JEE Advanced 2023 Paper-1

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?

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 $P: y^2 = 12x$. 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 $f:[0,1] \to [0,1]$ be the function defined by $f(x) = \frac{x^3}{3} - x^2 + \frac{5}{9}x + \frac{17}{36}$. Consider the square region $S = [0,1] \times [0,1]$. Let $G = \{(x,y) \in S : y > f(x)\}$ be called the green region and $R = \{(x,y) \in S : y < f(x)\}$ be called the red region. Let $L_h=\{(x,h) \in S : x \in [0,1]\}$ be the horizontal line drawn at a height $h \in [0,1]$. Then which of the following statements is(are) true?

Let $f : (0,1) \to \mathbb{R}$ be the function defined as $f(x) = \sqrt{n}$ if $x \in [\frac{1}{n+1}, \frac{1}{n})$ where $n \in \mathbb{N}$. Let $g : (0,1) \to \mathbb{R}$ be a function such that $\int_{x^2}^x \sqrt{\frac{1-t}{t}} dt < g(x) < 2\sqrt{x}$ for all $x \in (0,1)$. Then $\lim_{x \to 0} f(x)g(x)$

Let $Q$ be the cube with the set of vertices $\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \in \{0,1\}\}$. 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 $d(\ell_1, \ell_2)$ denote the shortest distance between them. Then the maximum value of $d(\ell_1, \ell_2)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is

Let $X = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : \frac{x^2}{8} + \frac{y^2}{20} < 1 \text{ and } y^2 < 5x\}$. Three distinct points $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is

Let $P$ be a point on the parabola $y^2 = 4ax$, where $a > 0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$, 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 $\tan^{-1}(x) \in (-\frac{\pi}{2}, \frac{\pi}{2})$, for $x \in \mathbb{R}$. Then the number of real solutions of the equation $\sqrt{1+\cos(2x)} = \sqrt{2} \tan^{-1}(\tan x)$ in the set $(-\frac{3\pi}{2}, -\frac{\pi}{2}) \cup (-\frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \frac{3\pi}{2})$ is equal to

Let $n \ge 2$ be a natural number and $f:[0,1] \to \mathbb{R}$ be the function defined by $$ f(x) = \begin{cases} n(1-2nx) & \text{if } 0 \le x \le \frac{1}{2n} \\ 2n(2nx-1) & \text{if } \frac{1}{2n} \le x \le \frac{3}{4n} \\ 4n(1-nx) & \text{if } \frac{3}{4n} \le x \le \frac{1}{n} \\ \frac{n}{n-1}(nx-1) & \text{if } \frac{1}{n} \le x \le 1 \end{cases} $$ 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

Let $\underbrace{75\dots57}_{r}$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \dots + \underbrace{75\dots57}_{98}$. If $S = \frac{\underbrace{75\dots57}_{99} + m}{n}$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m+n$ is

Let $A = \left\{ \frac{1967 + 1686i \sin \theta}{7 - 3i \cos \theta} : \theta \in \mathbb{R} \right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

Let $P$ be the plane $\sqrt{3}x + 2y + 3z = 16$ and let $S = \{ \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} : \alpha^2 + \beta^2 + \gamma^2 = 1 \text{ and the distance of } (\alpha, \beta, \gamma) \text{ from the plane } P \text{ is } \frac{7}{2} \}$. Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u} - \vec{v}| = |\vec{v} - \vec{w}| = |\vec{w} - \vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}, \vec{v}$ and $\vec{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $\left(ax^2 + \frac{70}{27bx}\right)^4$ is equal to the coefficient of $x^{-5}$ in the expansion of $\left(ax - \frac{1}{bx^2}\right)^7$, then the value of $2b$ is

Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations: $$x + 2y + z = 7$$ $$x + \alpha z = 11$$ $$2x - 3y + \beta z = \gamma$$ Match each entry in List-I to the correct entries in List-II. The correct option is:

Consider the given data with frequency distribution
$x_i$: 3 8 11 10 5 4
$f_i$: 5 2 3 2 4 4
Match each entry in List-I to the correct entries in List-II. The correct option is:

Let $\ell_1$ and $\ell_2$ be the lines $\vec{r_1} = \lambda(\hat{i} + \hat{j} + \hat{k})$ and $\vec{r_2} = (\hat{j} - \hat{k}) + \mu(\hat{i} + \hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_1$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the points of $\ell_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d(H_0)$ is the maximum value of $d(H)$ as $H$ varies over all planes in $X$.

