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 $P: y^{2}=12 x$.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 $\mathrm{A}_{4}$ and $\mathrm{A}_{3}$, 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 $L_h$ be the horizontal line $y = h$. Consider the statements:

Let $Q$ be the cube with the set of vertices $\left\{\left(x_{1}, x_{2}, x_{3}\right) \in R^{3}: x_{1}, x_{2}, x_{3} \in\{0,1\}\right\}$. 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 $\mathrm{d}\left(\ell_{1}, \ell_{2}\right)$ denote the shortest distance between them. Then the maximum value of

$\mathrm{d}\left(\ell_{1}, \ell_{2}\right)$ as $\ell_{1}$ varies over F and $\ell_{2}$ varies over S , is

Let $X:\left\{(x, y) \in Z \times Z: \frac{x^{2}}{8}+\frac{y^{2}}{20}<1\right.$ and $\left.y^{2}<5 x\right\}$. Three distinct points $P, Q$ and $R$ are randomly

chosen from X . Then the probability that $\mathrm{P}, \mathrm{Q}$ and R form a triangle whose area is a positive integer, is

Let P be a point on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$, where $\mathrm{a}>0$. 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 $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, for $x \in R$. 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 $\mathrm{n} \geq 2$ be a natural number and $\mathrm{f}:[0,1] \rightarrow \mathrm{R}$ be the function defined by

If n is such that the area of the region bounded by the curves $\mathrm{x}=0, \mathrm{x}=1, \mathrm{y}=0$ and $\mathrm{y}=\mathrm{f}(\mathrm{x})$ is 4 , then the

maximum value of the function $f$ is

Let $7 \overbrace{5 \cdots 5}^{r} 7$ 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 $\mathrm{S}=77+757+7557+\ldots+7 \overbrace{5 \cdots 5}^{98}7$. If $\mathrm{S}=\frac{7 \overbrace{5 \cdots 57}^{99}+\mathrm{m}}{\mathrm{n}}$, where m and n are natural numbers less than 3000 , then the value of $m+n$ is

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

Let $P$ be the plane $\sqrt{3} x+2 y+3 z=16$ and let $S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^{2}+\beta^{2}+\gamma^{2}=1\right.$ and the distance of

$(\alpha, \beta, \gamma)$ from the plane $P$ is $\left.\frac{7}{2}\right\}$. Let $\overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}}$ and $\overrightarrow{\mathrm{w}}$ be three distinct vectors in S such that

$|\overrightarrow{\mathrm{u}}-\overrightarrow{\mathrm{v}}|=|\overrightarrow{\mathrm{v}}-\overrightarrow{\mathrm{w}}|=|\overrightarrow{\mathrm{w}}-\overrightarrow{\mathrm{u}}|$. Let V be the volume of the parallelepiped determined by vectors, $\overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}}$ and $\overrightarrow{\mathrm{w}}$.

Then the value of $\frac{80}{\sqrt{3}} \mathrm{~V}$ is

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

the coefficient of $\mathrm{x}^{-5}$ in the expansion of $\left(\mathrm{ax}-\frac{1}{\mathrm{bx}^{2}}\right)^{7}$, then the value of 2 b is

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

Consider the given data with frequency distribution

$\mathrm{x}_{\mathrm{i}} \quad \begin{array}{llllll}3 & 8 & 11 & 10 & 5 & 4\end{array}$

$\mathrm{f}_{\mathrm{i}} \quad \begin{array}{llllll}5 & 2 & 3 & 2 & 4 & 4\end{array}$

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 $\vec{r}_{1}=\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}_{2}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}+\hat{\mathrm{k}})$, respectively. Let X be the set of all the planes H that contain the line $\ell_{1}$. For a plane H , let $\mathrm{d}(\mathrm{H})$ denote the smallest possible distance between the points of $\ell_{2}$ and H . Let $\mathrm{H}_{0}$ be a plane in X for which $\mathrm{d}\left(\mathrm{H}_{0}\right)$ is the maximum value of $\mathrm{d}(\mathrm{H})$ as H varies over all planes in X .Match each entry in List-I to the correct entries in List-II.

Let z be a complex number satisfying $|\mathrm{z}|^{3}+2 \mathrm{z}^{2}+4 \overline{\mathrm{z}}-8=0$, where $\overline{\mathrm{z}}$ 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.

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height 3 h 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 $\overrightarrow{\mathrm{u}}_{0}=\mathrm{u}_{0} \hat{\mathrm{x}}^{\wedge}$ 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 $\delta_{\text {min }}=30^{\circ}$ (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: $\mathrm{e}^{-1}=0.36$ ]

A bar of mass $\mathrm{M}=1.00 \mathrm{~kg}$ and length $\mathrm{L}=0.20 \mathrm{~m}$ 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 $\mathrm{m}=0.10 \mathrm{~kg}$ is moving on the same horizontal surface with $5.00 \mathrm{~ms}^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $\mathrm{L} / 2$ 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 $50 \mathrm{~cm} \times 5 \mathrm{~cm}$ and height 50 cm , as shown in the figure. It has two parallel electrically

conducting walls each of area $50 \mathrm{~cm} \times 50 \mathrm{~cm}$. 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 $\mathrm{cm}^{3} \mathrm{~s}^{-1}$. What is the

value of the capacitance of the container after 10 seconds? [Given: Permittivity of free space

$\varepsilon_{0}=9 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible

One mole of an ideal gas expands adiabatically from an initial state ( $\mathrm{T}_{\mathrm{A}}, \mathrm{V}_{0}$ ) to final state $\left(\mathrm{T}_{\mathrm{f}}, 5 \mathrm{~V}_{0}\right)$. Another mole of the same gas expands isothermally from a different initial state $\left(\mathrm{T}_{\mathrm{B}}, \mathrm{V}_{0}\right)$ to the same final state $\left(\mathrm{T}_{\mathrm{f}}, 5 \mathrm{~V}_{0}\right)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_{A} / T_{B}$ ?

