JEE Advanced 2022 Question Paper 1 Question Paper with Solutions | JEE Advanced Mock Test | JeeMate

Q 1PhysicsSection B • Numericalray optics

+3 / -0

A rod of length $2 \mathrm{~cm}$ makes an angle $\frac{2 \pi}{3} \mathrm{rad}$ with the principal axis of a thin convex lens. The lens has a focal length of $10 \mathrm{~cm}$ and is placed at a distance of $\frac{40}{3} \mathrm{~cm}$ from the object as shown in the figure. The height of the image is $\frac{30 \sqrt{3}}{13} \mathrm{~cm}$ and the angle made by it with respect to the principal axis is $\alpha$ rad. The value of $\alpha$ is $\frac{\pi}{n} r a d$ , where $n$ is __________ .

qImage-lenses-geometrical-optics-optics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 2PhysicsSection B • Numericalatoms and nuclei

+3 / -0

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction

{ }_{7}^{16} \mathrm{~N}+

{ }_{2}^{4} \mathrm{He} \rightarrow{ }_{1}^{1} \mathrm{H}+{ }_{8}^{19} \mathrm{O}

in a laboratory frame is

n

(in

M e V

. Assume that

{ }_{7}^{16} \mathrm{~N}

is at rest in the laboratory frame. The masses of

{ }_{7}^{16} \mathrm{~N},{ }_{2}^{4} \mathrm{He},{ }_{1}^{1} \mathrm{H}

and

{ }_{8}^{19} \mathrm{O}

can be taken to be

16.006 u, 4.003 u, 1.008 u

and

19.003 u

, respectively, where

1 u=930 \,\mathrm{MeVc}^{-2}

. The value of

n

is ________ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 3PhysicsSection B • Numericalatoms and nuclei

+4 / -2

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_{b}^{p}$ and the binding energy of a neutron be $E_{b}^{n}$ in the nucleus.

Which of the following statement(s) is(are) correct?

Q 4PhysicsSection A • MCQmagnetism

+3 / -1

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $x y$ -plane with the center of the loop always on the axis $\hat{n}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counterclockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{n}$ direction. List-I gives time dependences of $\hat{n}$ in terms of a constant angular frequency $\omega$ . List-II gives the torques experienced by the circular loop at time $t=\frac{\pi}{6 \omega}$ . Let $\alpha=\frac{A^{2} \mu_{0}^{2} m^{2} I^{2} \omega}{2 R}$ .

qImage-force-and-torque-on-current-carrying-conductor-magnetism-electricity-Jeemate

List-I	List-II
(I) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\jmath}+\cos \omega t \hat{k})$	(P) 0
(II) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{\jmath})$	(Q) $-\frac{\alpha}{4} \hat{\imath}$
(III) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{k})$	(R) $\frac{3 \alpha}{4} \hat{\imath}$
(IV) $\frac{1}{\sqrt{2}}(\cos \omega t \hat{\jmath}+\sin \omega t \hat{k})$	(S) $\frac{\alpha}{4} \hat{\jmath}$
	(T) $-\frac{3 \alpha}{4} \hat{\imath}$

Which one of the following options is correct?

A

I

\rightarrow

Q, II

\rightarrow

P, III

\rightarrow

S, IV

\rightarrow

T

B

\mathrm{I} \rightarrow \mathrm{S}, \mathrm{II} \rightarrow \mathrm{T}

, III

\rightarrow \mathrm{Q}

, IV

\rightarrow \mathrm{P}

C

\mathrm{I} \rightarrow \mathrm{Q}, \mathrm{II} \rightarrow \mathrm{P}

, III

\rightarrow \mathrm{S}

, IV

\rightarrow \mathrm{R}

D

\mathrm{I} \rightarrow \mathrm{T}

, II

\rightarrow \mathrm{Q}

, III

\rightarrow \mathrm{P}

, IV

\rightarrow \mathrm{R}

Q 5PhysicsSection B • Numericalcurrent electricity

+3 / -0

In the following circuit $C_{1}=12 \mu F, C_{2}=C_{3}=4 \mu F$ and $C_{4}=C_{5}=2 \mu F$ . The charge stored in $C_{3}$ is ____________ $\mu C$ .

qImage-combination-of-resistances-current-electricity-electricity-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 6PhysicsSection A • MCQheat and thermodynamics

+3 / -1

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

List-I	List-II
(I) $10^{-3} \mathrm{~kg}$ of water at $100^{\circ} \mathrm{C}$ is converted to steam at the same temperature, at a pressure of $10^{5} \mathrm{~Pa}$ . The volume of the system changes from $10^{-6} \mathrm{~m}^{3}$ to $10^{-3} \mathrm{~m}^{3}$ in the process. Latent heat of water $=2250\, \mathrm{~kJ} / \mathrm{kg}$ .	(P) $2 \mathrm{~kJ}$
(II) $0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{~K}$ undergoes an isobaric expansion to volume $3 \mathrm{~V}$ . Assume $R=8.0 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$ .	(Q) $7 k J$
(III) One mole of a monatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3} \mathrm{~m}^{3}$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$ .	(R) $4 \mathrm{~kJ}$
(IV) Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{~kJ}$ of heat and undergoes isobaric expansion.	(S) $5 \mathrm{~kJ}$
	(T) $3 \mathrm{~kJ}$

Which one of the following options is correct?

A

I

\rightarrow

T, II

\rightarrow

R, III

\rightarrow

S, IV

\rightarrow

Q

B

I

\rightarrow

S, II

\rightarrow

P, III

\rightarrow

T, IV

\rightarrow

P

C

I

\rightarrow

P, II

\rightarrow

R, III

\rightarrow

T, IV

\rightarrow

Q

D

I

\rightarrow

Q, II

\rightarrow

R, III

\rightarrow

S, IV

\rightarrow

T

Q 7PhysicsSection A • MCQray optics

+3 / -1

List I contains four combinations of two lenses (1 and 2) whose focal lengths (in $\mathrm{cm}$ ) are indicated in the figures. In all cases, the object is placed $20 \mathrm{~cm}$ from the first lens on the left, and the distance between the two lenses is $5 \mathrm{~cm}$ . List II contains the positions of the final images.

List-I	List-II
(I)	(P) Final image is formed at $7.5 \mathrm{~cm}$ on the right side of lens 2 .
(II)	(Q) Final image is formed at $60.0 \mathrm{~cm}$ on the right side of lens 2 .
(III)	(R) Final image is formed at $30.0 \mathrm{~cm}$ on the left side of lens $2 .$
(IV)	(S) Final image is formed at $6.0 \mathrm{~cm}$ on the right side of lens 2 .
	(T) Final image is formed at $30.0 \mathrm{~cm}$ on the right side of lens 2 .

Which one of the following options is correct?

A

(I)

\rightarrow

P; (II)

\rightarrow

R; (III)

\rightarrow

Q; (IV)

\rightarrow

T

B

(I)

\rightarrow

Q; (II)

\rightarrow

P; (III)

\rightarrow

T; (IV)

\rightarrow

S

C

(I)

\rightarrow \mathrm{P}

; (II)

\rightarrow \mathrm{T}

; (III)

\rightarrow \mathrm{R}

; (IV)

\rightarrow \mathrm{Q}

D

(I)

\rightarrow

T; (II)

\rightarrow

S; (III)

\rightarrow

Q; (IV)

\rightarrow

R

Q 8PhysicsSection B • Numericalgravitation

+3 / -0

Two spherical stars $A$ and $B$ have densities $\rho_{A}$ and $\rho_{B}$ , respectively. $A$ and $B$ have the same radius, and their masses $M_{A}$ and $M_{B}$ are related by $M_{B}=2 M_{A}$ . Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_{A}$ . The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_{A}$ . If $v_{A}$ and $v_{B}$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}$ . The value of $n$ is __________ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 9PhysicsSection B • Numericalray optics

+4 / -2

Three plane mirrors form an equilateral triangle with each side of length $L$ . There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

qImage-reflection-of-light-geometrical-optics-optics-Jeemate

Which of the following statement(s) is(are) correct?

