JEE Advanced 2023 Question Paper with Solutions

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $(\gamma=5 / 3)$ and one mole of an ideal diatomic gas $(\gamma=7 / 5)$ . Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. The change in its internal energy is _______ Joule.

Two point-like objects of masses $20 ~\mathrm{gm}$ and $30 ~\mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10 \mathrm{~cm}$ . This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{-8} \mathrm{~N} \mathrm{~m} ~\mathrm{rad}^{-1}$ . The angular frequency of the oscillations in $n \times 10^{-3} ~\mathrm{rad} ~\mathrm{s}^{-1}$ . The value of $n$ is _________ .

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm 0.1 \mathrm{~cm}$ and the distance of its real image from the lens is $20 \pm 0.2 \mathrm{~cm}$ . The error in the determination of focal length of the lens is $n \%$ . The value of $n$ is ______.

One mole of an ideal gas expands adiabatically from an initial state $\left(T_{\mathrm{A}}, V_0\right)$ to final state $\left(T_{\mathrm{f}}, 5 V_0\right)$ . Another mole of the same gas expands isothermally from a different initial state $\left(T_{\mathrm{B}}, V_0\right)$ to the same final state $\left(T_{\mathrm{f}}, 5 V_0\right)$ . The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$ . What is the ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$ ?

A person of height $1.6 \mathrm{~m}$ is walking away from a lamp post of height $4 \mathrm{~m}$ along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60 \mathrm{~cm} \mathrm{~s}^{-1}$ , the speed of the tip of the person's shadow on the ground with respect to the person is ____________ $\mathrm{cm}~ \mathrm{s}^{-1}$ .

A gas has a compressibility factor of 0.5 and a molar volume of $0.4 ~\mathrm{dm}^3 \mathrm{~mol}^{-1}$ at a temperature of $800 \mathrm{~K}$ and pressure $\mathbf{x}$ atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be $\mathbf{y} ~\mathrm{dm}^3 \mathrm{~mol}^{-1}$ . The value of $\mathbf{x} / \mathbf{y}$ is __________. [Use: Gas constant, $\mathrm{R}=8 \times 10^{-2} \mathrm{~L}$ atm $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ]

The stoichiometric reaction of $516 \mathrm{~g}$ of dimethyldichlorosilane with water results in a tetrameric cyclic product $\mathbf{X}$ in $75 \%$ yield. The weight (in g) of $\mathbf{X}$ obtained is _______. [Use, molar mass $\left(\mathrm{g} ~\mathrm{mol}^{-1}\right): \mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Si}=28, \mathrm{Cl}=35.5$ ]

Match the reactions in List-I with the features of their products in List-II and choose the correct option. .tg {border-collapse:collapse;border-spacing:0;width:100%} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-1wig{font-weight:bold;text-align:left;vertical-align:top} .tg .tg-0lax{text-align:left;vertical-align:top} List - I List - II (P) (1) Inversion of configuration (Q) (2) Retention of configuration (R) (3) Mixture of enantiomers (S) (4) Mixture of structural isomers (5) Mixture of diastereomers

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 $\mathrm{C}_3 \mathrm{O}_2$ . Count the number of atoms existing in their zero oxidation state in each molecule. Their sum is _______.

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 ________.

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$ . Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$ , respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$ , respectively. Then which of the following statements is(are) true?

Let $P$ be a point on the parabola $y^2=4 a x$ , where $a>0$ . The normal to the parabola at $P$ meets the $x$ -axis at a point $Q$ . The area of the triangle $P F Q$ , where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is

Let $S=(0,1) \cup(1,2) \cup(3,4)$ and $T=\{0,1,2,3\}$ . Then which of the following statements is(are) true?

Consider the given data with frequency distribution $\begin{array}{ccccccc}x_i & 3 & 8 & 11 & 10 & 5 & 4 \\f_i & 5 & 2 & 3 & 2 & 4 & 4\end{array}$ Match each entry in List-I to the correct entries in List-II . .tg {border-collapse:collapse;border-spacing:0;width:100%} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-1wig{font-weight:bold;text-align:left;vertical-align:top} .tg .tg-0lax{text-align:left;vertical-align:top} List - I List - II (P) The mean of the above data is (1) 2.5 (Q) The median of the above data is (2) 5 (R) The mean deviation about the mean of the above data is (3) 6 (S) The mean deviation about the median of the above data is (4) 2.7 (5) 2.4 The correct option is:

Consider an obtuse-angled triangle A B C in which the difference between the largest and the smallest angle is π 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 :

A Hydrogen-like atom has atomic number $Z$ . Photons emitted in the electronic transitions from level $n=4$ to level $n=3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is $1.95 \mathrm{eV}$ . If the photoelectric threshold wavelength for the target metal is $310 \mathrm{~nm}$ , the value of $Z$ is _________. [Given: $h c=1240 \mathrm{eV}-\mathrm{nm}$ and $R h c=13.6 \mathrm{eV}$ , where $R$ is the Rydberg constant, $h$ is the Planck's constant and $c$ is the speed of light in vacuum]

