JEE Main 2025 Apr 3 Shift 1 — Question Paper

Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let R be a relation on A defined by $x\mathrm{R}y$ if and only if $0 \le x^2 + 2y \le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to

If the domain of the function $f(x) = \log_e\left(\frac{2x-3}{5+4x}\right) + \sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha, \beta)$, then $\alpha^2 + 4\beta$ is equal to

Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to

Let $z \in \mathbb{C}$ be such that $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$. Then the sum of all possible values of $z^2$ is

Let $A$ be a matrix of order $3 \times 3$ and $|A| = 5$. If $\left|2\,adj\left(3A\,adj\left(2A\right)\right)\right| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha + \beta + \gamma$ is equal to

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$, then $24\,(a_1 + a_2 + a_3)$ is equal to

The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ upto 20 terms, is equal to

If $\sum_{r=1}^{9}\left(\frac{r+3}{2^r}\right) \cdot {}^{9}C_r = \alpha\left(\frac{3}{2}\right)^{9} - \beta$, $\alpha, \beta \in \mathbb{N}$, then $(\alpha + \beta)^2$ is equal to

The sum of all rational terms in the expansion of $\left(2 + \sqrt{3}\right)^{8}$ is

The number of solutions of the equation $2x + 3\tan x = \pi$, $x \in [-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$ is:

The radius of the smallest circle which touches the parabolas $y = x^2 + 2$ and $x = y^2 + 2$ is

A line passing through the point $P\left(\sqrt{5}, \sqrt{5}\right)$ intersects the ellipse $\frac{x^2}{36} + \frac{y^2}{25} = 1$ at $A$ and $B$ such that $(PA) \cdot (PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to :

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1 : 2x + y + 6 = 0$ and $\mathrm{L}_2 : 4x + 2y - p = 0$, $p > 0$, at the points A and B, respectively. If $\mathrm{AB} = \frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point A on the line $\mathrm{L}_2$ is M, then $\frac{\mathrm{AM}}{\mathrm{BM}}$ is equal to

Line $\mathrm{L}_1$ passes through the point $(1, 2, 3)$ and is parallel to $z$-axis. Line $\mathrm{L}_2$ passes through the point $(\lambda, 5, 6)$ and is parallel to $y$-axis. Let for $\lambda = \lambda_1, \lambda_2$, $\lambda_2 < \lambda_1$, the shortest distance between the two lines be 3. Then the square of the distance of the point $(\lambda_1, \lambda_2, 7)$ from the line $\mathrm{L}_1$ is

Let a line passing through the point $(4, 1, 0)$ intersect the line $\mathrm{L}_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ at the point $A(\alpha, \beta, \gamma)$ and the line $\mathrm{L}_2 : x - 6 = y = -z + 4$ at the point $B(a, b, c)$. Then $\begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix}$ is equal to

If $y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix}$, $x \in \mathbb{R}$, then $\frac{d^2y}{dx^2} + y$ is equal to

Let $f(x) = \begin{cases} (1+ax)^{1/x} & , \ x < 0 \\ 1+b & , \ x = 0 \\ \dfrac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , \ x > 0 \end{cases}$

be continuous at $x = 0$. Then $e^{a}bc$ is equal to:

Let $f(x) = \int x^3 \sqrt{3 - x^2}\, dx$. If $5f\left(\sqrt{2}\right) = -4$, then $f(1)$ is equal to

Let the domain of the function $f(x) = \log_2 \log_4 \log_6(3 + 4x - x^2)$ be $(a, b)$. If $\int_0^{b-a}\left[x^2\right] dx = p - \sqrt{q} - \sqrt{r}$, $p, q, r \in \mathbb{N}$, $\gcd(p, q, r) = 1$, where $[\cdot]$ is the greatest integer function, then $p + q + r$ is equal to

Let $g$ be a differentiable function such that $\int_0^x g(t)\, dt = x - \int_0^x t\, g(t)\, dt$, $x \ge 0$ and let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x \ g(x)$, $x \in \left[0, \frac{\pi}{2}\right)$. If $y(0) = 0$, then $y\left(\frac{\pi}{3}\right)$ is equal to

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^n$; $m, n \in \{1, 2, 3, \ldots, 9\}$, then $m + n$ is equal to __________

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $\mathrm{W_n}$. Let the probability $\mathrm{P(W_n)}$ of choosing the word $\mathrm{W_n}$ satisfy $\mathrm{P(W_n)} = 2\mathrm{P(W_{n-1})}$, $n > 1$.