Q 10PhysicsSection B • Numericalrotational motion

+3 / -0

At time

t=0

, a disk of radius

1 \mathrm{~m}

starts to roll without slipping on a horizontal plane with an angular acceleration of

\alpha=\frac{2}{3} \mathrm{rad} \,\mathrm{s}^{-2}

. A small stone is stuck to the disk. At

t=0

, it is at the contact point of the disk and the plane. Later, at time

t=\sqrt{\pi} \,s

, the stone detaches itself and flies off tangentially from the disk. The maximum height (in

m

) reached by the stone measured from the plane is

\frac{1}{2}+\frac{x}{10}

. The value of

x

is ____________ , [Take

g=10 \mathrm{~m} \mathrm{~s}^{-2}

.]

Answer Format:Enter numerical value rounded to the nearest integer

Official Correct Answer:0.48TO0.56

Step-by-Step Solution:

The angle rotated by disc in

t=\sqrt{\pi} \mathrm{s}

is

\begin{aligned} & \theta=\omega_0 t+\frac{1}{2} \alpha t^2 \\ & \Rightarrow \theta=\frac{1}{2} \times \frac{2}{3}(\sqrt{\pi})^2 \\ &=\frac{\pi}{3} \mathrm{rad} \end{aligned}

and the angular velocity of disc is

\begin{aligned} \omega & =\omega_0+\alpha t \\ = & \frac{2 \sqrt{\pi}}{3} \mathrm{rad} / \mathrm{s} \\ \text { and } v_{\mathrm{cm}} & =\omega R=\frac{2 \sqrt{\pi}}{3} \times 1 \\ & =\frac{2 \sqrt{\pi}}{3} \mathrm{~m} / \mathrm{s} \end{aligned}

So, at the moment it detaches the situation is

qImage-combined-translational-and-rotational-motion-rotational-motion-mechanics-Jeemate

v=\sqrt{(\omega R)^2+v_{\mathrm{cm}}^2+2(\omega R) v_{\mathrm{cm}} \cos 120^{\circ}}

=v_{\mathrm{cm}}=\frac{2 \sqrt{\pi}}{3} \mathrm{~m} / \mathrm{s}

and

\tan \theta=\frac{\omega R \sin 120^{\circ}}{v_{\mathrm{cm}}+\omega R \cos 120^{\circ}}

\Rightarrow \tan \theta=\sqrt{3}

\Rightarrow \theta=\frac{\pi}{3} \mathrm{rad}

So,

H_{\max }=\frac{u^2 \sin ^2 \theta}{2 g}

=\frac{\left(\frac{2 \sqrt{\pi}}{3}\right)^2 \times \sin ^2 60^{\circ}}{2 \times 10}

\begin{aligned} & =\frac{4 \pi \times 3}{9 \times 2 \times 10 \times 4} \\ & =\frac{\pi}{60} \mathrm{~m} \end{aligned}

So, height from ground will be

= R\left(1-\cos 60^{\circ}\right)+\frac{\pi}{60}=\frac{1}{2}+\frac{x}{10}

\Rightarrow x=\frac{\pi}{6}=0.52

Q 11PhysicsSection B • Numericalelectrostatics

+4 / -2

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$ , and the remaining one has charge $x$ . The perpendicular from each charge to the nearest hexagon side passes through the center 0 of the hexagon and is bisected by the side.

qImage-electric-flux-and-gauss-law-electrostatics-electricity-Jeemate

Which of the following statement(s) is(are) correct in SI units?

Q 12PhysicsSection A • MCQrotational motion

+3 / -1

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $m s^{-1}$ ) at time $t=\frac{\pi}{3} s$ .

List-I	List-II
(I) $A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{~m}$ with uniform angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$ . The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta=0$ and $\theta=\frac{\pi}{2}$ , respectively.	(P) $\frac{\sqrt{3}+1}{2}$
(II) Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{~s}$ respectively, with the same speed $v=\frac{5 \pi}{\sqrt{2}} \mathrm{~m} \mathrm{~s}^{-1}$ and at $45^{\circ}$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide. $\left(g=10 \mathrm{~ms}^{-2}\right)$ .	(Q) $\frac{\sqrt{3}-1}{\sqrt{2}}$
(III) Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A}=x_{0} \sin \frac{t}{t_{0}}$ and $x_{B}=x_{0} \sin \left(\frac{t}{t_{0}}+\frac{\pi}{2}\right)$ respectively, starting from $t=0$ . Take $x_{0}=1 \mathrm{~m}, t_{0}=1 \mathrm{~s}$ .	(R) $\sqrt{10}$
(IV) Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{~m}$ on the $x y$ plane, with constant angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$ . Particle $B$ is moving up at a constant speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S) $\sqrt{2}$
	(T) $\sqrt{25\pi^{2}+1}$

Which one of the following options is correct?

A

I

\rightarrow

R, II

\rightarrow

T, III

\rightarrow

P, IV

\rightarrow

S

B

I

\rightarrow

S, II

\rightarrow

P, III

\rightarrow

Q, IV

\rightarrow

R

C

I

\rightarrow

S, II

\rightarrow

T, III

\rightarrow

P, IV

\rightarrow

R

D

I

\rightarrow

T, II

\rightarrow

P, III

\rightarrow

R, IV

\rightarrow

S

Q 13PhysicsSection B • Numericalmotion

+3 / -0

A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$ . When the acceleration due to gravity is $g$ , the range of the projectile is $d$ . If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\prime}=\frac{g}{0.81}$ , then the new range is $d^{\prime}=n d$ . The value of $n$ is ___________ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 14PhysicsSection B • Numericalelectromagnetic induction

+3 / -0

Consider an LC circuit, with inductance

L=0.1 \,\mathrm{H}

and capacitance

C=10^{-3} \mathrm{~F}

, kept on a plane. The area of the circuit is

1 \mathrm{~m}^{2}

. It is placed in a constant magnetic field of strength

B_{0}

which is perpendicular to the plane of the circuit. At time

t=0

, the magnetic field strength starts increasing linearly as

B=B_{0}+\beta t

with

\beta=0.04 \,\mathrm{T\,s}^{-1}

. The maximum magnitude of the current in the circuit is _________

m A

.

Answer Format:Enter numerical value rounded to the nearest integer

Q 15PhysicsSection B • Numericalcapacitor

+4 / -2

A medium having dielectric constant $K>1$ fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is $d$ . The capacitor is connected to a battery of voltage $V$ , as shown in Figure (a). Now, both the plates are moved by a distance of $\frac{d}{2}$ from their original positions, as shown in Figure (b).

qImage-capacitors-with-dielectric-capacitor-electricity-Jeemate

In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

Q 16PhysicsSection B • Numericalcurrent electricity

+4 / -2

The figure shows a circuit having eight resistances of $1 \Omega$ each, labelled $R_{1}$ to $R_{8}$ , and two ideal batteries with voltages $\varepsilon_{1}=12 \mathrm{~V}$ and $\varepsilon_{2}=6 \mathrm{~V}$ .

qImage-kirchhoffs-circuit-laws-current-electricity-electricity-Jeemate

Which of the following statement(s) is(are) correct?

