JEE Main 2025 Jan 22 Shift 2 — Question Paper

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

Let $\alpha_{\theta}$ and $\beta_{\theta}$ be the distinct roots of $2x^2 + (\cos\theta)x - 1 = 0$, $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_{\theta}^{4} + \beta_{\theta}^{4}$, then $16(M + m)$ equals :

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

If the system of equations :

$x + y + 2z = 6$,

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

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

where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ 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 $|A| = \dfrac{1}{2}$ and trace $(A) = 3$. If $B = \mathrm{adj}(\mathrm{adj}(2A))$, then the value of $|B| + \text{trace}(B)$ equals :

Suppose that the number of terms in an A.P. is $2k$, $k \in \mathbb{N}$. 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 :

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 $B_1$ and $B_2$ are not adjacent to each other, is :

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 :

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

Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ 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 :

Let $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ and $H : \dfrac{x^2}{A^2} - \dfrac{y^2}{B^2} = 1$. Let the distance between the foci of $E$ and the foci of $H$ be $2\sqrt{3}$. If $a - A = 2$, and the ratio of the eccentricities of $E$ and $H$ is $\dfrac{1}{3}$, then the sum of the lengths of their latus rectums is equal to :

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 $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\dfrac{\pi}{3}$. 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 :

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

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

If $\lim\limits_{x \to \infty} \left( \left( \dfrac{e}{1-e} \right) \left( \dfrac{1}{e} - \dfrac{x}{1+x} \right) \right)^{x} = \alpha$, then the value of $\dfrac{\log_e \alpha}{1 + \log_e \alpha}$ equals :

Let $f(x) = \displaystyle\int_{0}^{x^2} \dfrac{t^2 - 8t + 15}{e^t} \, dt$, $x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :

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

If $x = f(y)$ is the solution of the differential equation

$$\left(1 + y^2\right) + \left(x - 2e^{\tan^{-1} y}\right) \dfrac{dy}{dx} = 0, \; y \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$$

with $f(0) = 1$, then $f\left(\dfrac{1}{\sqrt{3}}\right)$ is equal to :

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

If $\displaystyle\sum_{r=1}^{30} \dfrac{r^2 \left({}^{30}C_r\right)^2}{{}^{30}C_{r-1}} = \alpha \times 2^{29}$, then $\alpha$ is equal to ________.

Let $A(6, 8)$, $B(10\cos\alpha, -10\sin\alpha)$ and $C(-10\sin\alpha, 10\cos\alpha)$, be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ be its orthocenter and centroid respectively, then $(5a - 3h + 6k + 100\sin 2\alpha)$ is equal to ________.

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

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

Which one of the following is the correct dimensional formula for the capacitance in F ? M, L, T and C stand for unit of mass, length, time and charge,

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

[Given, mass of wire $= (0.60 \pm 0.003)\,\mathrm{g}$

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

length of wire $= (10.00 \pm 0.05)\,\mathrm{cm}$]

The torque due to the force $\left(2\hat{i} + \hat{j} + 2\hat{k}\right)$ about the origin, acting on a particle whose position vector is $\left(\hat{i} + \hat{j} + \hat{k}\right)$, would be

A ball of mass 100 g is projected with velocity 20 m/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

A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s. The ratio of its kinetic energies at point B and C is :

[Figure: a vertical circle with centre O; A is the lowest point, D the highest point; C lies above the horizontal diameter and B below it, with OB making $30^{\circ}$ with the horizontal and angle between OC and OB equal to $90^{\circ}$]

(Take acceleration due to gravity as $10\ \mathrm{m/s^2}$)

A force $\vec{F} = 2\hat{i} + b\hat{j} + \hat{k}$ is applied on a particle and it undergoes a displacement $\hat{i} - 2\hat{j} - \hat{k}$. What will be the value of b, if work done on the particle is zero.

[Figure: a tapering tube; water enters the wider end marked (1) and leaves the narrower end marked (2)]

A 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 m/s, then velocity of water leaving the point (2) will be

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)

For a diatomic gas, if $\gamma_1 = \left(\frac{C_p}{C_v}\right)$ for rigid molecules and $\gamma_2 = \left(\frac{C_p}{C_v}\right)$ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ?

($C_p$ and $C_v$ are specific heats of the gas at constant pressure and volume)

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 options given below :

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

Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet.

Reason (R) : The mass of the pendulum remains unchanged at Earth and the other planet.

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

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

[Figure: dipole P at origin O pointing along the positive $x$-axis; point A on the $x$-axis at distance r from O; point B on the $y$-axis at distance 2r from O]

A rectangular metallic loop is moving out of a region of uniform magnetic field into a magnetic field-free region with a constant velocity. When the loop is partially inside the magnetic field, the plot of induced emf ($\varepsilon$) with time (t) is given by :

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

Given below are two statements. One 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 the above statements, choose the correct answer from the options given below :

[Figure: logic circuit — inputs A and B are fed directly to one AND gate; A and B are also fed through NOT gates to a second AND gate; the outputs of both AND gates are fed to a gate G whose output is Y]

A	B	Y
0	0	1
0	1	0
1	0	0
1	1	1

To obtain the given truth table, the following logic gate should be placed at G :

A tube of length 1m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is $\sqrt{\dfrac{F}{\alpha M}}$ in SI unit. The value of $\alpha$ is __________.