In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{~A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$ . Which of the following statement(s) is(are) correct? [Given: $e^{-1}=0.36$ ]

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option. .tg {border-collapse:collapse;border-spacing:0;width:100%} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-1wig{font-weight:bold;text-align:left;vertical-align:top} .tg .tg-0lax{text-align:left;vertical-align:top} List - I List - II (P) ${ }_{92}^{238} U \rightarrow{ }_{91}^{234} \mathrm{~Pa}$ (1) one $\alpha$ particle and one $\beta^{+}$ particle (Q) ${ }_{82}^{214} \mathrm{~Pb} \rightarrow{ }_{82}^{210} \mathrm{~Pb}$ (2) three $\beta^{-}$ particles and one $\alpha$ particle (R) ${ }_{81}^{210} \mathrm{Tl} \rightarrow{ }_{82}^{206} \mathrm{~Pb}$ (3) two $\beta^{-}$ particles and one $\alpha$ particle (S) ${ }_{91}^{228} \mathrm{~Pa} \rightarrow{ }_{88}^{224} \mathrm{Ra}$ (4) one $\alpha$ particle and one $\beta^{-}$ particle (5) one $\alpha$ particle and two $\beta^{+}$ particles

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?

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 _________.

Let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , for $x \in \mathbb{R}$ . Then the number of real solutions of the equation $\sqrt{1+\cos (2 x)}=\sqrt{2} \tan ^{-1}(\tan x)$ in the set $\left(-\frac{3 \pi}{2},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$ is equal to :

Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $\begin{aligned}& x+2 y+z=7 \\\\& x+\alpha z=11 \\\\& 2 x-3 y+\beta z=\gamma\end{aligned}$ Match each entry in List-I to the correct entries in List-II . .tg {border-collapse:collapse;border-spacing:0;width:100%} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-1wig{font-weight:bold;text-align:left;vertical-align:top} .tg .tg-0lax{text-align:left;vertical-align:top} List - I List - II (P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$ , then the system has (1) a unique solution (Q) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$ , then the system has (2) no solution (R) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$ , then the system has (3) infinitely many solutions (S) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$ , then the system has (4) $x=11, y=-2$ and $z=0$ as a solution (5) $x=-15, y=4$ and $z=0$ as a solution The correct option is:

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$ . Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int\limits_{x^2}^x \sqrt{\frac{1-t}{t}} d t < g(x) < 2 \sqrt{x}$ for all $x \in(0,1)$ . Then $\lim\limits_{x \rightarrow 0} f(x) g(x)$

Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1, x_2, x_3 \in\{0,1\}\right\}$ . Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$ . Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$ ; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$ . For lines $\ell_1$ and $\ell_2$ , let $d\left(\ell_1, \ell_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left(\ell_1, \ell_2\right)$ , as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$ , is :

Let $n \geq 2$ be a natural number and $f:[0,1] \rightarrow \mathbb{R}$ be the function defined by $f(x)= \begin{cases}n(1-2 n x) & \text { if } 0 \leq x \leq \frac{1}{2 n} \\\\ 2 n(2 n x-1) & \text { if } \frac{1}{2 n} \leq x \leq \frac{3}{4 n} \\\\ 4 n(1-n x) & \text { if } \frac{3}{4 n} \leq x \leq \frac{1}{n} \\\\ \frac{n}{n-1}(n x-1) & \text { if } \frac{1}{n} \leq x \leq 1\end{cases}$ If $n$ is such that the area of the region bounded by the curves $x=0, x=1, y=0$ and $y=f(x)$ is 4 , then the maximum value of the function $f$ is :

In the scheme given below, $\mathbf{X}$ and $\mathbf{Y}$ , respectively, are

Match the electronic configurations in List-I with appropriate metal complex ions in List-II and choose the correct option. [Atomic Number: $\mathrm{Fe}=26, \mathrm{Mn}=25, \mathrm{Co}=27$ ] .tg {border-collapse:collapse;border-spacing:0;width:100%} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-1wig{font-weight:bold;text-align:left;vertical-align:top} .tg .tg-0lax{text-align:left;vertical-align:top} List - I List - II (P) $t_{2 g}^6 e_g^0$ (1) $\left[\mathrm{Fe}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{2+}$ (Q) $t_{2 g}^3 e_g^2$ (2) $\left[\mathrm{Mn}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{2+}$ (R) $\mathrm{e}^2 \mathrm{t}_2^3$ (3) $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_6\right]^{3+}$ (S) $t_{2 g}^4 e_g^2$ (4) $\left[\mathrm{FeCl}_4\right]^{-}$ (5) $\left[\mathrm{CoCl}_4\right]^{2-}$

In the following reactions, $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ , and $\mathbf{S}$ are the major products. The correct statement(s) about $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ , and $\mathbf{S}$ is(are) :

The correct statement(s) related to processes involved in the extraction of metals is(are) :

The total number of $s p^2$ hybridised carbon atoms in the major product $\mathbf{P}$ (a non-heterocyclic compound) of the following reaction is ______.