Q 17PhysicsSection B • Numericalrotational motion

+3 / -0

A solid sphere of mass $1 \mathrm{~kg}$ and radius $1 \mathrm{~m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta=30^{\circ}$ from the horizontal. Two forces of magnitude $1 \mathrm{~N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{~m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is _________ $m \,s^{-2} .\left(\right.$ Take $g=10\, m s^{-2}$ )

qImage-combined-translational-and-rotational-motion-rotational-motion-mechanics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 18PhysicsSection B • Numericalproperties of matter

+4 / -2

An ideal gas of density $\rho=0.2 \mathrm{~kg} \mathrm{~m}^{-3}$ enters a chimney of height $h$ at the rate of $\alpha=$ $0.8 \mathrm{~kg} \mathrm{~s}^{-1}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_{1}=0.1 \mathrm{~m}^{2}$ and the upper end is $A_{2}=0.4 \mathrm{~m}^{2}$ . The pressure and the temperature of the gas at the lower end are $600 \mathrm{~Pa}$ and $300 \mathrm{~K}$ , respectively, while its temperature at the upper end is $150 \mathrm{~K}$ . The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and the ratio of specific heats of the gas $\gamma=2$ . Ignore atmospheric pressure.

qImage-fluid-flow-bernoullis-principle-and-viscosity-properties-of-matter-mechanics-Jeemate

Which of the following statement(s) is(are) correct?

Official Correct Answer:Declared in Solution

Step-by-Step Solution:

Given,

\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2: \gamma=2

The process described is adiabatic. In an adiabatic process (a process during which no heat is gained or lost), the following relationship holds:

P_1^{1-\gamma} \cdot T_1^\gamma = P_2^{1-\gamma} \cdot T_2^\gamma

Where :

-

P_1

and

P_2

are the initial and final pressures,

-

T_1

and

T_2

are the initial and final temperatures,

-

\gamma

is the ratio of specific heats (given as 2).

Given that

P_1 = 600 \, \text{Pa}

,

T_1 = 300 \, \text{K}

,

T_2 = 150 \, \text{K}

, and

\gamma = 2

, solve for

P_2

:

\frac{P_2}{P_1} = \left(\frac{T_1}{T_2}\right)^{\gamma/(1-\gamma)}

\Rightarrow P_2 = P_1 \left(\frac{T_1}{T_2}\right)^{\gamma/(1-\gamma)}

\Rightarrow P_2 = 600 \, \text{Pa} \times \left(\frac{300 \, \text{K}}{150 \, \text{K}}\right)^{2/(1-2)}

\Rightarrow P_2 = 150 \, \text{Pa}

Using the ideal gas law in the form

\rho = \frac{PM}{RT}

(where R is the specific gas constant), and rearranging to find the relationship between

\rho_1

and

\rho_2

:

\frac{\rho_1}{\rho_2} = \frac{P_1}{P_2} \cdot \frac{T_2}{T_1}

\Rightarrow \rho_2 = \rho_1 \cdot \frac{P_2}{P_1} \cdot \frac{T_1}{T_2}

\Rightarrow \rho_2 = 0.2 \, \text{kg/m}^3 \cdot \frac{150 \, \text{Pa}}{600 \, \text{Pa}} \cdot \frac{300 \, \text{K}}{150 \, \text{K}}

$\Rightarrow \rho_2 = 0.1 \, \text{kg/m} ^3$

Next, use the principle of conservation of mass flow rate to determine the velocity of the gas at the lower and upper ends.

Mass flow rate,

\frac{d m}{d t} = \rho_1 \cdot A_1 \cdot v_1 = \rho_2 \cdot A_2 \cdot v_2

,

The mass flow rate is conserved, and given by

\alpha = \rho \cdot A \cdot v

. Thus, you have:

\alpha = \rho_1 \cdot A_1 \cdot v_1

\Rightarrow 0.8 \, \text{kg/s} = 0.2 \, \text{kg/m}^3 \cdot 0.1 \, \text{m}^2 \cdot v_1

\Rightarrow v_1 = \frac{0.8 \, \text{kg/s}}{0.2 \, \text{kg/m}^3 \cdot 0.1 \, \text{m}^2} = 40 \, \text{m/s}

Similarly, for the upper end :

\alpha = \rho_2 \cdot A_2 \cdot v_2

\Rightarrow v_2 = \frac{0.8 \, \text{kg/s}}{0.1 \, \text{kg/m}^3 \cdot 0.4 \, \text{m}^2} = 20 \, \text{m/s}

Work done on the gas

=

total energy

=\Delta \mathrm{K}+\Delta \mathrm{U}+

internal energy

\begin{aligned} & \mathrm{P}_1 \mathrm{~A}_1 \Delta x_1-\mathrm{P}_2 \mathrm{~A}_2 \Delta x_2=\frac{1}{2} \Delta m v_2^2-\frac{1}{2} \Delta m v_1^2+\Delta m g h+\frac{f}{2} \\ &\left(\mathrm{P}_2 \Delta \mathrm{V}_2-\mathrm{P}_1 \Delta \mathrm{V}_1\right) 2 \mathrm{P}_1 \frac{\Delta \mathrm{v}_1}{\Delta m}-2 \mathrm{P}_2 \frac{\Delta \mathrm{v}_2}{\Delta m}=\frac{\mathrm{v}_2^2-\mathrm{v}_1^2}{2}+g h \\ & \frac{2 \times 600}{0.2}-\frac{2 \times 150}{0.1}=\frac{20^2-40^2}{2}+10 \mathrm{~h} \\ & 6000-3000=\frac{400-1600}{2}+10 \mathrm{~h} \\ & 3000=-600+10 \mathrm{~h} \\ & 10 \mathrm{~h}=3000+600 \\ & \mathrm{~h}=360 \mathrm{~m} \end{aligned}

Q 19ChemistrySection A • MCQchemical kinetics and nuclear chemistry

+3 / -1

Match the rate expressions in LIST-I for the decomposition of $X$ with the corresponding profiles provided in LIST-II. $X_{\mathrm{s}}$ and $\mathrm{k}$ are constants having appropriate units.

List-I	List-II
(I) rate $=\frac{\mathrm{k}[\mathrm{X}]}{\mathrm{X}_{\mathrm{s}}+[\mathrm{X}]}$ under all possible initial concentrations of $\mathrm{X}$	(P)
(II) rate $=\frac{k[X]}{X_{s}+[X]}$ where initial concentrations of $X$ are much less than $X_{s}$	(Q)
(III) rate $=\frac{k[X]}{X_{s}+[X]}$ where initial concentrations of $\mathrm{X}$ are much higher than $X_{s}$	(R)
(IV) rate $=\frac{k[X]^{2}}{X_{s}+[X]}$ where initial concentration of $X$ is much higher than $\mathrm{X}_{\mathrm{s}}$	(S)
	(T)

A

\mathrm{I} \rightarrow \mathrm{P}

; II

\rightarrow

Q; III

\rightarrow \mathrm{S}

; IV

\rightarrow \mathrm{T}

B

\mathrm{I} \rightarrow \mathrm{R}

; II

\rightarrow \mathrm{S}

; III

\rightarrow \mathrm{S}

; IV

\rightarrow \mathrm{T}

C

\mathrm{I} \rightarrow \mathrm{P}

; II

\rightarrow

Q; III

\rightarrow

Q; IV

\rightarrow

R

D

\mathrm{I} \rightarrow \mathrm{R}

; II

\rightarrow \mathrm{S}

; III

\rightarrow

Q; IV

\rightarrow \mathrm{R}

Q 20ChemistrySection A • MCQcompounds containing nitrogen

+3 / -1

Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option.

