic colloids are not formed by simple mixing of dispersed phase and dispersion medium.\n(II) For emulsions, both the dispersed phase and the dispersion medium are liquid.\n(III) Micelles are produced by dissolving a surfactant in any solvent at any temperature.\n(IV) Tyndall effect can be observed from a colloidal solution with dispersed phase having the same refractive index as that of the dispersion medium.\n\nThe option with the correct set of statements is (A) (I) and (II) (B) (II) and (III) (C) (III) and (IV) (D) (II) and (IV)	Available	Available
37	View ExplanationExplain	In the following reactions, $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ , and $\mathbf{S}$ are the major products.\n\n\n\n\n\nThe correct statement about $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ , and $\mathbf{S}$ is (A) $\mathbf{P}$ is a primary alcohol with four carbons. (B) $\mathbf{Q}$ undergoes Kolbe's electrolysis to give an eight-carbon product. (C) $\mathbf{R}$ has six carbons and it undergoes Cannizzaro reaction. (D)	Diagram Question	Available
38	View ExplanationExplain	A disaccharide $\mathbf{X}$ cannot be oxidised by bromine water. The acid hydrolysis of $\mathbf{X}$ leads to a laevorotatory solution. The disaccharide $\mathbf{X}$ is (A) (B) (C) (D)	Diagram Question	Available
39	View ExplanationExplain	The complex(es), which can exhibit the type of isomerism shown by $\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Br}_{2}\right]$ , is(are) [ $\mathrm{en}=\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$ ] (A) $\left[\operatorname{Pt}(\mathrm{en})(\mathrm{SCN})_{2}\right]$ (B) $\left[\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]$ (C) $\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{4}\right]$ (D) $\left[\mathrm{Cr}(\mathrm{en})_{2}\left(\mathrm{H}_{2} \mathrm{O}\right)\left(\mathrm{SO}_{4}\right)\right]^{+}$	Available	Available
40	View ExplanationExplain	Atoms of metals $x, y$ , and $z$ form face-centred cubic (fcc) unit cell of edge length $L_{x}$ , body-centred cubic (bcc) unit cell of edge length $\mathrm{L}_{\mathrm{y}}$ , and simple cubic unit cell of edge length $\mathrm{L}_{\mathrm{z}}$ , respectively.\n\nIf $r_{z}=\frac{\sqrt{3}}{2} r_{y} ; r_{y}=\frac{8}{\sqrt{3}} r_{x} ; M_{z}=\frac{3}{2} M_{y}$ and $M_{z}=3 M_{x}$ , then the correct statement(s) is(are)\n[Given: $M_{x}, M_{y}$ , and $M_{z}$ are molar masses of metals $x, y$ , and $z$ , respectively. $r_{x}, r_{y}$ , and $r_{z}$ are atomic radii of metals $x, y$ , and $z$ , respectively.] (A) Packing efficiency of unit cell of $x>$ Packing efficiency of unit cell of $y>$ Packing efficiency of unit cell of $z$ (B) $\mathrm{L}_{\mathrm{y}}>\mathrm{L}_{\mathrm{z}}$ (C) $\mathrm{L}_{\mathrm{x}}>\mathrm{L}_{\mathrm{y}}$ (D) Density of $\mathrm{x}>$ Density of y	Available	Available
41	View ExplanationExplain	In the following reactions, $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ , and $\mathbf{S}$ are the major products.\n\n\n\n\n\nThe correct statement(s) about $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ , and $\mathbf{S}$ is(are) (A) $\mathbf{P}$ and $\mathbf{Q}$ are monomers of polymers dacron and glyptal, respectively. (B) $\mathbf{P}, \mathbf{Q}$ , and $\mathbf{R}$ are dicarboxylic acids. (C) Compounds $\mathbf{Q}$ and $\mathbf{R}$ are the same. (D) $\mathbf{R}$ does not undergo aldol condensation and $\mathbf{S}$ does not undergo Cannizzaro reaction.	Diagram Question	Available
42	View ExplanationExplain	Among $\left[\mathrm{I}_{3}\right]^{+},\left[\mathrm{SiO}_{4}\right]^{4-}, \mathrm{SO}_{2} \mathrm{Cl}_{2}, \mathrm{XeF}_{2}, \mathrm{SF}_{4}, \mathrm{ClF}_{3}, \mathrm{Ni}(\mathrm{CO})_{4}, \mathrm{XeO}_{2} \mathrm{~F}_{2},\left[\mathrm{PtCl}_{4}\right]^{2-}, \mathrm{XeF}_{4}$ , and $\mathrm{SOCl}_{2}$ , the total number of species having $s p^{3}$ hybridised central atom is $\qquad$ .	Available	Available
43	View ExplanationExplain	Consider the following molecules: $\mathrm{Br}_{3} \mathrm{O}_{8}, \mathrm{~F}_{2} \mathrm{O}, \mathrm{H}_{2} \mathrm{~S}_{4} \mathrm{O}_{6}, \mathrm{H}_{2} \mathrm{~S}_{5} \mathrm{O}_{6}$ , and $\mathrm{C}_{3} \mathrm{O}_{2}$ .\n\nCount the number of atoms existing in their zero oxidation state in each molecule.\nTheir sum is $\qquad$ .	Available	Available
