JEE Main 2025 — 22 January, Morning Shift

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that

$f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y})$ for all $\mathrm{x}, \mathrm{y} \in \mathbf{R}$. If $f^{\prime}(0)=4 \mathrm{a}$ and $f$ staisfies $f^{\prime \prime}(\mathrm{x})-3 \mathrm{a} f^{\prime}(\mathrm{x})-f(\mathrm{x})=0$, $\mathrm{a}>0$,

then the area of the region $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}) \mid 0 \leq \mathrm{y} \leq f(\mathrm{ax}), 0 \leq \mathrm{x} \leq 2\}$ is :

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the centroid of $\triangle P Q R$ is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

Let $\mathrm{z}_{\mathrm{l}}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_{1}\right)=\frac{-\pi}{4}, \arg \left(z_{2}\right)=0$ and $\arg \left(z_{3}\right)=\frac{\pi}{4}$. If $\left|z_{1} \overline{\mathrm{z}}_{2}+\mathrm{z}_{2} \overline{\mathrm{z}}_{3}+\mathrm{z}_{3} \overline{\mathrm{z}}_{1}\right|^{2}=\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathrm{Z}$, then the value of $\alpha^{2}+\beta^{2}$ is:

Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values

of $16\left(\left(\sec ^{-1} x\right)^{2}+\left(\operatorname{cosec}^{-1} x\right)^{2}\right)$ is :

A coin is tossed three times. Let X denote the number of times a tail follows a head. If $\mu$ and $\sigma^{2}$ denote the mean and variance of X , then the value of $64\left(\mu+\sigma^{2}\right)$ is :

Let $a_{1}, a_{2}, a_{3} \ldots$. be a G.P. of increasing positive terms. If $\mathrm{a}_{1} \mathrm{a}_{5}=28$ and $\mathrm{a}_{2}+\mathrm{a}_{4}=29$, the $\mathrm{a}_{6}$ is equal to

Let $L_{1}: \frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-3}{4}$ and $L_{2}: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ be two lines. Then which of the

following points lies on the line of the shortest distance between $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ ?

The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{e} x\right)^{2}+3}=\mathrm{x}^{8}, x>0$, is :

If $\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{T}_{\mathrm{r}}=\frac{(2 \mathrm{n}-1)(2 \mathrm{n}+1)(2 \mathrm{n}+3)(2 \mathrm{n}+5)}{64}$, then $\lim _{n \rightarrow \infty} \sum_{r=1}^{n}\left(\frac{1}{T_{r}}\right)$ is equal to :

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

Let $\mathrm{x}=\mathrm{x}(\mathrm{y})$ be the solution of the differential equation $y^{2} d x+\left(x-\frac{1}{y}\right) d y=0$. If $x(1)=1$, then

$x\left(\frac{1}{2}\right)$ is :

Let the parabola $y = x^2 + px - 3$ meet the coordinate axes at the points $P, Q$ and $R.$ If the circle $C$ with centre at $(-1, -1)$ passes through the points $P, Q$ and $R$ then the area of $\triangle PQR$ is:

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle C at exactly two points. If the set of all possible values of $r$ is the interval $(\alpha, \beta)$, then $3 \beta-2 \alpha$ is equal to :

Let for $f(\mathrm{x})=7 \tan ^{8} \mathrm{x}+7 \tan ^{6} \mathrm{x}-3 \tan ^{4} \mathrm{x}-3 \tan ^{2} \mathrm{x}$, $\mathrm{I}_{1}=\int_{0}^{\pi / 4} f(\mathrm{x}) \mathrm{dx}$ and

$\mathrm{I}_{2}=\int_{0}^{\pi / 4} \mathrm{x} f(\mathrm{x}) \mathrm{dx}$. Then $7 \mathrm{I}_{1}+12 \mathrm{I}_{2}$ is equal to :