List-I	List-II
(I) Aniline	(P) Sodium fusion extract of the compound on boiling with $\mathrm{FeSO}_{4}$ , followed by acidification with conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$ , gives Prussian blue color.
(II) $o$ -Cresol	(Q) Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
(III) Cysteine	(R) Addition of the compound to a saturated solution of $\mathrm{NaHCO}_{3}$ results in effervescence.
(IV) Caprolactam	(S) The compound reacts with bromine water to give a white precipitate.
	(T) Treating the compound with neutral $\mathrm{FeCl}_{3}$ solution produces violet color.

A

\mathrm{I} \rightarrow \mathrm{P}, \mathrm{Q}

; II

\rightarrow \mathrm{S} ; \mathrm{III} \rightarrow \mathrm{Q}, \mathrm{R}

; IV

\rightarrow \mathrm{P}

B

\mathrm{I} \rightarrow \mathrm{P}

; II

\rightarrow \mathrm{R}

,

\mathrm{S}

; III

\rightarrow \mathrm{R}

;

\mathrm{IV} \rightarrow \mathrm{Q}, \mathrm{S}

C

\mathrm{I} \rightarrow \mathrm{Q}, \mathrm{S}

; II

\rightarrow \mathrm{P}, \mathrm{T}

; III

\rightarrow \mathrm{P}

; IV

\rightarrow \mathrm{S}

D

\mathrm{I} \rightarrow \mathrm{P}

, S; II

\rightarrow \mathrm{T}

; III

\rightarrow \mathrm{Q}, \mathrm{R}

;

\mathrm{IV} \rightarrow \mathrm{P}

Q 21ChemistrySection B • Numericalpractical organic chemistry

+3 / -0

Consider the following reaction.

qImage--practical-organic-chemistry-organic-chemistry-Jeemate

On estimation of bromine in $1.00 \mathrm{~g}$ of $\mathbf{R}$ using Carius method, the amount of $\mathrm{AgBr}$ formed (in $\mathrm{g}$ ) is ___________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{P}=31, \mathrm{Br}=80, \mathrm{Ag}=108]$

Answer Format:Enter numerical value rounded to the nearest integer

Q 22ChemistrySection B • Numericald and f block elements

+4 / -2

The treatment of galena with

\mathrm{HNO}_{3}

produces a gas that is

Q 23ChemistrySection B • Numericalhydrocarbons

+3 / -0

If the reaction sequence given below is carried out with 15 moles of acetylene, the amount of the product $\mathbf{D}$ formed (in $\mathrm{g}$ ) is ___________ .

qImage-aromatic-hydrocarbons-hydrocarbons-organic-chemistry-Jeemate

The yields of $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ are given in parentheses.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Cl}=35$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 24ChemistrySection B • Numericalthermodynamics

+3 / -0

$2 \mathrm{~mol} \,\mathrm{of}\, \mathrm{Hg}(\mathrm{g})$ is combusted in a fixed volume bomb calorimeter with excess of $\mathrm{O}_{2}$ at $298 \mathrm{~K}$ and 1 atm into $\mathrm{HgO}(s)$ . During the reaction, temperature increases from $298.0 \mathrm{~K}$ to $312.8 \mathrm{~K}$ . If heat capacity of the bomb calorimeter and enthalpy of formation of $\mathrm{Hg}(g)$ are $20.00 \mathrm{~kJ} \mathrm{~K}^{-1}$ and $61.32 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ at $298 \mathrm{~K}$ , respectively, the calculated standard molar enthalpy of formation of $\mathrm{HgO}(s)$ at 298 $\mathrm{K}$ is $\mathrm{X}\, \mathrm{kJ}\, \mathrm{mol}^{-1}$ . The value of $|\mathrm{X}|$ is _________ .

[Given: Gas constant $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 25ChemistrySection B • Numericalsome basic concepts of chemistry

+3 / -0

The treatment of an aqueous solution of $3.74 \mathrm{~g}$ of $\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$ with excess KI results in a brown solution along with the formation of a precipitate. Passing $\mathrm{H}_{2} \mathrm{~S}$ through this brown solution gives another precipitate $\mathbf{X}$ . The amount of $\mathbf{X}$ (in $g$ ) is ___________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~K}=39, \mathrm{Cu}=63, \mathrm{I}=127$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 26ChemistrySection B • Numericalatomic structure

+4 / -2

For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap of two

2 p_{z}

orbitals is(are)

Q 27ChemistrySection B • Numericalp block elements

+3 / -0

Dissolving $1.24 \mathrm{~g}$ of white phosphorous in boiling $\mathrm{NaOH}$ solution in an inert atmosphere gives a gas $\mathbf{Q}$ . The amount of $\mathrm{CuSO}_{4}$ (in g) required to completely consume the gas $\mathbf{Q}$ is _________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{P}=31, \mathrm{~S}=32, \mathrm{Cu}=63$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 28ChemistrySection B • Numericalsurface chemistry

+4 / -2

The correct option(s) related to adsorption processes is(are)

Q 29ChemistrySection B • Numericalaldehydes ketones and carboxylic acids

+4 / -2

Considering the reaction sequence given below, the correct statement(s) is(are)

qImage-preparation-properties-and-uses-of-carboxylic-acids-aldehydes-ketones-and-carboxylic-acids-organic-chemistry-Jeemate

Q 30ChemistrySection B • Numericalisolation of elements

+4 / -2

The electrochemical extraction of aluminum from bauxite ore involves

Q 31ChemistrySection B • Numericalelectrochemistry

+3 / -0

The reduction potential $\left(E^{0}\right.$ , in $\left.\mathrm{V}\right)$ of $\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})$ is __________.

[Given: $E_{\left(\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{MnO}_{2}(\mathrm{~s})\right)}^{0}=1.68 \mathrm{~V} ; E_{\left(\mathrm{MnO}_{2}(\mathrm{~s}) / \mathrm{Mn}^{2+}(\mathrm{aq})\right)}^{0}=1.21 \mathrm{~V} ; E_{\left(\mathrm{Mn}^{2+}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})\right)}^{0}=-1.03 \mathrm{~V}$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 32ChemistrySection B • Numericalcompounds containing nitrogen

+4 / -2

Considering the following reaction sequence,

qImage-diazonium-salts-and-other-nitrogen-containing-compounds-compounds-containing-nitrogen-organic-chemistry-Jeemate

the correct option(s) is(are)

Q 33ChemistrySection B • Numericalpractical organic chemistry

+3 / -0

The weight percentage of hydrogen in $\mathbf{Q}$ , formed in the following reaction sequence, is ________.

qImage--practical-organic-chemistry-organic-chemistry-Jeemate

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{Cl}=35$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 34ChemistrySection A • MCQcoordination compounds

+3 / -1

LIST-I contains metal species and LIST-II contains their properties.

