JEE Main 2025 — 22 January, Evening Shift

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^{7}, x^{5}, x^{3}$ and $x$ respectively in the expansion of

$\left(x+\sqrt{x^{3}-1}\right)^{5}+\left(x-\sqrt{x^{3}-1}\right)^{5}, x>1$. If $u$ and $v$ satisfy the equations

$\alpha u+\beta v=18$, $\gamma u+\delta v=20$, then $u+v$ equals :

In a group of 3 girls and 4 boys, there are two boys $B_{1}$ and $B_{2}$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $\mathrm{B}_{1}$ and $\mathrm{B}_{2}$ are not adjacent to each other, is :

Let $P(4,4 \sqrt{3})$ be a point on the parabola $y^{2}=4 \mathrm{ax}$ and PQ be a focal chord of the parabola. If $M$ and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of M . Let A be a $3 \times 3$ matrix such that $|\mathrm{A}|=\frac{1}{2}$ and trace $(\mathrm{A})=3$. If $B=\operatorname{adj}(\operatorname{adj}(2 A))$, then the value of $|B|+$ trace $(B)$ equals:

Suppose that the number of terms in an A.P. is 2 k , $\mathrm{k} \in \mathrm{N}$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to :

Let a line pass through two distinct points $\mathrm{P}(-2,-1,3)$ and Q , and be parallel to the vector $3 \hat{i}+2 \hat{j}+2 k$. If the distance of the point $Q$ from the point $R(1,3,3)$ is 5 , then the square of the area of $\triangle \mathrm{PQR}$ is equal to:

If $\lim _{x \rightarrow \infty}\left(\left(\frac{e}{1-e}\right)\left(\frac{1}{e}-\frac{x}{1+x}\right)\right)^{x}=\alpha$, then the value of $\frac{\log _{\mathrm{e}} \alpha}{1+\log _{\mathrm{e}} \alpha}$ equals :

Let $f(x)=\int_{0}^{x^{2}} \frac{t^{2}-8 t+15}{e^{t}} d t, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of

f, respectively, are :

The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $P(2,-10,1)$, is:

If $x=f(y)$ is the solution of the differential equation $\left(1+y^{2}\right)+\left(x-2 e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :

If $\int \mathrm{e}^{\mathrm{x}}\left(\frac{\mathrm{x} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}}+\frac{\sin ^{-1} \mathrm{x}}{\left(1-\mathrm{x}^{2}\right)^{3 / 2}}+\frac{\mathrm{x}}{1-\mathrm{x}^{2}}\right) \mathrm{dx}=\mathrm{g}(\mathrm{x})+\mathrm{C}$,where C is the constant of integration, then $\mathrm{g}\left(\frac{1}{2}\right)$ equals :

Let $\alpha_{\theta}$ and $\beta_{\theta}$ be the distinct roots of $2 x^{2}+(\cos \theta) x-1=0, \theta \in(0,2 \pi)$. If $m$ and $M$ are the

minimum and the maximum values of $\alpha_{\theta}^{4}+\beta_{\theta}^{4}$, then $16(\mathrm{M}+\mathrm{m})$ equals :

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $f: A \rightarrow B$ such that $1 \in f(\mathrm{~A})$ is equal to :

If the system of linear equations :

$x+y+2 z=6$,

$2 x+3 y+a z=a+1$,

$-x-3 y+b z=2 b$,

where $\mathrm{a}, \mathrm{b} \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to $:$

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$ and $3 \vec{a}-\lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is :

Let $\mathrm{E}: \frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1, \mathrm{a}>\mathrm{b}$ and $\mathrm{H}: \frac{\mathrm{x}^{2}}{\mathrm{~A}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~B}^{2}}=1$. Let the distance between the foci of E and the

foci of $H$ be $2 \sqrt{3}$. If $\mathrm{a}-\mathrm{A}=2$, and the ratio of the eccentricities of E and H is $\frac{1}{3}$, then the sum of the

lengths of their latus rectums is equal to :

If A and B are two events such that $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12 x^{2}-7 x+1=0$, then the value of $\frac{P(\overline{\mathrm{~A}} \cup \overline{\mathrm{~B}})}{\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})}$ is:

The sum of all values of $\theta \in[0,2 \pi]$ satisfying $2 \sin ^{2} \theta=\cos 2 \theta$ and $2 \cos ^{2} \theta=3 \sin \theta$ is

Let the curve $z(1+i) + \bar{z}(1-i) = 4, z \in C,$ divide the region $|z-3| \le 1$ into two parts of areas $\alpha$ and $\beta.$ Then $|\alpha - \beta|$ equals:

The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is:

Let y = f(x) be the solution of the differential equation $\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, -1 < x < 1$ such that $[f(0) = 0.$ If $6 \int_{-\frac{1}{2}}^{\frac{1}{2}} f(x) dx = 2\pi - \alpha$ then $\alpha^2$ is equal to $\underline{\hspace{2cm}}$

Let $\mathrm{A}(6,8), \mathrm{B}(10 \cos \alpha,-10 \sin \alpha)$ and C $(-10 \sin \alpha, 10 \cos \alpha)$, be the vertices of a triangle. If

$\mathrm{L}(\mathrm{a}, 9)$ and $\mathrm{G}(\mathrm{h}, \mathrm{k})$ be its orthocenter and centroid respectively, then $(5 \mathrm{a}-3 \mathrm{~h}+6 \mathrm{k}+100 \sin 2 \alpha)$ is

equal to $\_\_\_\_\_$

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then $(\mathrm{QR})^{2}$ is equal to $\_\_\_\_\_$ .