Match each entry in List-I to the correct entries in List-II. The correct option is:

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.

Match each entry in List-I to the correct entries in List-II. The correct option is:

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec{u}_0 = u_0 \hat{x}$ and falls on the ground at a distance $d$ from the building making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1/\sqrt{3}$. Which of the following statement(s) is(are) correct?

A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism. The angle of deviation of the emergent ray is $\delta = 60^{\circ}$ (see Figure-1). The angle of minimum deviation for red light from the same prism is $\delta_{\min} = 30^{\circ}$ (see Figure-2). The refractive index of the prism material for blue light is $\sqrt{3}$. Which of the following statement(s) is(are) correct?

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 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: $e^{-1} = 0.36$]

A bar of mass $M = 1.00$ kg and length $L = 0.20$ m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m = 0.10$ kg is moving on the same horizontal surface with 5.00 m s$^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed $v$. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?

A container has a base of 50 cm $\times$ 5 cm and height 50 cm, as shown in the figure. It has two parallel electrically conducting walls each of area 50 cm $\times$ 50 cm. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of 250 cm$^3$ s$^{-1}$. What is the value of the capacitance of the container after 10 seconds?
[Given: Permittivity of free space $\epsilon_0 = 9 \times 10^{-12}$ C$^2$N$^{-1}$m$^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_A/T_B$?

Two satellites P and Q are moving in different circular orbits around the Earth (radius $R$). The heights of P and Q from the Earth surface are $h_P$ and $h_Q$, respectively, where $h_P = R/3$. The accelerations of P and Q due to Earth’s gravity are $g_P$ and $g_Q$, respectively. If $g_P/g_Q = 36/25$, what is the value of $h_Q$?

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 eV. If the photoelectric threshold wavelength for the target metal is 310 nm, the value of $Z$ is _______.
[Given: $hc = 1240$ eV-nm and $Rhc = 13.6$ eV, where $R$ is the Rydberg constant, $h$ is the Planck’s constant and $c$ is the speed of light in vacuum]

An optical arrangement consists of two concave mirrors M$_1$ and M$_2$, and a convex lens L with a common principal axis, as shown in the figure. The focal length of L is 10 cm. The radii of curvature of M$_1$ and M$_2$ are 20 cm and 24 cm, respectively. The distance between L and M$_2$ is 20 cm. A point object S is placed at the mid-point between L and M$_2$ on the axis. When the distance between L and M$_1$ is $n/7$ cm, one of the images coincides with S. 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 $10 \pm 0.1$ cm and the distance of its real image from the lens is $20 \pm 0.2$ cm. The error in the determination of focal length of the lens is $n \%$. The value of $n$ is _______.

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.

A person of height 1.6 m is walking away from a lamp post of height 4 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 cm s$^{-1}$, the speed of the tip of the person’s shadow on the ground with respect to the person is _______ cm s$^{-1}$.

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 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 $1.2 \times 10^{-8}$ N m rad$^{-1}$. The angular frequency of the oscillations in $n \times 10^{-3}$ rad s$^{-1}$. The value of $n$ is _____.

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.

Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option.
[Given: Wien’s constant as $2.9 \times 10^{-3}$ m-K and $\frac{hc}{e} = 1.24 \times 10^{-6}$ V-m]

A series LCR circuit is connected to a $45 \sin(\omega t)$ Volt source. The resonant angular frequency of the circuit is $10^5$ rad s$^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8 \times 10^4$ rad s$^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L = 50$ mH, match each entry in List-I with an appropriate value from List-II and choose the correct option.

A thin conducting rod MN of mass 20 gm, length 25 cm and resistance 10 $\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $\vec{B}_0 = 4$ T directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t = 0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-I with an appropriate value from List-II, and choose the correct option.
[Given: The acceleration due to gravity $g = 10$ m s$^{-2}$ and $e^{-1} = 0.4$]

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

In the following reactions, $\textbf{P}$, $\textbf{Q}$, $\textbf{R}$, and $\textbf{S}$ are the major products.