List-I	List-II
(I) $\left[\mathrm{Cr}(\mathrm{CN})_{6}\right]^{4-}$	(P) $t_{2 \mathrm{g}}$ orbitals contain 4 electrons
(II) $\left[\mathrm{RuCl}_{6}\right]^{2-}$	(Q) $\mu$ (spin-only $)=4.9 \mathrm{BM}$
(III) $\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$	(R) low spin complex ion
(IV) $\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$	(S) metal ion in $4+$ oxidation state
	(T) $d^{4}$ species

[Given: Atomic number of $\mathrm{Cr}=24, \mathrm{Ru}=44, \mathrm{Fe}=26$ ]

Match each metal species in LIST-I with their properties in LIST-II, and choose the correct option

A

I

\rightarrow

R, T; II

\rightarrow

P, S; III

\rightarrow

Q, T; IV

\rightarrow

P, Q

B

I

\rightarrow

R, S; II

\rightarrow

P, T; III

\rightarrow

P, Q; IV

\rightarrow

Q, T

C

I

\rightarrow

P, R; II

\rightarrow

R, S; III

\rightarrow

R, T; IV

\rightarrow

P, T

D

I

\rightarrow

Q, T; II

\rightarrow

S, T; III

\rightarrow

P, T; IV

\rightarrow

Q, R

Q 35ChemistrySection B • Numericalionic equilibrium

+3 / -0

A solution is prepared by mixing $0.01 \mathrm{~mol}$ each of $\mathrm{H}_{2} \mathrm{CO}_{3}, \mathrm{NaHCO}_{3}, \mathrm{Na}_{2} \mathrm{CO}_{3}$ , and $\mathrm{NaOH}$ in $100 \mathrm{~mL}$ of water. $p \mathrm{H}$ of the resulting solution is _________.

[Given: $p \mathrm{~K}_{\mathrm{a} 1}$ and $p \mathrm{~K}_{\mathrm{a} 2}$ of $\mathrm{H}_{2} \mathrm{CO}_{3}$ are $6.37$ and 10.32, respectively; $\log 2=0.30$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 36ChemistrySection A • MCQhydrogen

+3 / -1

LIST-I contains compounds and LIST-II contains reactions

List-I	List-II
(I) $\mathrm{H}_{2} \mathrm{O}_{2}$	(P) $\mathrm{Mg}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$
(II) $\mathrm{Mg}(\mathrm{OH})_{2}$	(Q) $\mathrm{BaO}_{2}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow$
(III) $\mathrm{BaCl}_{2}$	(R) $\mathrm{Ca}(\mathrm{OH})_{2}+\mathrm{MgCl}_{2} \rightarrow$
(IV) $\mathrm{CaCO}_{3}$	(S) $\mathrm{BaO}_{2}+\mathrm{HCl} \rightarrow$
	(T) $\mathrm{Ca}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$

Match each compound in LIST-I with its formation reaction(s) in LIST-II, and choose the correct option

A

I

\rightarrow

Q; II

\rightarrow

P; III

\rightarrow

S; IV

\rightarrow

R

B

I

\rightarrow

T; II

\rightarrow

P; III

\rightarrow

Q; IV

\rightarrow

R

C

I

\rightarrow

T; II

\rightarrow

R; III

\rightarrow

Q; IV

\rightarrow

P

D

I

\rightarrow

Q; II

\rightarrow

R; III

\rightarrow

S; IV

\rightarrow

P

Q 37MathematicsSection B • Numericalpermutations and combinations

+3 / -0

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits

0,2,3,4,6,7

is _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 38MathematicsSection B • Numericalsequences and series

+3 / -0

Let

l_{1}, l_{2}, \ldots, l_{100}

be consecutive terms of an arithmetic progression with common difference

d_{1}

, and let

w_{1}, w_{2}, \ldots, w_{100}

be consecutive terms of another arithmetic progression with common difference

d_{2}

, where

d_{1} d_{2}=10

. For each

i=1,2, \ldots, 100

, let

R_{i}

be a rectangle with length

l_{i}

, width

w_{i}

and area

A_{i}

. If

A_{51}-A_{50}=1000

, then the value of

A_{100}-A_{90}

is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Official Correct Answer:18900

Step-by-Step Solution:

Given,

${l_1},{l_2},\,.......,\,{l_{100}}$ are in A.P with common difference ${d_1}$ .

So from property of A.P we can say,

${l_2} = {l_1} + {d_1}$

${l_3} = {l_1} + 2{d_1}$

$\vdots$

${l_{100}} = {l_1} + 99{d_1}$

Also given,

${w_1},{w_2},\,......\,,\,{w_{100}}$ are in A.P with common difference ${d_2}$ .

$\therefore$ From the property of A.P we can say,

${w_2} = {w_1} + {d_2}$

${w_3} = {w_1} + 2{d_2}$

$\vdots$

${w_{100}} = {w_1} + 99{d_2}$

Now, also given,

${d_1}{d_2} = 10$

and ${R_i}$ is a rectangle whose length is ${l_i}$ and width is ${w_i}$ and area ${A_i}$ .

$\therefore$ We know, area of rectangle

${A_i} = {l_i} \times {w_i}$

$\therefore$ ${A_{51}} = {l_{51}} \times {w_{51}}$

and ${A_{50}} = {l_{50}} \times {w_{50}}$

Given, ${A_{51}} - {A_{50}} = 1000$

$\Rightarrow ({l_1} + 50{d_1})({w_1} + 50{d_2}) - ({l_1} + 49{d_1})({w_1} + 49{d_2}) = 1000$

$\Rightarrow [{l_1}{w_1} + 2500{d_1}{d_2} + 50({l_1}{d_2} + {d_1}{w_1})] - [{l_1}{w_1} + 49 \times 49{d_1}{d_2} + 49({l_1}{d_2} + {w_1}{d_1})] = 1000$

$\Rightarrow [{(50)^2}{d_1}{d_2} - {(49)^2}{d_1}{d_2}] + (50 - 49)({l_1}{d_2} + {d_1}{w_1}) = 1000$

$\Rightarrow (99 \times 1){d_1}{d_2} + {l_1}{d_2} + {d_1}{w_1} = 1000$

$\Rightarrow 99 \times 10 + {l_1}{d_2} + {w_1}{d_1} = 1000$

$\Rightarrow {l_1}{d_2} + {w_1}{d_1} = 10$

Now,

${A_{100}} - {A_{90}}$

$= {l_{100}}\,.\,{w_{100}} - {l_{90}}\,.\,{w_{90}}$

$= ({l_1} + 99{d_1})({w_1} + 99{d_2}) - ({l_1} + 89{d_1})({w_1} + 89{d_2})$

$= [{l_1}{w_1} + {(99)^2}{d_1}{d_2} + 99({l_1}{d_2} + {w_1}{d_1})] - [{l_1}{w_1} + {(89)^2}{d_1}{d_2} + 89({l_1}{d_2} + {w_1}{d_1})]$

$= [{(99)^2} - {89^2}]{d_1}{d_2} + 10({l_1}{d_2} + {w_1}{d_1})$

$= 188 \times 10 \times {d_1}{d_2} + 10 \times 10$

$= 188 \times 10 \times 10 + 100$

$= 18800 + 100$

$= 18900$

Q 39MathematicsSection B • Numericalcomplex numbers

+3 / -0

Let

\bar{z}

denote the complex conjugate of a complex number

z

and let

i=\sqrt{-1}

. In the set of complex numbers, the number of distinct roots of the equation

\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)

is _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 40MathematicsSection B • Numericalinverse trigonometric functions

+3 / -0

Considering only the principal values of the inverse trigonometric functions, the value of

\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^{2}}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^{2}}+\tan ^{-1} \frac{\sqrt{2}}{\pi}

is

Answer Format:Enter numerical value rounded to the nearest integer

Q 41MathematicsSection B • Numericallimits continuity and differentiability

+3 / -0

Let

\alpha

be a positive real number. Let

f: \mathbb{R} \rightarrow \mathbb{R}

and

g:(\alpha, \infty) \rightarrow \mathbb{R}

be the functions defined by

f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)} .

Then the value of

\lim \limits_{x \rightarrow \alpha^{+}} f(g(x))

is

Answer Format:Enter numerical value rounded to the nearest integer

Q 42MathematicsSection B • Numericalsequences and series

+4 / -2

Let $a_{1}, a_{2}, a_{3}, \ldots$ be an arithmetic progression with $a_{1}=7$ and common difference 8. Let $T_{1}, T_{2}, T_{3}, \ldots$ be such that $T_{1}=3$ and $T_{n+1}-T_{n}=a_{n}$ for $n \geq 1$ . Then, which of the following is/are TRUE ?