Let $f(\mathrm{x})$ be a real differentiable function such that $f(0)=1$ and $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f^{\prime}(\mathrm{y})+f^{\prime}(\mathrm{x}) f(\mathrm{y})$ for all $x, y \in \mathbf{R}$. Then $\sum_{\mathrm{n}=1}^{100} \log _{\mathrm{e}} f(\mathrm{n})$ is equal to :

Let $\mathrm{A}=\{1,2,3, \ldots \ldots ., 10\}$ and $B = \left\{ \frac{m}{n} : m, n \in A, m < n, \ \text{ and } \ \ \ \text{gcd}(m, n) = 1 \right\}$. Then n(B) is equal to:

The area of the region, inside the circle $(x-2 \sqrt{3})^{2}+y^{2}=12$ and outside the parabola $y^{2}=2 \sqrt{3} x$ is

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to :

Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its latus-rectum is :

Let the function,

Be differentiable for all $\mathrm{x} \in \mathbf{R}$, where

$\mathrm{a}>1, \mathrm{~b} \in \mathbf{R}$.

If the area of the region enclosed by $y=f(x)$ and the line

$y=-20$ is $\alpha+\beta \sqrt{3}, \alpha, \beta, \in Z$, then

the value of $\alpha+\beta$ is $\_\_\_\_\_$

If $\sum_{\mathrm{r}=0}^{5} \frac{{ }^{11} \mathrm{C}_{2 \mathrm{r}+1}}{2 \mathrm{r}+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to $\_\_\_\_\_$

Let $A$ be a square matrix of order 3 such that $\operatorname{det}(A)=-2$ and $\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{m+n} \cdot 3^{m n}$, $\mathrm{m}>\mathrm{n}$. Then $4 \mathrm{~m}+2 \mathrm{n}$ is equal to $\_\_\_\_\_$ .

Let $\mathrm{L}_{1}: \frac{\mathrm{x}-1}{3}=\frac{\mathrm{y}-1}{-1}=\frac{\mathrm{z}+1}{0}$ and $L_{2}: \frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}}{0}=\frac{\mathrm{z}+4}{\alpha}, \alpha \in \mathrm{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $\mathrm{A}(1,1,-1)$ on $\mathrm{L}_{2}$, then the value of $26 \alpha(\mathrm{~PB})^{2}$ is $\_\_\_\_\_$

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is $\_\_\_\_\_$ .

A solution of aluminium chloride is electrolysed for 30 minutes using a current of 2 A . The amount of the aluminium deposited at the cathode is____. [Given : molar mass of aluminium and chlorine are $27 \mathrm{~g} \mathrm{~mol}^{-1}$ and $35.5 \mathrm{~g} \mathrm{~mol}^{-1}$ respectively, Faraday constant $\left.=96500 \mathrm{C} \mathrm{mol}^{-1}\right]$.

Which of the following statement is not true for radioactive decay?

How many different stereoisomers are possible for the given molecule?

Which of the following electronegativity order is incorrect?

Lanthanoid ions with $4 f^{7}$ configuration are :

(A) $\mathrm{Eu}^{2+}$ (B) $\mathrm{Gd}^{3+}$ (C) $\mathrm{Eu}^{3+}$

(D) $\mathrm{Tb}^{3+}$ (E) $\mathrm{Sm}^{2+}$

Choose the correct answer from the options given below:

Match the List-I with List-II and choose the correct answer from the options given below :

Which of the following acids is a vitamin ?

A liquid when kept inside a thermally insulated closed vessel at $25^{\circ} \mathrm{C}$ was mechanically stirred from outside. What will be the correct option for the following thermodynamic parameters?

Radius of the first excited state of helium ion is $\_\_\_\_\_\_$.

(Given: $a_0$ = radius of first stationary state of hydrogen atom)

Given below are two statements : one is labelled as Statement I and the other is labelled as Statement II :

Statement I : One mole of propyne reacts with excess of sodium to liberate half a mole of $\mathrm{H}_{2}$ gas.