If $\sum_{\mathrm{r}=1}^{30} \frac{\mathrm{r}^{2}{ }^{\left({ }^{30} \mathrm{C}_{\mathrm{r}}\right)^{2}}}{{ }^{30} \mathrm{C}_{\mathrm{r}-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to $\_\_\_\_\_$ .

Let $\mathrm{A}=\{1,2,3\}$. The number of relations on A , containing $( 1,2 )$ and $(2,3)$, which are reflexive and transitive but not symmetric, is $\_\_\_\_\_$ .

Arrange the following compounds in increasing order of their dipole moment :

$\mathrm{HBr}, \mathrm{H}_{2} \mathrm{~S}, \mathrm{NF}_{3}$ and $\mathrm{CHCl}_{3}$

Identify the number of structure/s from the following which can be correlated to D-glyceraldehyde.

The maximum covalency of a non-metallic group 15 element ' E ' with weakest $\mathrm{E}-\mathrm{E}$ bond is :

The species which does not undergo disproportionation reaction is :

Match the Compounds (List-I) with the appropriate Catalyst/Reagents (List-II) for their reduction into corresponding amines.

Choose the correct answer from the options given below:

The maximum number of RBr producing 2-methylbutane by above sequence of reactions is___. (Consider the structural isomers only)

Match List-I with List-II.

Choose the correct answer from the options given below:

The correct order of the following complexes in terms of their crystal field stabilization energies is :

Density of $3\ \mathrm{M}$ $\mathrm{NaCl}$ solution is $1.25\ \mathrm{g\ mL^{-1}}$. The molality of the solution is:

The molar solubility(s) of zirconium phosphate with molecular formula $\left(\mathrm{Zr}^{4+}\right)_{3}\left(\mathrm{PO}_{4}^{3-}\right)_{4}$ is given by relation :

The most stable carbocation from the following is :

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

Statement (I) : An element in the extreme left of the periodic table forms acidic oxides.

Statement (II) : Acid is formed during the reaction between water and oxide of a reactive element present in the extreme right of the periodic table.

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

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

Statement (I): A spectral line will be observed for a $2 \mathrm{p}_{\mathrm{x}} \rightarrow 2 \mathrm{p}_{\mathrm{y}}$ transition.

Statement (II): $2 p_{x}$ and $2 p_{y}$ are degenerate orbitals.

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

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

Statement (I) : Nitrogen, sulphur, halogen and phosphorus present in an organic compound are detected by Lassaigne's Test.

Statement (II) : The elements present in the compound are converted from covalent form into ionic form by fusing the compound with Magnesium in Lassaigne's test.

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

Identify the homoleptic complex(es) that is/are low spin.

(A) $\left[\mathrm{Fe}(\mathrm{CN})_{5} \mathrm{NO}\right]^{2-}$

(B) $\left[\mathrm{CoF}_{6}\right]^{3-}$

(C) $\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}$

(D) $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$

(E) $\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$

Choose the correct answer from the options given below :

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

Statement (I) : Corrosion is an electrochemical phenomenon in which pure metal acts as an anode and impure metal as a cathode.

Statement (II) : The rate of corrosion is more in alkaline medium than in acidic medium.

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

The alkane from below having two secondary hydrogens is :

The compound with molecular formula $\mathrm{C}_{6} \mathrm{H}_{6}$, which gives only one monobromo derivative and takes up four moles of hydrogen per mole for complete hydrogenation has____ $\pi$ electrons.

Niobium $(\mathrm{Nb})$ and ruthenium ( Ru ) have " $x$ " and " $y$ " number of electrons in their respective $4 d$ orbitals. The value of $x+y$ is____

The complex of $\mathrm{Ni}^{2+}$ ion and dimethyl glyoxime contains____number of Hydrogen $(\mathrm{H})$ atoms.

Consider the following cases of standard enthalpy of reaction $\left(\Delta \mathrm{H}_{\mathrm{r}}^{\mathrm{o}}\right.$ in $\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$

$\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{~g})+\frac{7}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_{2}(\mathrm{~g})+3 \mathrm{H}_{2} \mathrm{O}(\ell) \Delta \mathrm{H}_{1}^{\mathrm{o}}=-1550$

C (graphite) $+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g}) \Delta \mathrm{H}_{2}^{\mathrm{o}}=-393.5$

$\mathrm{H}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{H}_{2} \mathrm{O}(\ell) \Delta \mathrm{H}_{3}^{\mathrm{o}}=-286$

The magnitude of $\Delta \mathrm{H}_{\mathrm{f}_{2} \mathrm{H}_{6}(\mathrm{~g})}^{\mathrm{o}}$ is____ $\mathrm{kJ} \mathrm{mol}^{-1}$ (Nearest integer).