Q 43MathematicsSection B • Numericaldefinite integration

+4 / -2

Consider the equation

$\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty)$

Which of the following statements is/are TRUE?

Q 44MathematicsSection B • Numerical3d geometry

+4 / -2

Let

P_{1}

and

P_{2}

be two planes given by

\begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned}

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on

P_{1}

and

P_{2}

?

Official Correct Answer:Declared in Solution

Step-by-Step Solution:

\mathrm{P}_1

and

\mathrm{P}_2

be two planes given by

\begin{aligned} & P_1: 10 x+15 y+12 z-60=0 \\ & P_2:-2 x+5 y+4 z-20=0 \end{aligned}

Now finding line of intersection of both the planes,

Let

z=\lambda

, then

\begin{aligned} 10 x+15 y & =60-12 \lambda .........(1) \\ -2 x+5 y & =20-4 \lambda .........(2) \end{aligned}

Now solving the eq. (1) and (2) we get,

\frac{x}{0}=\frac{y-4}{-4}=\frac{z}{5} \quad(\because \lambda=z)

Now any skew line with the line of intersection of given plane can be edge of tertrahedron.

Now using above concept we will solve all options.

For option (A)

\frac{x-1}{0}=\frac{y-1}{0}=\frac{z-1}{5}

Point is

(1,1,5 \lambda+1)

Now satisfying this point in given plane we have,

\begin{aligned} & 10 \times 1+15 \times 1+12 \times(5 \lambda+1)-60=0 \\ & \Rightarrow 60 \lambda=23 \\ & \Rightarrow \lambda=\frac{23}{60} \end{aligned}

Now we can see line is intersecting the plane

\mathrm{P}_1

, at some point.

Now checking for plane

\left(\mathrm{P}_2\right)

\begin{aligned} & -2 \times 1+5 \times 1 +4(5 \lambda+1)=20 \\ & \Rightarrow 20 \lambda =13 \\ & \Rightarrow \lambda =\frac{13}{20} \end{aligned}

Also intersecting plane

\left(\mathrm{P}_2\right)

Hence, it can be the edge of tetrahedron.

For option (B)

\frac{x-6}{-5}=\frac{y}{2}=\frac{z}{3} \text { point is }(-5 \lambda+6,2 \lambda, 3 \lambda)

this point is sytisfying plane

P_1

\begin{aligned} & 10(-5 \lambda+6)+15 \times 2 \lambda+12 \times 3 \lambda=60 \\ & \Rightarrow -50 \lambda+60+30 \lambda+36 \lambda=60 \\ & \Rightarrow 16 \lambda =0 \\ & \Rightarrow \lambda =0 \end{aligned}

Now checking for plane

\mathrm{P}_2

\begin{aligned} & -2(-5 \lambda+6)+5 \times 2 \lambda+4 \times 3 \lambda=20 \\ & \Rightarrow +10 \lambda-12+10 \lambda+12 \lambda=20 \\ & \Rightarrow 32 \lambda=32 \\ & \Rightarrow \lambda =1 \end{aligned}

Hence, it can be the edge of tetrahedron.

For option (C)

\frac{x}{-2}=\frac{y-4}{5}=\frac{z}{4}

point is

(-2 \lambda, 5 \lambda+4,4 \lambda)

Similarly checking in plane

\mathrm{P}_1

we get.

\begin{aligned} & \lambda \text { for } P_1=0 \\ & \lambda \text { for } P_2=0 \end{aligned}

Hence, it can not be the edge of tetrahedron

For option (D),

\frac{x}{1}=\frac{y-4}{-2}=\frac{z}{3}

point

(\lambda,-2 \lambda+4,3 \lambda)

and for

\lambda=0

point will be

(0,-4,0)

which is lying on line of intersection and DR of plane

\mathrm{P}_2

is

(-2,5,4)

and DR of line is

(1,-2,3)

Now line is lying completely on

\mathrm{P}_2

Hence, it can be the edge of tetrahedron.

Q 45MathematicsSection B • Numericalcircle

+3 / -0

Let

A B C

be the triangle with

A B=1, A C=3

and

\angle B A C=\frac{\pi}{2}

. If a circle of radius

r>0

touches the sides

A B, A C

and also touches internally the circumcircle of the triangle

A B C

, then the value of

r

is __________ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 46MathematicsSection B • Numericalcomplex numbers

+3 / -0

Let

z

be a complex number with a non-zero imaginary part. If

\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}

is a real number, then the value of

|z|^{2}

is _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 47MathematicsSection B • Numericalprobability

+3 / -0

In a study about a pandemic, data of 900 persons was collected. It was found that

190 persons had symptom of fever,

220 persons had symptom of cough,

220 persons had symptom of breathing problem,

330 persons had symptom of fever or cough or both,

350 persons had symptom of cough or breathing problem or both,

340 persons had symptom of fever or breathing problem or both,

30 persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 48MathematicsSection A • MCQprobability

+3 / -1

Two players, $P_{1}$ and $P_{2}$ , play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_{1}$ and $P_{2}$ , respectively. If $x>y$ , then $P_{1}$ scores 5 points and $P_{2}$ scores 0 point. If $x=y$ , then each player scores 2 points. If $x < y$ , then $P_{1}$ scores 0 point and $P_{2}$ scores 5 points. Let $X_{i}$ and $Y_{i}$ be the total scores of $P_{1}$ and $P_{2}$ , respectively, after playing the $i^{\text {th }}$ round.

List-I	List-II
(I) Probability of $\left(X_{2} \geq Y_{2}\right)$ is	(P) $\frac{3}{8}$
(II) Probability of $\left(X_{2}>Y_{2}\right)$ is	(Q) $\frac{11}{16}$
(III) Probability of $\left(X_{3}=Y_{3}\right)$ is	(R) $\frac{5}{16}$
(IV) Probability of $\left(X_{3}>Y_{3}\right)$ is	(S) $\frac{355}{864}$
	(T) $\frac{77}{432}$

The correct option is:

A

(I)

\rightarrow

(Q); (II)

\rightarrow

(R); (III)

\rightarrow

(T); (IV)

\rightarrow(S)

B

(I)

\rightarrow

(Q); (II)

\rightarrow

(R); (III)

\rightarrow

(T); (IV)

\rightarrow

(T)

C

(I)

\rightarrow

(P); (II)

\rightarrow

(R); (III)

\rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{S})

D

(I)

\rightarrow

(P); (II)

\rightarrow

(R); (III)

\rightarrow

(Q); (IV)

\rightarrow

(T)

Official Correct Answer:Option (A)

Step-by-Step Solution:

Given,

\begin{aligned} & P(\text { draw in } 1 \text { round })=\frac{6}{36}=\frac{1}{6} \\ & P(\text { win in } 1 \text { round })=\frac{1}{2}\left(1-\frac{1}{6}\right)=\frac{5}{12} \\ & P(\text { loss in } 1 \text { round })=\frac{5}{12} \end{aligned}

Now finding the probability of all we get.

