JEE Main 2025 Question Paper with Solutions

A body of mass $4 \text{ kg}$ is placed on a plane at a point $P$ with coordinates $(3, 4) \text{ m}$. Under the action of a constant force $\vec{F} = (2\hat{i} + 3\hat{j}) \text{ N}$, the body moves to a new point $Q$ with coordinates $(6, 10) \text{ m}$ in $4 \text{ seconds}$.Determine the ratio of the average power to the instantaneous power at the end of the $4 \text{ s}$ interval.

A balloon and its content having mass M is moving up with an acceleration ‘a’. The mass that must be released from the content so that the balloon starts moving up with an acceleration ‘3a’ will be (Take ‘g’ as acceleration due to gravity)

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

Consider two different diatomic gases. For the first gas, the molecules are rigid and the ratio of specific heats is given by $\gamma_1$. For the second gas, the molecules possess additional vibrational modes and the ratio of specific heats is given by $\gamma_2$. If $C_p$ and $C_v$ are the specific heats at constant pressure and constant volume respectively, which of the following options correctly describes the relationship between $\gamma_1$ and $\gamma_2$?

A tube of length 1 m is filled completely with an ideal liquid of mass 2 M , 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 αM F in SI unit. The value of α is _________.

Given below are two statements : Statement (I) : A spectral line will be observed for a $2 p_x \rightarrow 2 p_y$ transition. Statement (II) : $2 \mathrm{p}_x$ and $2 \mathrm{p}_y$ are degenerate orbitals. 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 most stable carbocation from the following is :

Match List - I with List - II. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List - I (Partial Derivatives) List - II (Thermodynamic Quantity) (A) $\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{P}}$ (I) Cp (B) $\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{P}}$ (II) $-$ S (C) $\left(\frac{\partial \mathrm{G}}{\partial \mathrm{P}}\right)_{\mathrm{T}}$ (III) Cv (D) $\left(\frac{\partial \mathrm{U}}{\partial \mathrm{T}}\right)_{\mathrm{V}}$ (IV) V Choose the correct answer from the options given below :

The species which does not undergo disproportionation reaction is :

Let $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}$ and $\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1$ . 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 $\frac{1}{3}$ , then the sum of the lengths of their latus rectums is equal to :

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 $16(M+m)$ equals :

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

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 :

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

A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of 2×10 5 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×10 4 N/C. The value of x is _________. Take the mass of the proton =1.6×10 −27 kg.

A symmetric thin biconvex lens is cut into four equal parts by two planes AB and CD as shown in figure. If the power of original lens is $4\text{ D}$ then the power of a part of the divided lens is

A transparent film of refractive index, 2.0 is coated on a glass slab of refractive index, 1.45. What is the minimum thickness of transparent film to be coated for the maximum transmission of Green light of wavelength $550\text{ nm}$ . [Assume that the light is incident nearly perpendicular to the glass surface.]

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

A symmetric thin biconvex lens is cut into four equal parts by two planes $A B$ and $C D$ as shown in figure. If the power of original lens is 4D then the power of a part of the divided lens is

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

If the system of linear equations : $\begin{aligned}& x+y+2 z=6 \\& 2 x+3 y+\mathrm{az}=\mathrm{a}+1 \\& -x-3 y+\mathrm{b} z=2 \mathrm{~b}\end{aligned}$ where $a, b \in \mathbf{R}$ , has infinitely many solutions, then $7 a+3 b$ 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 $|A|=\frac{1}{2}$ and trace $(A)=3$ . If $B=\operatorname{adj}(\operatorname{adj}(2 A))$ , then the value of $|B|+$ trace $(B)$ equals :

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

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 $\mathrm{R}(1,3,3)$ is 5 , then the square of the area of $\triangle P Q R$ is equal to :

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

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

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 :

Residue (A) + HCl (dil) $\rightarrow$ Compound (B) Structure of residue (A) and compound (B) formed respectively is :

Consider the given figure and choose the correct option :

A ball of mass 100 g is projected with velocity 20 m/s at 60 ∘ with horizontal. The decrease in kinetic energy of the ball during the motion from point of projection to highest point is