The correct statement(s) about $\textbf{P}$, $\textbf{Q}$, $\textbf{R}$, and $\textbf{S}$ is(are)

Consider the following reaction scheme and choose the correct option(s) for the major products $\textbf{Q}$, $\textbf{R}$ and $\textbf{S}$.

$$ \text{Styrene} \xrightarrow[\text{(ii) NaOH, H}_2\text{O}_2, \text{H}_2\text{O}]{\text{(i) B}_2\text{H}_6} \mathbf{P} \xrightarrow[\substack{\text{(ii) Cl}_2, \text{Red phosphorus} \\ \text{(iii) H}_2\text{O}}]{\text{(i) CrO}_3, \text{H}_2\text{SO}_4} \mathbf{Q} $$ $$ \mathbf{P} \xrightarrow[\substack{\text{(ii) NaCN} \\ \text{(iii) H}_3\text{O}^+, \Delta}]{\text{(i) SOCl}_2} \mathbf{R} \xrightarrow{\text{conc. H}_2\text{SO}_4} \mathbf{S} $$

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

Plotting $1/\Lambda_m$ against $c\Lambda_m$ for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line with y-axis intercept of P and slope of S. The ratio P/S is

[$\Lambda_m$ = molar conductivity
$\Lambda_m^\circ$ = limiting molar conductivity
c = molar concentration
$K_a$ = dissociation constant of HX]

On decreasing the $pH$ from 7 to 2, the solubility of a sparingly soluble salt (MX) of a weak acid (HX) increased from $10^{-4} \text{ mol L}^{-1}$ to $10^{-3} \text{ mol L}^{-1}$. The $pK_a$ of HX is

In the given reaction scheme, $\textbf{P}$ is a phenyl alkyl ether, $\textbf{Q}$ is an aromatic compound; $\textbf{R}$ and $\textbf{S}$ are the major products.

$$ \textbf{P} \xrightarrow{\text{HI}} \textbf{Q} \xrightarrow[\text{(ii) CO}_2 \text{ (iii) H}_3\text{O}^+]{\text{(i) NaOH}} \textbf{R} \xrightarrow[\text{(ii) H}_3\text{O}^+]{\text{(i) (CH}_3\text{CO)}_2\text{O}} \textbf{S} $$
The correct statement about $\textbf{S}$ is

The stoichiometric reaction of 516 g of dimethyldichlorosilane with water results in a tetrameric cyclic product $\textbf{X}$ in 75% yield. The weight (in g) of $\textbf{X}$ obtained is ___.

[Use, molar mass (g mol$^{-1}$): H = 1, C = 12, O = 16, Si = 28, Cl = 35.5]

A gas has a compressibility factor of 0.5 and a molar volume of $0.4 \text{ dm}^3 \text{ mol}^{-1}$ at a temperature of 800 K and pressure x atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be y $\text{dm}^3 \text{ mol}^{-1}$. The value of x/y is ___.

[Use: Gas constant, $R = 8 \times 10^{-2} \text{ L atm K}^{-1} \text{ mol}^{-1}$]

The plot of $\log k_f$ versus $1/T$ for a reversible reaction $\text{A (g)} \rightleftharpoons \text{P (g)}$ is shown.
Pre-exponential factors for the forward and backward reactions are $10^{15} \text{ s}^{-1}$ and $10^{11} \text{ s}^{-1}$, respectively. If the value of $\log K$ for the reaction at 500 K is 6, the value of $|\log k_b|$ at 250 K is ___.

[$K$ = equilibrium constant of the reaction
$k_f$ = rate constant of forward reaction
$k_b$ = rate constant of backward reaction]

One mole of an ideal monoatomic gas undergoes two reversible processes ($\text{A} \to \text{B}$ and $\text{B} \to \text{C}$) as shown in the given figure:
$\text{A} \to \text{B}$ is an adiabatic process. If the total heat absorbed in the entire process ($\text{A} \to \text{B}$ and $\text{B} \to \text{C}$) is $RT_2 \ln 10$, the value of $2 \log V_3$ is ___.

[Use, molar heat capacity of the gas at constant pressure, $C_{p,m} = \frac{5}{2}R$]

In a one-litre flask, 6 moles of A undergoes the reaction $\text{A (g)} \rightleftharpoons \text{P (g)}$. The progress of product formation at two temperatures (in Kelvin), $T_1$ and $T_2$, is shown in the figure:
If $T_1 = 2T_2$ and $(\Delta G^\Theta_2 - \Delta G^\Theta_1) = RT_2 \ln x$, then the value of x is ___.

[$\Delta G^\Theta_1$ and $\Delta G^\Theta_2$ are standard Gibb’s free energy change for the reaction at temperatures $T_1$ and $T_2$, respectively.]

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

Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.

Match the electronic configurations in List-I with appropriate metal complex ions in List-II and choose the correct option.
[Atomic Number: Fe = 26, Mn = 25, Co = 27]

Match the reactions in List-I with the features of their products in List-II and choose the correct option.

The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.