\begin{aligned} & \mathrm{P}\left(\mathrm{X}_2>\mathrm{Y}_2\right)=\mathrm{P}(10,0)+\mathrm{P}(7,2) \\ & =\frac{5}{12} \times \frac{5}{12}+\frac{5}{12} \times \frac{1}{6} \times 2 \\ & =\frac{45}{144}=\frac{5}{16} \rightarrow \mathrm{R} \\ & \mathrm{P}\left(\mathrm{X}_2=\mathrm{Y}_2\right)=\mathrm{P}(5,5)+\mathrm{P}(4,4) \\ & =\frac{5}{12} \times \frac{5}{12} \times 2+\frac{1}{6} \times \frac{1}{6} \\ & =\frac{25+2}{72}=\frac{3}{8} \\ & \mathrm{P}\left(\mathrm{X}_3=\mathrm{Y}_3\right)=\mathrm{P}(6,6)+\mathrm{P}(7,7) \\ & =\frac{1}{6 \times 6 \times 6}+\frac{5}{12} \times \frac{1}{6} \times \frac{5}{12} \times \frac{1}{6} \\ & =\frac{77}{432} \rightarrow \mathrm{F} \\ & \mathrm{P}\left(\mathrm{X}_3>\mathrm{Y}_3\right)=\frac{1}{2}\left(1-\frac{77}{432}\right)=\frac{355}{864} \rightarrow \mathrm{S} \\ & P\left(X_2 \geq Y_2\right)=\frac{5}{16}+\frac{3}{8}=\frac{11}{16} \rightarrow \mathrm{Q} \end{aligned}

Q 49MathematicsSection A • MCQellipse

+3 / -1

Consider the ellipse

$' in math mode at position 1:$ ̲ \frac{x^{2}}{4…" style="color:#cc0000">$ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$

Let $H(\alpha, 0), 0<\alpha<2$ , be a point. A straight line drawn through $H$ parallel to the $y$ -axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$ -axis at a point $G$ . Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$ -axis.

List-I	List-II
(I) If $\phi=\frac{\pi}{4}$ , then the area of the triangle $F G H$ is	(P) $\frac{(\sqrt{3}-1)^{4}}{8}$
(II) If $\phi=\frac{\pi}{3}$ , then the area of the triangle $F G H$ is	(Q) 1
(III) If $\phi=\frac{\pi}{6}$ , then the area of the triangle $F G H$ is	(R) $\frac{3}{4}$
(IV) If $\phi=\frac{\pi}{12}$ , then the area of the triangle $F G H$ is	(S) $\frac{1}{2 \sqrt{3}}$
	(T) $\frac{3 \sqrt{3}}{2}$

The correct option is:

A

(\mathrm{I}) \rightarrow(\mathrm{R}) ;(\mathrm{II}) \rightarrow(\mathrm{S}) ;(\mathrm{III}) \rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})

B

(I)

\rightarrow

(R); (II)

\rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})

C

(I)

\rightarrow(\mathrm{Q}) ;(\mathrm{II}) \rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})

D

(I)

\rightarrow

(Q); (II)

\rightarrow

(S); (III)

\rightarrow

(Q); (IV)

\rightarrow

(P)

Q 50MathematicsSection A • MCQmatrices and determinants

+3 / -1

Let $p, q, r$ be nonzero real numbers that are, respectively, the $10^{\text {th }}, 100^{\text {th }}$ and $1000^{\text {th }}$ terms of a harmonic progression. Consider the system of linear equations

$' in math mode at position 1:$ ̲ \begin{gathere…" style="color:#cc0000">$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered}$

List-I	List-II
(I) If $\frac{q}{r}=10$ , then the system of linear equations has	(P) $x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$ as a solution
(II) If $\frac{p}{r} \neq 100$ , then the system of linear equations has	(Q) $x=\frac{10}{9}, y=-\frac{1}{9}, z=0$ as a solution
(III) If $\frac{p}{q} \neq 10$ , then the system of linear equations has	(R) infinitely many solutions
(IV) If $\frac{p}{q}=10$ , then the system of linear equations has	(S) no solution
	(T) at least one solution

The correct option is:

A

(I)

\rightarrow

(T); (II)

\rightarrow

(R); (III)

\rightarrow

(S); (IV)

\rightarrow

(T)

B

(I)

\rightarrow

(Q); (II)

\rightarrow

(S); (III)

\rightarrow

(S); (IV)

\rightarrow

(R)

C

(I)

\rightarrow(\mathrm{Q})

; (II)

\rightarrow

(R); (III)

\rightarrow(\mathrm{P})

; (IV)

\rightarrow

(R)

D

(I)

\rightarrow

(T); (II)

\rightarrow

(S); (III)

\rightarrow

(P); (IV)

\rightarrow

(T)

Official Correct Answer:Option (B)

Step-by-Step Solution:

Given:

\begin{gathered} x+y+z=1 ..........(1)\\ 10 x+100 y+1000 z=0 .........(2)\\ q r x+p r y+p q z=0 ............(3) \end{gathered}

Now equation (3) can be re-written as

\frac{x}{p}=\frac{y}{q}=\frac{z}{r}=0 \quad \because\{p, q, r \neq 0\}

Now given

p, q

and

r

are

10^{\text {th }}, 100^{\text {th }}

and

1000^{\text {th }}

term of an. H.P.,

So let

p=\frac{y}{a+9 d}, q=\frac{1}{a+99 d}, r=\frac{1}{a+999 d}

Now from equation (3)

(a+9 d) x+(a+99 d) y+(a+999 d) z=0

Now from and (1), (2) and (3) we get

\begin{aligned} \Delta & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 10 & 100 & 1000 \\ a+9 d & a+99 d & a+999 d \end{array}\right|=0 \\ \Delta x & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 100 & 1000 \\ 0 & a+99 d & a+999 d \end{array}\right|=900(d-a) \\ \Delta y & =\left|\begin{array}{ccc} 10 & 0 & 1000 \\ 1+9 d & 0 & a+999 d \end{array}\right|=990(a-d) \\ \Delta z & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 10 & 100 & 0 \\ a+9 d & a+99 d & 0 \end{array}\right|=90(d-a) \end{aligned}

(I) If

\frac{q}{r}=10 \Rightarrow a=d

\Delta=\Delta_x=\Delta_y=\Delta_z=0

And equation (1) and equation (2) represents non-parallel plane equation (2) and equation (3) represents same plane

\Rightarrow

Infinitely many solutions.

Now finding solution by taking

z=\lambda

then

\begin{aligned} & x+y =1-\lambda \\ & x+10 y =-100 \lambda \\ & x =\frac{10}{9}+100 \lambda \\ & y =\frac{-1}{9}-11 \lambda \\ & \Rightarrow (x, y, z) \in\left(\frac{10}{9}+10 \lambda, \frac{-1}{9}-11 \lambda, \lambda\right) \end{aligned}

So

\mathrm{P}

is not valid for any value of

\lambda \rightarrow \mathrm{Q}

(II)

\frac{p}{r} \neq 100 \Rightarrow a \neq d

\Delta=0 \,\text{and}\, \Delta_{x^{\prime}} \Delta_{y^{\prime}} \Delta_z \neq 0

So no solution.

\text { II } \rightarrow \mathrm{S}

(III) If

\frac{p}{q} \neq 10 \Rightarrow a \neq d \text { then } \Delta_z \neq 0

So no solution.

\text { III } \rightarrow \mathrm{S}

(IV) If

\Rightarrow a=d \text { then } \Delta_z=0 \Rightarrow \Delta_x=\Delta_y=0

So infinitely many solutions.

\mathrm{IV} \rightarrow \mathrm{R}

Q 51MathematicsSection B • Numericalfunctions

+4 / -2

Let $|M|$ denote the determinant of a square matrix $M$ . Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be the function defined by

$g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$

where

$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right| .$

Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$ , and $p(2)=2-\sqrt{2}$ . Then, which of the following is/are TRUE ?