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 $\mathrm{n}=4$ 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 [0pt] [Given: hc $=1240 \mathrm{eV}-\mathrm{nm}$ and $\mathrm{Rhc}=13.6 \mathrm{eV}$, where R is the Rydberg constant, h is the Planck's constant and $c$ is the speed of light in vacuum]

An optical arrangement consists of two concave mirrors $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$, 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 $n / 7 \mathrm{~cm}$, one of the images coincides with $S$. The value of $n$ is $\qquad$

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 $\qquad$

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 $\qquad$ 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 $60 \mathrm{~cm} \mathrm{~s}^{-1}$, The speed of the tip of the person's shadow on the ground with respect to the person is $\qquad$ $\mathrm{cm} \mathrm{s}{ }^{-1}$

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 $1.2 \times 10^{-8} \mathrm{Nm} \mathrm{rad}^{-1}$. The angular frequency of the oscillations in $\mathrm{n} \times 10^{-3} \mathrm{rad}$ $\mathrm{s}^{-1}$. The value of n is $\qquad$

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 $2.9 \times 10^{-3} \mathrm{~m}-\mathrm{K}$ and $\frac{\mathrm{hc}}{\mathrm{e}}=1.24 \times 10^{-6} \mathrm{~V}-\mathrm{m}$ ]

A series LCR circuit is connected to a $45 \sin (\omega t)$ Volt source. The resonant angular frequency of the circuit is $10^{5} \mathrm{rad} \mathrm{s}^{-1}$ and current amplitude at resonance is $\mathrm{I}_{0}$. When the angular frequency of the source is $\omega=8 \times$ $10^{4} \mathrm{rad} \mathrm{s}^{-1}$, the current amplitude in the circuit is $0.05 \mathrm{I}_{0}$. If $\mathrm{L}=50 \mathrm{mH}$, 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 $\mathrm{B}_{0}=4 \mathrm{~T}$ 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.[0pt] [Given: The acceleration due to gravity $\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $\mathrm{e}^{-1}=0.4$ ]

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

Plotting $1 / \Lambda_{\mathrm{m}}$ against $\mathrm{c} \Lambda_{\mathrm{m}}$ 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 $\mathrm{P} / \mathrm{S}$ is [ $\Lambda_{\mathrm{m}}=$ molar conductivity $\Lambda_{\mathrm{m}}^{0}=$ limiting molar conductivity $\mathrm{c}=$ molar concentration $\mathrm{K}_{\mathrm{a}}=$ 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 $10^{-4}\; \mathrm{~mol} \mathrm{~L}^{-1}$ to $10^{-3} \;\mathrm{~mol} \;\mathrm{~L}^{-1}$. The $\mathrm{pK}_{\mathrm{a}}$ of HX is

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

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

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

The plot of $\log k_{f}$ versus $1 / \mathrm{T}$ for a reversible reaction $\mathrm{A}(\mathrm{g}) \rightleftharpoons \mathrm{P}(\mathrm{g})$ is shown Pre-exponential factors for the forward and backward reactions are $10^{15} \mathrm{~s}^{-1}$ and $10^{11} \mathrm{~s}^{-1}$, respectively. If the value of $\log \mathrm{K}$ for the reaction at 500 K is 6 , the value of $\left|\log k_{b}\right|$ 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 ( $A \rightarrow B$ and $B \rightarrow C$ ) as shown in the given figure: $A \rightarrow B$ is an adiabatic process. If the total heat absorbed in the entire process ( $A \rightarrow B$ and $B \rightarrow C$ ) is $\mathrm{RT}_{2} \ln 10$, the value of $2 \log \mathrm{~V}_{3}$ is $\_\_\_\_\_$ —. [Use, molar heat capacity of the gas at constant pressure, $\mathrm{C}_{\mathrm{p}, \mathrm{m}}=\frac{5}{2} \mathrm{R}$ ]

In a one-litre flask, 6 moles of A undergoes the reaction $\mathrm{A}(\mathrm{g}) \rightleftharpoons \mathrm{P}$ (g). The progress of product formation at two temperatures (in Kelvin), $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$, is shown in the figure: If $T_{1}=2 T_{2}$ and $\left(\Delta G_{2}^{\theta}-\Delta G_{1}^{\theta}\right)=R T_{2} \ln x$, then the value of $x$ is $\ldots .$. [ $\left[\Delta G_{1}^{\theta}\right.$ and $\Delta G_{2}^{\theta}$ are standard Gibb's free energy change for the reaction at temperatures $T_{1}$ and $T_{2}$, respectively.]

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.