$20\ \mathrm{mL}$ of $2\ \mathrm{M}$ $\mathrm{NaOH}$ solution is added to $400\ \mathrm{mL}$ of $0.5\ \mathrm{M}$ $\mathrm{NaOH}$ solution. The final concentration of the solution is $\_\_\_\_ \times 10^{-2}\ \mathrm{M}$ (nearest integer).

A small rigid spherical ball of mass $M$ is dropped in a long vertical tube

containing glycerine. The velocity of the ball becomes constant after some

time. If the density of glycerine is half of the density of the ball, then the viscous

force acting on the ball will be (consider g as

acceleration due to gravity)

The maximum percentage error in the measurment of density of a wire

is[0pt] [Given, mass of wire $=(0.60 \pm 0.003) \mathrm{g}$ radius of

wire $=(0.50 \pm 0.01) \mathrm{cm}$ length of wire

$(10.00 \pm 0.05) \mathrm{cm}$ ]

A series LCR circuit is connected to an alternating source of emf E .

The current amplitude at resonant frequency is $I_{0}$. If the value of

resistance $R$ becomes twice of its initial value then amplitude of current

at resonance will be

For a short dipole placed at origin O , the dipole moment P is along x -axis,

as shown in the figure. If the electric potential and electric field at A are

$V_{0}$ and $E_{0}$, respectively, then the correct combination of the

electric potential and electric field, respectively, at point $B$ on the $y$-axis is given by

Which one of the following is the correct dimensional formula for the

capacitance in F? M, $\mathrm{L}, \mathrm{T}$ and C stand for unit of

mass, length, time and charge,

An electron projected perpendicular to a uniform magnetic field B moves in a circle. If Bohr's quantization is applicable, then the radius of the electronic orbit in the first excited state is :

For a diatomic gas, if $\gamma_{1}=\left(\frac{\mathrm{Cp}}{\mathrm{Cv}}\right)$ for rigid molecules and $\gamma_{2}=\left(\frac{\mathrm{Cp}}{\mathrm{Cv}}\right)$ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ? ( Cp and Cv are specific heats of the gas at constant pressure and volume)

A light source of wavelength $\lambda$ illuminates a metal surface and electrons

are ejected with maximum kinetic energy of 2 eV . If the same surface is illuminated

by a light source of wavelength $\frac{\lambda}{2}$, then the maximum kinetic energy of

ejected electrons will be (The work function of metal is 1 eV )

The torque due to the force $(2 \hat{i}+\hat{j}+2 \hat{k})$ about the origin,

acting on a particle whose position vector is

$(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$, would be

A force $\vec{F}=2 \hat{i}+b \hat{j}+\hat{k}$ is applied on a particle and it

undergoes a displacement $\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$.

What will be the value of $b$, if work done on the particle is zero.

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

Assertion (A) : In Young's double slit experiment, the fringes produced by red light are closer as compared to those produced by blue light.

Reason (R) : The fringe width is directly proportional to the wavelength of light.

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

A ball of mass 100 g is projected with velocity $20 \mathrm{~m} / \mathrm{s}$

at $60^{\circ}$ with horizontal. The decrease in kinetic energy of the ball during

the motion from point of projection to highest point is :

The tube of length $L$ is shown in the figure. The radius of cross section at the

point (1) is 2 cm and at the point (2) is 1 cm , respectively. If the velocity of water

entering at point (1) is $2 \mathrm{~m} / \mathrm{s}$, then velocity of water

leaving the point (2) will be :

Given are statements for certain thermodynamic variables, (A) Internal energy,

volume (V) and mass (M) are extensive variables. (B) Pressure (P), temperature (T)

and density ( $\rho$ ) are intensive variables. (C) Volume (V), temperature (T) and

density ( $\rho$ ) are intensive variables. (D) Mass (M), temperature (T) and internal

energy are extensive variables. Choose the correct answer from the points given below :

Two long parallel wires $X$ and $Y$, separated by a distance of 6 cm , carry

currents of 5 A and 4 A , respectively, in opposite directions as shown in the figure.

Magnitude of the resultant magnetic field at point P at a distance of 4 cm from

wire Y is $\mathrm{x} \times 10^{-5} \mathrm{~T}$. The value of $x$ is $\qquad$ .

Take permeability of free space as $\mu_{0}=4 \pi \times 10^{-7}$ SI units.

A parallel plate capacitor of area $A=16 \mathrm{~cm}^{2}$ and separation

between the plates 10 cm , is charged by a DC current. Consider a hypothetical

plane surface of area $A_{0}=3.2 \mathrm{~cm}^{2}$ inside the capacitor and

parallel to the plates. At an instant, the current through the circuit is 6 A . At the

same instant the displacement current through $\mathrm{A}_{0}$ is $\qquad$ mA .