Statement II : Four g of propyne reacts with $\mathrm{NaNH}_{2}$ to liberate $\mathrm{NH}_{3}$ gas which occupies 224 mL at STP.

In the light of the above statements, choose the most appropriate answer from the options given below:

A vessel at 1000 K contains $\mathrm{CO}_{2}$ with a pressure of 0.5 atm . Some of $\mathrm{CO}_{2}$ is converted into CO on addition of graphite. If total pressure at equilibrium is 0.8 atm , then $\mathrm{K}_{\mathrm{P}}$ is :

The IUPAC name of the following compound is :

Which of the following electrolyte can be sued to obtain $\mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{8}$ by the process of electrolysis?

The compounds which give positive Fehling's test are:

Choose the CORRECT answer from the options given below :

In which of the following complexes the CFSE, $\Delta_{0}$ will be equal to zero?

Arrange the following solutions in order of their increasing boiling points.

(i) $10^{-4} \mathrm{M} \mathrm{NaCl}$

(ii) $10^{-4} \mathrm{M}$ Urea

(iii) $10^{-3} \mathrm{M} \mathrm{NaCl}$

(iv) $10^{-2} \mathrm{M} \mathrm{NaCl}$

From the magnetic behaviour of $\left[\mathrm{NiCl}_{4}\right]^{2-}$ (paramagnetic) and $\left[\mathrm{Ni}(\mathrm{CO})_{4}\right]$ (diamagnetic), choose the correct geometry and oxidation state.

The incorrect statements regarding geometrical isomerism are :

(A) Propene shows geometrical isomerism.

(B) Trans isomer has identical atoms/groups on the opposite sides of the double bond.

(C) Cis-but-2-ene has higher dipole moment than trans-but-2-ene.

(D) 2-methylbut-2-ene shows two geometrical isomers

(E) Trans-isomer has lower melting point that cis isomer.

Choose the CORRECT answer from the options given below :

Some $\mathrm{CO}_{2}$ gas was kept in a sealed container at a pressure of 1 atm and at 273 K . This entire amount of $\mathrm{CO}_{2}$ gas was later passed through an aqueous solution of $\mathrm{Ca}(\mathrm{OH})_{2}$. The excess unreacted $\mathrm{Ca}(\mathrm{OH})_{2}$ was later neutralized with 0.1 M of 40 mL HCl . If the volume of the sealed container of $\mathrm{CO}_{2}$ was x , then x is _____ $\mathrm{cm}^{3}$ (nearest integer).

[Given : The entire amount of $\mathrm{CO}_{2}(\mathrm{~g})$ reacted with exactly half the initial amount of $\mathrm{Ca}(\mathrm{OH})_{2}$ present in the aqueous solution.]

The number of molecules/ions that show linear geometry among the following is ____.

$\mathrm{SO}_{2}, \mathrm{BeCl}_{2}, \mathrm{CO}_{2}, \mathrm{~N}_{3}^{-}, \mathrm{NO}_{2}, \mathrm{~F}_{2} \mathrm{O}, \mathrm{XeF}_{2}, \mathrm{NO}_{2}^{+}, \mathrm{I}_{3}^{-}, \mathrm{O}_{3}$

$\mathrm{A} \rightarrow \mathrm{B}$ The molecule A changes into its isomeric form $B$ by following a first order kinetics at a

temperature of 1000 K . If the energy barrier with respect to reactant energy for such isomeric

transformation is $191.48 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and the frequency factor is $10^{20}$, the time required

for $50 \%$, molecules of A to become B is _____ picoseconds (nearest integer).

$\left[\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right.$ ]

A line charge of length $\frac{\mathrm{a} \text { ' }}{2}$ is kept at the center

of an edge BC of a cube ABCDEFGH having edge length ' $a$ ' as shown in the figure.