44	View ExplanationExplain	For $\mathrm{He}^{+}$ , a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm . The wavelength (in nm ) of the emitted photon during the transition is $\qquad$ .\n[Use:\nBohr radius, $\mathrm{a}=52.9 \mathrm{pm}$ \nRydberg constant, $R_{\mathrm{H}}=2.2 \times 10^{-18} \mathrm{~J}$ \nPlanck's constant, $\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$ \nSpeed of light, $\mathrm{c}=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$ ]\nQ. 1250 mL of 0.2 molal urea	Available	Available
45	View ExplanationExplain	The reaction of 4-methyloct-1-ene ( $\mathbf{P}, 2.52 \mathrm{~g}$ ) with HBr in the presence of ( $\left.\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CO}\right)_{2} \mathrm{O}_{2}$ gives two isomeric bromides in a $9: 1$ ratio, with a combined yield of $50 \%$ . Of these, the entire amount of the primary alkyl bromide was reacted with an appropriate amount of diethylamine followed by treatment with aq. $\mathrm{K}_{2} \mathrm{CO}_{3}$ to give a non-ionic product $\mathbf{S}$ in $100 \%$ yield.\n\nThe mass (in mg ) of $\mathbf{S}$ obtained is $\qquad$ .\n[Use molar mass (in $\mathrm{g} \mathrm{mol}^{-1}$ ): $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{Br}=80$ ]\n\n\section*{SECTION 4 (Maximum Marks: 12)}\n- This section contains TWO (02) paragraphs.\n- Based on each paragraph, there are TWO (02) questions.\n- The	Available	Available
46	View ExplanationExplain	The value of entropy change, $\mathrm{S}_{\beta}-\mathrm{S}_{\alpha}\left(\right.$ in $\left.\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right)$ , at 300 K is $\qquad$ .\n[Use: ln 2 = 0.69\nGiven: $\mathrm{S}_{\beta}-\mathrm{S}_{\alpha}=0$ at 0 K ]\n\n\section*{"PARAGRAPH I"}\n\nThe entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given. $S_{\mathrm{T}}$ and $S_{0}$ are entropies of the phases at temperatures T and 0 K , respectively.\n\n\nThe transition temperature for $\alpha$ to $\beta$ phase change is 600 K and $C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}=1 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ . Assume ( $C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}$ ) is independent of temperature in the range of 200 to $700 \mathrm{~K} . C_{\mathrm{p}, \alpha}$ and $C_{\mathrm{p}, \beta}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.	Diagram Question	Available
47	View ExplanationExplain	The value of enthalpy change, $\mathrm{H}_{\beta}-\mathrm{H}_{\alpha}\left(\right.$ in $\left.\mathrm{J} \mathrm{mol}^{-1}\right)$ , at 300 K is $\qquad$ .\n\n\section*{"PARAGRAPH II"}\n\nA trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\mathrm{Sn} / \mathrm{HCl}$ gives a major product, which on treatment with an excess of $\mathrm{NaNO}_{2} / \mathrm{HCl}$ at $0^{\circ} \mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$ , upon treatment with excess of $\mathrm{H}_{2} \mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$ . Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$ . The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$ .\nThe molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{~g} \mathrm{~mol}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{~g} \mathrm{~mol}^{-1}$ .	Available	Available
48	View ExplanationExplain	The number of heteroatoms present in one molecule of $\mathbf{R}$ is $\qquad$ .\n[Use: Molar mass (in g mol ${ }^{-1}$ ): $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{Br}=80, \mathrm{Cl}=35.5$ \nAtoms other than C and H are considered as heteroatoms]\n\n\section*{"PARAGRAPH II"}\n\nA trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\mathrm{Sn} / \mathrm{HCl}$ gives a major product, which on treatment with an excess of $\mathrm{NaNO}_{2} / \mathrm{HCl}$ at $0^{\circ} \mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$ , upon treatment with excess of $\mathrm{H}_{2} \mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$ . Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$ . The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$ .\nThe molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{~g} \mathrm{~mol}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{~g} \mathrm{~mol}^{-1}$ .	Available	Available
49	View ExplanationExplain	The total number of carbon atoms and heteroatoms present in one molecule of $\mathbf{S}$ is $\qquad$ .\n[Use: Molar mass (in $\mathrm{g} \mathrm{mol}^{-1}$ ): $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{Br}=80, \mathrm{Cl}=35.5$ \nAtoms other than C and H are considered as heteroatoms]	Available	Available