Official Correct Answer:Declared in Solution

Step-by-Step Solution:

Given,

$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right|$

Here $\cos \left(\theta+\frac{\pi}{4}\right)=-\sin \left(\theta-\frac{\pi}{4}\right)$

and $\tan \left(\theta-\frac{\pi}{4}\right)=-\cot \left(\theta+\frac{\pi}{4}\right)$

and $\log _e\left(\frac{4}{\pi}\right)=-\log _e\left(\frac{\pi}{4}\right)$

Also $\sin \pi=-\cos \frac{\pi}{2}=\tan \pi=0$

$\Rightarrow$ $f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 2 & \sin \theta & 1 \\ 0 & 1 & \sin \theta \\ 0 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} 0 & -\sin \left(\theta-\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & 0 & \log _{\mathrm{e}}\left(\frac{4}{\pi}\right) \\ -\tan \left(\theta-\frac{\pi}{4}\right) & -\log _{\mathrm{e}}\left(\frac{4}{\pi}\right) & 0 \end{array}\right|$

$\begin{aligned} & f(\theta)=\left(1+\sin ^2 \theta\right)+0 \text { (skew symmetric) } \\ & g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1} \\ & =|\sin \theta|+|\cos \theta| \quad \\ & g(\theta) \in[1, \sqrt{2}] \text { for } \theta \in\left[0, \frac{\pi}{2}\right] \end{aligned}$

Again let $\mathrm{P}(\mathrm{x})=\mathrm{k}(\mathrm{x}-\sqrt{2})(\mathrm{x}-1)$

$\begin{aligned} & 2-\sqrt{2}=\mathrm{k}(2-\sqrt{2})(2-1) \\ & \Rightarrow \mathrm{k}=1 \quad(\mathrm{P}(2)=2-\sqrt{2} \text { given }) \\ & \therefore \mathrm{P}(\mathrm{x})=(\mathrm{x}-\sqrt{2})(\mathrm{x}-1) \end{aligned}$

For option (A) $P\left(\frac{3+\sqrt{2}}{4}\right)<0$ Correct.

For option (B) $P\left(\frac{1+3 \sqrt{2}}{4}\right)<0$ Incorrect.

For option (C) $P\left(\frac{5 \sqrt{2}-1}{4}\right)>0$ Correct.

For option (D) $P\left(\frac{5-\sqrt{2}}{4}\right)>0$ Incorrect.

Q 52MathematicsSection A • MCQtrigonometric functions and equations

+3 / -1

Consider the following lists :

List-I	List-II
(I) $\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$	(P) has two elements
(II) $\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\}$	(Q) has three elements
(III) $\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}$	(R) has four elements
(IV) $\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$	(S) has five elements
	(T) has six elements

The correct option is:

A

(I)

\rightarrow(\mathrm{P})

; (II)

\rightarrow(\mathrm{S})

; (III)

\rightarrow(\mathrm{P})

; (IV)

\rightarrow(\mathrm{S})

B

(I)

\rightarrow

(P); (II)

\rightarrow

(P); (III)

\rightarrow

(T); (IV)

\rightarrow

(R)

C

(I)

\rightarrow

(Q); (II)

\rightarrow(\mathrm{P})

; (III)

\rightarrow

(T); (IV)

\rightarrow

(S)

D

(I)

\rightarrow(\mathrm{Q})

; (II)

\rightarrow(\mathrm{S}) ;

(III)

\rightarrow(\mathrm{P})

; (IV)

\rightarrow(\mathrm{R})

Official Correct Answer:Option (B)

Step-by-Step Solution:

Solving all question one by one we get,

\begin{aligned} \text { (i) } & \left\{x \in\left[\frac{-2 \pi}{3}, \frac{2 \pi}{3}\right], \cos x+\sin x=1\right\} \\ & \cos x+\sin x=1 \\ \Rightarrow \quad & \sin \left(\frac{\pi}{4}+x\right)=\frac{1}{\sqrt{2}} \\ \Rightarrow & \frac{\pi}{4}+x=n \pi+(-1)^n \frac{\pi}{4} \end{aligned}

\begin{aligned} & \Rightarrow x=n \pi+(-1)^n \frac{\pi}{4}-\frac{\pi}{4} \\ & \text { So, } x \in\left\{0, \frac{\pi}{2}\right\} \\ & \therefore x \text { has } 2 \text { elements } \rightarrow P \end{aligned}

\begin{aligned} & \text { (ii) }\left\{x \in\left(\frac{-5 \pi}{18}, \frac{5 \pi}{18}\right), \sqrt{3} \tan 3 x=1\right\} \\ & \Rightarrow \tan 3 x=\frac{1}{\sqrt{3}} \\ & \Rightarrow x=n \pi+\frac{\pi}{6} \\ & \Rightarrow x=\frac{n \pi}{3}+\frac{\pi}{18} \\ & \text { So } x \in\left\{\frac{\pi}{18}, \frac{-5 \pi}{18}\right\} \\ & \therefore x \text { has 2 elements } \rightarrow P \end{aligned}

\begin{aligned} & \text { (iii) }\left\{x \in\left[\frac{-6 \pi}{5}, \frac{6 \pi}{5}\right], 2 \cos 2 x=\sqrt{3}\right\} \\ & \Rightarrow 2 \cos 2 x=\sqrt{3} \\ & \Rightarrow \cos 2 x=\frac{\sqrt{3}}{2} \\ & \Rightarrow 2 x=2 n \pi \pm \frac{\pi}{6} \\ & \Rightarrow x=n \pi \pm \frac{\pi}{12} \\ & \text { So } x \in\left\{ \pm \frac{\pi}{12}, \pi \pm \frac{\pi}{12},-\pi \pm \frac{\pi}{12}\right\} \\ & \therefore x \text { has } 6 \text { elements } \rightarrow \mathrm{T} \end{aligned}

\begin{aligned} & \text { (iv) }\left\{x \in\left[\frac{-7 \pi}{4}, \frac{7 \pi}{4}\right], \sin x-\cos x=1\right\} \\ & \Rightarrow \sin x-\cos x=1 \\ & \Rightarrow \sin \left(x-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}} \\ & \Rightarrow x-\frac{\pi}{4}=n \pi+(-1)^n \frac{\pi}{4} \\ & \Rightarrow x=n \pi+(-1)^n \frac{\pi}{4}+\frac{\pi}{4} \\ & \text { So, } \\ & x \in\left\{\frac{\pi}{2}, \frac{-3 \pi}{2}, \pi,-\pi\right\} \\ & \therefore x \text { has } 4 \text { elements } \rightarrow \mathrm{R} \end{aligned}

Q 53MathematicsSection B • Numerical3d geometry

+4 / -2

Let

S

be the reflection of a point

Q

with respect to the plane given by

\vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k}

where

t, p

are real parameters and

\hat{\imath}, \hat{\jmath}, \hat{k}

are the unit vectors along the three positive coordinate axes. If the position vectors of

Q

and

S

are

10 \hat{\imath}+15 \hat{\jmath}+20 \hat{k}

and

\alpha \hat{\imath}+\beta \hat{\jmath}+\gamma \hat{k}

respectively, then which of the following is/are TRUE ?

Q 54MathematicsSection B • Numericalparabola

+4 / -2

Consider the parabola $y^{2}=4 x$ . Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_{1}$ and $P_{2}$ . Let $Q_{1}$ and $Q_{2}$ be points on the lines $S P_{1}$ and $S P_{2}$ respectively such that $P Q_{1}$ is perpendicular to $S P_{1}$ and $P Q_{2}$ is perpendicular to $S P_{2}$ . Then, which of the following is/are TRUE?

JEE Advanced 2022 Paper 1 Question Paper with Solutions