If the density of line is $\lambda \mathrm{C}$ per unit length, then the total electric

flux through all the faces of the cube will be $\qquad$ _. (Take, $\epsilon_{0}$ as the free space permittivity)

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion A: If Young's double slit experiment is performed in an optically denser medium than air, then the consecutive fringes come closer.

Reason R : The speed of light reduces in an optically denser medium than air while its frequency does not change.

In the light of the above statements, choose the most appropriate answer from the options given below :

Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in same atmosphere. The temperature of the smaller body is 800 K and temperature of bigger body is 400 K . If the energy radiate from the smaller body is E , the energy radiated from the bigger body is (assume, effect of the surrounding to be negligible)

An amount of ice of mass $10^{-3} \mathrm{~kg}$ and temperature

$-10^{\circ} \mathrm{C}$ is transformed to vapour of temperature $110^{\circ}$ by applying heat. The total amount of work required for this conversion is, (Take, specific heat of ice $=2100 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}$, specific heat of water $=4180 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}$, specific heat of steam $=1920 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}$, Latent heat of ice $=3.35 \times 10^{5} \mathrm{Jkg}^{-1}$ and Latent heat of steam $=2.25 \times 10^{6} \mathrm{Jkg}^{-1}$ )

An electron is made to enters symmetrically between two parallel and equally

but oppositely charged metal plates, each of 10 cm length. The electron emerges

out of the field region with a horizontal component of velocity

$10^{6} \mathrm{~m} / \mathrm{s}$. If the magnitude of the electric

between the plates is 9.1 $\mathrm{V} / \mathrm{cm}$, then the v

ertical component of velocity of electron is (mass of electron

$=9.1 \times 10^{-31} \mathrm{~kg}$ and charge of electron $=1.6 \times 10^{-19} \mathrm{C}$ )

A closed organ and an open organ tube filled by two different gases having same bulk

modulus but different densities $\rho_{1}$ and $\rho_{2}$ respectively. The frequency

of $9^{\text {th }}$ harmonic of closed tube is identical with $4^{\text {th }}$ harmonic

of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases

is $\rho_{1}: \rho_{2}=1: 16$, then the length of the open tube is :

A uniform circular disc of radius ' $R$ ' and mass ' $M$ ' is rotating about

an axis perpendicular to its plane and passing through its centre. A small

circular part of radius $\mathrm{R} / 2$ is removed from the original disc

as shown in the figure. Find the moment of inertia of the remaining part of the

original disc about the axis as given above.

A small point of mass $m$ is placed at a distance $2 R$ from the centre ' O ' of a

big uniform solid sphere of mass $M$ and radius R. The gravitational force on

' $m$ ' due to $M$ is $F_{1}$. A spherical part of radius $R / 3$ is removed

from the big sphere as shown in the figure and the gravitational force on $m$

due to remaining part of M is found to be $\mathrm{F}_{2}$. The value of ratio

$F_{1}: F_{2}$ is

The work functions of cesium (Cs) and lithium (Li) metals are 1.9 eV and 2.5 eV , respectively. If we incident a light of wavelength 550 nm on these two metal surface, then photo-electric effect is possible for the case of

If B is magnetic field and $\mu_{0}$ is permeability of free space, then the dimensions of $\left(\mathrm{B} / \mu_{0}\right)$ is

A bob of mass $m$ is suspended at a point $O$ by a light string of length $l$

and left to perform vertical motion (circular) as shown in figure. Initially, by applying

horizontal velocity $v_{0}$ at the point ' A '. the string becomes slack when, the bob

reaches at the point ' $D$ '. The ratio of the kinetic energy of the bob at the

points B and C is $\qquad$ .

Given below are two statements : one is labelled as Statement I and the other is labelled as Statement II :

Statement-I : The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs.

Statement-II : The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries.

In the light of the above statements, choose the correct answer from the options given below.