If $\mathrm{P(CDBEA)} = \frac{2^{\alpha}}{2^{\beta} - 1}$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to : __________

Let the product of the focal distances of the point $\mathrm{P}\left(4, 2\sqrt{3}\right)$ on the hyperbola $\mathrm{H}: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be 32. Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$. Then $p^2 + q^2$ is equal to __________

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$, $\vec{c} = \lambda\hat{j} + \mu\hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a} \times \hat{d} = \vec{b} \times \hat{d}$ and $\vec{c} \cdot \hat{d} = 1$. If $\vec{c}$ is perpendicular to $\vec{a}$, then $\left|3\lambda\hat{d} + \mu\vec{c}\right|^2$ is equal to __________

The area of the region bounded by the curve $y = \max\{|x|,\ x|x-2|\}$, the $x$-axis and the lines $x = -2$ and $x = 4$ is equal to __________

A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	Gravitational constant	I.	$[\mathrm{LT^{-2}}]$
B.	Gravitational potential energy	II.	$[\mathrm{L^2T^{-2}}]$
C.	Gravitational potential	III.	$[\mathrm{ML^2T^{-2}}]$
D.	Acceleration due to gravity	IV.	$[\mathrm{M^{-1}L^3T^{-2}}]$

Choose the correct answer from the options given below:

A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.

The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as

Which of the following curves possibly represent one-dimensional motion of a particle?

A. [Figure: phase vs time — a straight line through the origin with positive slope]

B. [Figure: velocity vs displacement — a circle centred at the origin]

C. [Figure: velocity vs time — a circle centred at the origin]

D. [Figure: total distance vs time — a non-decreasing curve rising in the first quadrant]

Choose the correct answer from the options given below:

A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is

[Figure: a solid sphere of radius r resting on a rough horizontal surface with a horizontal force F = 49 N applied tangentially at its topmost point]

Consider a completely full cylindrical water tank of height 1.6 m and of cross-sectional area 0.5 $\mathrm{m^2}$. It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load 50 kg is applied at the top surface of the water in the tank then the velocity of the water coming out at the instant when the hole is opened is:

($g = 10 \ \mathrm{m/s^2}$)

During the melting of a slab of ice at 273 K at atmospheric pressure:

A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of 800 $\mathrm{cm^3}$ and temperature 27°C. The change in temperature when the gas is adiabatically compressed to 200 $\mathrm{cm^3}$ is:

(Take $\gamma = 1.5$ ; $\gamma$ is the ratio of specific heats at constant pressure and at constant volume)

[Figure: a closed vertical chamber; a piston (initially at height $L_0$ from the bottom) hangs from a spring attached to the top; after heating the piston rises to height $L_1$]

A piston of mass M is hung from a massless spring whose restoring force law goes as $F = -kx^3$, where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with 'n' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height $L_0$ to $L_1$, the total energy delivered by the filament is :(Assume spring to be in its natural length before heating)

Two blocks of masses m and M, (M > m), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then

($\mu$ = coefficient of friction between the two blocks)

[Figure: block M on a frictionless table connected to a wall by a spring of constant k; smaller block m rests on top of M with friction coefficient $\mu$ between them]

A. The time period of small oscillation of the two blocks is $T = 2\pi\sqrt{\frac{(m+M)}{k}}$

B. The acceleration of the blocks is $a = -\frac{kx}{M+m}$ ($x$ = displacement of the blocks from the mean position)

C. The magnitude of the frictional force on the upper block is $\frac{m\mu|x|}{M+m}$

D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu(M+m)g}{k}$

E. Maximum frictional force can be $\mu(M + m)g$.

Choose the correct answer from the options given below:

A wire of length 25 m and cross-sectional area 5 $\mathrm{mm^2}$ having resistivity of $2 \times 10^{-6} \ \Omega$ m is bent into a complete circle. The resistance between diametrically opposite points will be

A parallel plate capacitor is filled equally(half) with two dielectrics of dielectric constants $\varepsilon_1$ and $\varepsilon_2$, as shown in figures. The distance between the plates is d and area of each plate is A. If capacitance in first configuration and second configuration are $C_1$ and $C_2$ respectively, then $\frac{C_1}{C_2}$ is :

First Configuration

[Figure: parallel plate capacitor with the two dielectric slabs stacked horizontally — $\varepsilon_2$ occupying the upper half and $\varepsilon_1$ the lower half of the gap (dielectrics in series)]

Second Configuration

[Figure: parallel plate capacitor with the two dielectric slabs side by side — $\varepsilon_2$ filling the left half and $\varepsilon_1$ the right half of the gap (dielectrics in parallel)]

The electrostatic potential on the surface of uniformly charged spherical shell of radius R = 10 cm is 120 V. The potential at the centre of shell, at a distance r = 5 cm from centre, and at a distance r = 15 cm from the centre of the shell respectively, are:

The radiation pressure exerted by a 450 W light source on a perfectly reflecting surface placed at 2m away from it, is

Consider following statements for refraction of light through prism, when angle of deviation is minimum.

A. The refracted ray inside prism becomes parallel to the base.

B. Larger angle prisms provide smaller angle of minimum deviation.

C. Angle of incidence and angle of emergence becomes equal.

D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.

E. Angle of refraction becomes double of prism angle.

Choose the correct answer from the options given below:

The radii of curvature for a thin convex lens are 10 cm and 15 cm respectively. The focal length of the lens is 12 cm. The refractive index of the lens material is

The work function of a metal is 3 eV. The color of the visible light that is required to cause emission of photoelectrons is

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\mathrm{^{1}_{0}n + \, ^{235}_{92}U \to \, ^{140}_{54}Xe + \, ^{94}_{38}Sr + 2\,^{1}_{0}n}$	I.	Chemical reaction
B.	$\mathrm{2H_2 + O_2 \to 2H_2O}$	II.	Fusion with +ve Q value
C.	$\mathrm{^{2}_{1}H + \, ^{2}_{1}H \to \, ^{3}_{2}He + \, ^{1}_{0}n}$	III.	Fission
D.	$\mathrm{^{1}_{1}H + \, ^{3}_{1}H \to \, ^{2}_{1}H + \, ^{2}_{1}H}$	IV.	Fusion with -ve Q value

Choose the correct answer from the options given below:

Choose the correct logic circuit for the given truth table having inputs A and B.

Inputs		Output
A	B	Y
0	0	0
0	1	0
1	0	1
1	1	1

Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length $a$. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length $2a$ and also the masses of the spheres are made $2m$, then they will collide after __________ seconds.

A 4.0 cm long straight wire carrying a current of 8A is placed perpendicular to a uniform magnetic field of strength 0.15 T. The magnetic force on the wire is __________ mN.

A loop ABCDA, carrying current $I = 12$ A, is placed in a plane, consists of two semi-circular segments of radius $R_1 = 6\pi$ m and $R_2 = 4\pi$ m. The magnitude of the resultant magnetic field at center O is $k \times 10^{-7}$ T. The value of k is ________.

(Given $\mu_0 = 4\pi \times 10^{-7} \ \mathrm{Tm\,A^{-1}}$)

[Figure: two concentric semicircles above the line A–B–O–C–D; outer semicircle of radius $R_1$ from A to D and inner semicircle of radius $R_2$ from B to C, joined by straight segments A–B and C–D along the diameter, with current I flowing along the loop ABCDA]

In the figure shown below, a resistance of 150.4 $\Omega$ is connected in series to an ammeter A of resistance 240 $\Omega$. A shunt resistance of 10 $\Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.

[Figure: a 20 V battery in series with a 150.4 Ω resistor and an ammeter A; a 10 Ω shunt resistor is connected in parallel with the ammeter]

Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is $x$I. The value of $x$ is __________.

Among $10^{-9}$ g (each) of the following elements, which one will have the highest number of atoms?

Element: Pb, Po, Pr and Pt

Which of the following postulate of Bohr's model of hydrogen atom is not in agreement with quantum mechanical model of an atom?

Which of the following properties will change when system containing solution 1 will become solution 2?

[Figure: a box labelled “10 mol of solute x + 10 L of water” (Solution 1) with an arrow to a box labelled “1 L of solution 1 + 1 mol of solute x + 1 L of water” (Solution 2)]

2 moles each of ethylene glycol and glucose are dissolved in 500 g of water. The boiling point of the resulting solution is:

(Given : Ebullioscopic constant of water $= 0.52~\mathrm{K~kg~mol^{-1}}$)

In the following system, $\mathrm{PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)}$ at equilibrium, upon addition of xenon gas at constant T & p, the concentration of

Correct order of limiting molar conductivity for cations in water at 298 K is :

Given below are two statements:

Statement I : A catalyst cannot alter the equilibrium constant ($K_c$) of the reaction, temperature remaining constant.

Statement II : A homogenous catalyst can change the equilibrium composition of a system, temperature remaining constant.

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

In a reaction A + B $\rightarrow$ C, initial concentrations of A and B are related as $[A]_0 = 8[B]_0$. The half lives of A and B are 10 min and 40 min, respectively. If they start to disappear at the same time, both following first order kinetics, after how much time will the concentration of both the reactants be same ?

Match the LIST-I with LIST-II

	LIST-I (Molecules/ ion)		LIST-II (Hybridisation of central atom)
A.	$\mathrm{PF_5}$	I.	$dsp^2$
B.	$\mathrm{SF_6}$	II.	$sp^3d$
C.	$\mathrm{Ni(CO)_4}$	III.	$sp^3d^2$
D.	$\mathrm{[PtCl_4]^{2-}}$	IV.	$sp^3$

Choose the correct answer from the options given below:

Which of the following statements are correct ?

A. The process of adding an electron to a neutral gaseous atom is always exothermic. B. The process of removing an electron from an isolated gaseous atom is always endothermic. C. The 1$^{st}$ ionization energy of boron is less than that of beryllium. D. The electronegativity of C is 2.5 in $\mathrm{CH_4}$ and $\mathrm{CCl_4}$ E. Li is the most electropositive among elements of group I.

Choose the correct answer from the options given below:

Given below are two statements:

Statement I : The N - N single bond is weaker and longer than that of P - P single bond.

Statement II : Compounds of group 15 elements in + 3 oxidation states readily undergo disproportionation reactions.

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

The metal ions that have the calculated spin-only magnetic moment value of 4.9 B.M. are :

A. $\mathrm{Cr^{2+}}$ B. $\mathrm{Fe^{2+}}$ C. $\mathrm{Fe^{3+}}$ D. $\mathrm{Co^{2+}}$ E. $\mathrm{Mn^{3+}}$

Choose the correct answer from the options given below:

The correct order of the complexes $\mathrm{[Co(NH_3)_5(H_2O)]^{3+}}$ (A), $\mathrm{[Co(NH_3)_6]^{3+}}$ (B), $\mathrm{[Co(CN)_6]^{3-}}$ (C) and $\mathrm{[CoCl(NH_3)_5]^{2+}}$ (D) in terms of wavelength of light absorbed is

Identify the correct statements from the following.

[Figure: statements A–D show pairs of skeletal structures]

A. $\mathrm{CH_3CH_2COCH_2CH_3}$ (pentan-3-one) and $\mathrm{CH_3COCH_2CH_2CH_3}$ (pentan-2-one) are metamers B. $\mathrm{CH_3CH_2CH_2{-}CN}$ and $\mathrm{CH_3CH_2CH_2{-}NC}$ are functional isomers C. $\mathrm{CH_3CH_2CH_2CH_2CH_2OH}$ (pentan-1-ol) and $\mathrm{CH_3CH(OH)CH_2CH_2CH_3}$ (pentan-2-ol) are position isomers D. $\mathrm{CH_3CH_2CH_2CH_2NH_2}$ (butan-1-amine) and $\mathrm{CH_3CH_2{-}NH{-}CH_2CH_2CH_3}$ (N-ethylpropan-1-amine) are homologous

Choose the correct answer from the options given below:

Which compound would give 3-methyl-6-oxoheptanal upon ozonolysis?

Identify [A], [B] and [C], respectively in the following reaction sequence:

$$[\mathrm{A}] \xrightarrow[273-278~\mathrm{K}]{\mathrm{NaNO_2,~HCl}} \mathrm{C_6H_5\overset{+}{N_2}Cl^-} \xrightarrow{\mathrm{KI}} [\mathrm{B}] \xrightarrow[\text{Dry ether}]{2\mathrm{Na}} [\mathrm{C}]$$

[Figure: the intermediate is drawn as benzenediazonium chloride]

The least acidic compound, among the following is:

[Figure: (A) 3-ethoxyphenol (benzene ring with OEt and OH meta to each other); (B) benzenesulfonic acid ($\mathrm{C_6H_5SO_3H}$); (C) butanoic acid ($\mathrm{CH_3CH_2CH_2COOH}$); (D) ethyl propiolate ($\mathrm{EtO_2C{-}C{\equiv}C{-}H}$)]

Number of molecules from below which cannot give iodoform reaction is:

Ethanol, Isopropyl alcohol, Bromoacetone, 2−Butanol, 2−Butanone, Butanal, 2−Pentanone, 3−Pentanone, Pentanal and 3−Pentanol.

In the following reactions, which one is NOT correct?

[Figure: each option shows benzenediazonium chloride ($\mathrm{C_6H_5\overset{+}{N_2}Cl^-}$) converted to a product]

Which of the following is the correct structure of L−Fructose?

Given :

$\Delta H^{\ominus}_{\mathrm{sub}}\,[\mathrm{C\,(graphite)}] = 710~\mathrm{kJ~mol^{-1}}$

$\Delta_{\mathrm{C-H}}H^{\ominus} = 414~\mathrm{kJ~mol^{-1}}$

$\Delta_{\mathrm{H-H}}H^{\ominus} = 436~\mathrm{kJ~mol^{-1}}$

$\Delta_{\mathrm{C=C}}H^{\ominus} = 611~\mathrm{kJ~mol^{-1}}$

The $\Delta H_f^{\ominus}$ for $\mathrm{CH_2=CH_2}$ is __________ $\mathrm{kJ~mol^{-1}}$ (nearest integer value)

The number of optical isomers exhibited by the iron complex (A) obtained from the following reaction is __________.

$\mathrm{FeCl_3 + KOH + H_2C_2O_4 \rightarrow A}$

Consider the following reactions

$\underset{\text{Little amount}}{\mathrm{A}} + \mathrm{NaCl} + \mathrm{H_2SO_4} \rightarrow \mathrm{CrO_2Cl_2} + \text{Side Products}$

$\mathrm{CrO_2Cl_2}_{\,\text{(Vapour)}} + \mathrm{NaOH} \rightarrow \mathrm{B} + \mathrm{NaCl} + \mathrm{H_2O}$

$\mathrm{B} + \mathrm{H^+} \rightarrow \mathrm{C} + \mathrm{H_2O}$

The number of terminal 'O' present in the compound 'C' is __________.

0.5 g of an organic compound on combustion gave 1.46 g of $\mathrm{CO_2}$ and 0.9 g of $\mathrm{H_2O}$. The percentage of carbon in the compound is __________. (Nearest integer)

[Given : Molar mass (in g mol$^{-1}$) C : 12, H : 1, O : 16]

During estimation of nitrogen by Dumas' method of compound X (0.42 g)

[Figure: structure of X — piperazine, a six-membered saturated ring with N–H groups at positions 1 and 4 ($\mathrm{C_4H_{10}N_2}$)]

__________ mL of $\mathrm{N_2}$ gas will be liberated at STP. (nearest integer)

(Given molar mass in g mol$^{-1}$ : C : 12, H : 1, N : 14)