A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of $2 \times 10^5\ \mathrm{ms^{-1}}$. When the electric field is switched off, the proton moves along a circular path of radius 2 cm. The magnitude of electric field is $x \times 10^4\ \mathrm{N/C}$. The value of $x$ is __________. Take the mass of the proton $= 1.6 \times 10^{-27}\ \mathrm{kg}$.

Given below are two statements :

Statement (I) : A spectral line will be observed for a $2p_x \to 2p_y$ transition.

Statement (II) : $2p_x$ and $2p_y$ are degenerate orbitals.

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

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

The species which does not undergo disproportionation reaction is :

Consider the given figure and choose the correct option :

[Figure: energy vs reaction coordinate diagram — the curve rises from Reactant to a peak labelled Activated complex, then falls to Product; $E_1$ marks the drop from the peak down to the product level and $E_2$ the gap between the product and reactant levels, the product level lying above the reactant level]

Arrange the following compounds in increasing order of their dipole moment : $\mathrm{HBr}$, $\mathrm{H_2S}$, $\mathrm{NF_3}$ and $\mathrm{CHCl_3}$

Given below are two statements :

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 :

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

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

(A) $\mathrm{[Fe(CN)_5NO]^{2-}}$ (B) $\mathrm{[CoF_6]^{3-}}$ (C) $\mathrm{[Fe(CN)_6]^{4-}}$ (D) $\mathrm{[Co(NH_3)_6]^{3+}}$ (E) $\mathrm{[Cr(H_2O)_6]^{2+}}$

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 :

Given below are two statements :

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 :

When sec-butyl cyclohexane reacts with bromine in the presence of sunlight, the major product is :

The most stable carbocation from the following is :

The alkane from below having two secondary hydrogens is :

$\mathrm{RBr} \xrightarrow{\text{(i) Mg, dry ether ; (ii) } \mathrm{H_2O}}$ 2-Methylbutane

[Figure: skeletal structure of 2-methylbutane drawn as the product]

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

Toluene (excess) $\xrightarrow{\text{(i) } \mathrm{CrO_2Cl_2,\ CS_2}\ \text{(ii) } \mathrm{H_3O^+}\ \text{(iii) } \mathrm{NaHSO_3}}$ Filter $\longrightarrow$ Residue (A)

Residue (A) $+$ HCl(dil) $\rightarrow$ Compound (B)

Structure of residue (A) and compound (B) formed respectively is :

[A] and [B]

Match the compounds (List-I) with the appropriate catalyst/reagents (List-II) for their reduction into corresponding amines :

List-I (Compound)	List-II (Catalyst/Reagent)
(A) $\mathrm{R{-}\overset{\displaystyle O}{\overset{\|}{C}}{-}NH_2}$	(I) NaOH (aqueous)
(B) [Structure: nitrobenzene, $\mathrm{C_6H_5{-}NO_2}$]	(II) $\mathrm{H_2/Ni}$
(C) $\mathrm{R{-}C{\equiv}N}$	(III) $\mathrm{LiAlH_4,\ H_2O}$
(D) [Structure: N-alkyl phthalimide, N–R]	(IV) $\mathrm{Sn,\ HCl}$

Choose the correct answer from the options given below :

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

[Figure: four Fischer projections — (A) CHO at top with $-$OH on the right at all three carbons below; (B) CHO at top, then $-$OH (right), HO$-$ (left), $-$OH (right), $-$OH (right); (C) CHO at top, then $-$OH (right), HO$-$ (left), HO$-$ (left), $-$OH (right); (D) drawn inverted with CHO at the bottom and, from top, HO$-$ (left), HO$-$ (left), $-$OH (right), HO$-$ (left)]

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

Consider the following cases of standard enthalpy of reaction $\left(\Delta H_r^{\circ}\ \text{in kJ mol}^{-1}\right)$

$$\mathrm{C_2H_6(g)} + \frac{7}{2}\mathrm{O_2(g)} \rightarrow 2\mathrm{CO_2(g)} + 3\mathrm{H_2O(l)} \quad \Delta H_1^{\circ} = -1550$$

$$\mathrm{C(graphite)} + \mathrm{O_2(g)} \rightarrow \mathrm{CO_2(g)} \quad \Delta H_2^{\circ} = -393.5$$

$$\mathrm{H_2(g)} + \frac{1}{2}\mathrm{O_2(g)} \rightarrow \mathrm{H_2O(l)} \quad \Delta H_3^{\circ} = -286$$

The magnitude of $\Delta H^{\circ}_{f\,\mathrm{C_2H_6(g)}}$ is __________ kJ mol$^{-1}$ (Nearest integer).

The number of electrons present in the $4d$ orbitals of Niobium (Nb) and Ruthenium (Ru) are "$x$" and "$y$" respectively. The value of $x + y$ is __________.

The compound with molecular formula $\mathrm{C_6H_6}$, which gives only one monobromo derivative and takes up four moles of hydrogen per mole for complete hydrogenation has __________ $\pi$ electrons.

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