A parallel-plate capacitor of capacitance $40 \mu \mathrm{~F}$ is connected

to a 100 V power supply. Now the intermediate space between the plates is filled

with a dielectric material of dielectric constant $\mathrm{K}=2$. Due to the

introduction of dielectric material, the extra charge and the change in the electrostatic

energy in the capacitor, respectively, are -

Given is a thin convex lens of glass (refractive index $\mu$ ) and each side

having radius of curvature R. One side is polished for complete reflection. At what

distance from the lens, an object be placed on the optic axis so that the image gets

formed on the object itself.

Two soap bubbles of radius 2 cm and 4 cm , respectively, are in contact with

each other. The radius of curvature of the common surface, in cm , is $\qquad$ .

The driver sitting inside a parked car is watching vehicles approaching from

behind with the help of his side view mirror, which is a convex mirror with radius

of curvature $\mathrm{R}=2 \mathrm{~m}$. Another car approaches him from

behind with a uniform speed of $90 \mathrm{~km} / \mathrm{hr}$. When the

car is at a distance of 24 m from him, the magnitude of the acceleration of the

image of the side view mirror is ' $a$ '. The value of 100a is $\qquad$ $\mathrm{m} / \mathrm{s}^{2}$.

Three conductions of same length having thermal conductivity $\mathrm{k}_{1}, \mathrm{k}_{2}$ and $\mathrm{k}_{3}$ are connected as shown in figure. Area of cross sections of $1^{\text {st }}$ and $2^{\text {nd }}$ conductor are same and for $3^{\text {rd }}$ conductor it is double of the $1^{\text {st }}$ conductor. The temperatures are given in the figure. In steady state condition, the value of $\theta$ is $\qquad$ ${ }^{\circ} \mathrm{C}$. (Given : $\mathrm{k}_{1}=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_{2}=120 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_{3}=$ $135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}$ )

The position vectors of two 1 kg particles, (A) and (B), are given by

$\vec{r}_{A}=\left(\alpha_{1} t^{2} \hat{i}+\alpha_{2} t \hat{j}+\alpha_{3} t \hat{k}\right) m$ and $\vec{r}_{B}=\left(\beta_{1} \hat{t}+\beta_{2} \mathrm{t}^{2} \hat{j}+\beta_{3} t \hat{k}\right) \mathrm{m}$, respectively ;

$\left(\alpha_{1}=1 \mathrm{~m} / \mathrm{s}^{2}, \alpha_{2}=3 \mathrm{~nm} / \mathrm{s}, \alpha_{3}=2 \mathrm{~m} / \mathrm{s}, \beta_{1}=2 \mathrm{~m} / \mathrm{s}\right.$,

$\beta_{2}=-1 \mathrm{~m} / \mathrm{s}^{2}, \beta_{3}=4 \mathrm{p} \mathrm{m} / \mathrm{s}$ ),

where t is time, n and p are constants, At $\mathrm{t}=1 \mathrm{~s},\left|\overrightarrow{\mathrm{~V}}_{\mathrm{A}}\right|=\left|\overrightarrow{\mathrm{V}}_{\mathrm{B}}\right|$ and velocities $\vec{V}_{A}$ and $\vec{V}_{B}$

of the particles are orthogonal to each other. At $t=1 \mathrm{~s}$, the magnitude of angular

momentum of particle (A) with respect to the position of particle (B) is $\sqrt{\mathrm{L}} \mathrm{kgm}^{2} \mathrm{~s}^{-1}$. The value of $L$ is $\qquad$ .

A particle is projected at an angle of $30^{\circ}$ from horizontal at a speed of

$60 \mathrm{~m} / \mathrm{s}$. The height traversed by the particle in the first

second is $h_{0}$ and height traversed in the last second, before it reaches the maximum height,

is $h_{1}$. The ratio $h_{0}: h_{1}$ is

$\qquad$ .[0pt] [Take, $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ ]