JEE Main 2025 · 22 Jan 2025 · Shift 2

Q1.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 4D then the power of a part of the divided lens is

Q2.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 …

Q3.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 $(10.00\pm 0.05)\mathrm{cm}]$

Q4.A series LCR circuit is connected to an alternating source of emf E. The current amplitude at resonant frequency is ${\mathrm{I}}_0$. If the value of resistance R becomes twice of its initial value then amplitude of current at resonance will be

Q5.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 ${\mathrm{V}}_0$ and ${\mathrm{E}}_0$, respectively, then the correct combination of the electric potential and electric fie …

Q6.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,

Q7.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 :

Q8.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 correc …

Q9.A rectangular metallic loop is moving out of a uniform magnetic field region to a field free region with a constant speed. When the loop is partially inside the magnate field, the plot of magnitude of induced emf ( $\epsilon$ ) with time ( $t$ ) is given by

Q10.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 …

Q11.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 …

Q12.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 $(\hat{i}+\hat{j}+\hat{k})$, would be

Q13.To obtain the given truth table, following logic gate should be placed at G :

Q14.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.

Q15.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 d …

Q16.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 :

Q17.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 nm . [Assume that the light is incident nearly perpendicul …

Q18.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 :

Q19.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 v …

Q20.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 $10\mathrm{m}/\mathrm{s}$. The ratio of its kinetic energies at point B and C is :
(Take acceleration due to gravity as …

Q21.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 …

Q22.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 …

Q23.A parallel plate capacitor of area $\mathrm{A}=16{\mathrm{cm}}^2$ and separation between the plates 10 cm , is charged by a DC current. Consider a hypothetical plane surface of area ${\mathrm{A}}_0=3.2{\mathrm{cm}}^2$ inside the capacitor and parallel to the plates. At an ins …

Q24.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 …

Q25.The net current flowing in the given circuit is
______ A.

Q1.Arrange the following compounds in increasing order of their dipole moment :
$\mathrm{HBr}$, ${\mathrm{H}}_2\mathrm{S}$, ${\mathrm{NF}}_3$ and ${\mathrm{CHCl}}_3$

Q2.Identify the number of structure/s from the following which can be correlated to D-glyceraldehyde.

Q3.The maximum covalency of a non-metallic group 15 element 'E' with weakest $\mathrm{E}-\mathrm{E}$ bond is :

Q4.Consider the given figure and choose the correct option :

Q5.When sec-butylcyclohexane reacts with bromine in the presence of sunlight, the major product is :

Q6.The species which does not undergo disproportionation reaction is :

Q7.Match the Compounds (List-I) with the appropriate Catalyst/Reagents (List-II) for their reduction into corresponding amines.

List-I (Compounds):
(A) (see structure)
(B) (see structure)
(C) $\mathrm{R}-\mathrm{C}\equiv \mathrm{N}$
(D) (see structure)

List-II (Catalyst/Reagents) …

Q8.The maximum number of RBr producing 2-methylbutane by above sequence of reactions is ______. (Consider the structural isomers only)

Q9.Match List-I with List-II.

List-I (Partial Derivatives):
(A) ${\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)}_{\mathrm{P}}$
(B) ${\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)}_{\mathrm{P}}$
(C) …

Q10.The correct order of the following complexes in terms of their crystal field stabilization energies is :

Q11.Density of 3 M NaCl solution is $1.25\mathrm{g}/\mathrm{mL}$. The molality of the solution is :

Q12.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 :

Q13.The most stable carbocation from the following is :

Q14.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 t …

Q15.Given below are two statements :

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

Statement (II): $2{\mathrm{p}}_{\mathrm{x}}$ and $2{\mathrm{p}}_{\mathrm{y}}$ are degenerate orbitals.

In th …

Q16.Given below are two statement :

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 compou …

Q17.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) …

Q18.Residue (A) +HCl (dil.) → Compound (B)

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

Q19.Given below are two statements :

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 st …

Q20.The alkane from below having two secondary hydrogens is :

Q21.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.

Q22.Niobium ( Nb ) and ruthenium ( Ru ) have " x " and "y" number of electrons in their respective 4 d orbitals. The value of $\mathrm{x}+\mathrm{y}$ is ____

Q23.The complex of ${\mathrm{Ni}}^{2+}$ ion and dimethyl glyoxime contains ____ number of Hydrogen (H) atoms.

Q24.Consider the following cases of standard enthalpy of reaction ( $\Delta {\mathrm{H}}_{\mathrm{r}}^{\mathrm{o}}$ in $\mathrm{kJ}{\mathrm{mol}}^{-1}$ ) …

Q25.20 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).

Q1.Let $\alpha ,\beta ,\gamma$ and $\delta$ be the coefficients of ${\mathrm{x}}^7,{\mathrm{x}}^5,{\mathrm{x}}^3$ and $x$ respectively in the expansion of ${\left(\mathrm{x}+\sqrt{{\mathrm{x}}^3-1}\right)}^5+{\left(\mathrm{x}-\sqrt{{\mathrm{x}}^3-1}\right)}^5,\mathrm{x}>1$. If u …

Q2.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 …

Q3.Let $\mathrm{P}(4,4\sqrt{3})$ be a point on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ 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 …

Q4.For a $3\times 3$ matrix M , let trace $(\mathrm{M})$ 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 $B=\mathrm{adj}(\mathrm{adj}(2A))$, then the value of $|B|+$ t …

Q5.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

Q6.Let a line pass through two distinct points $\mathrm{P}(-2,-1,3)$ and Q , and be parallel to the vector $3\hat{i}+2\hat{j}+2\hat{k}$. If the distance of the point $Q$ from the point $\mathrm{R}(1,3,3)$ is 5 , then the square of the area of $△\mathrm{PQR}$ is equal to:

Q7.If ${\lim }_{\mathrm{x}\to \infty }{\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{\mathrm{x}}{1+\mathrm{x}}\right)\right)}^{\mathrm{x}}=\alpha$, then the value of $\frac{{\log }_{\mathrm{e}}\alpha }{1+{\log }_{\mathrm{e}}\alpha }$ equals :

Q8.Let $f(x)={\int }_0^{x^2}\frac{t^2-8t+15}{e^t}dt,x\in \mathbf{R}$. Then the numbers of local maximum and local minimum points of f, respectively, are :

Q9.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:

Q10.If $\mathrm{x}=\mathrm{f}(\mathrm{y})$ is the solution of the differential equation
$\left(1+{\mathrm{y}}^2\right)+\left(\mathrm{x}-2{\mathrm{e}}^{{\tan }^{-1}\mathrm{y}}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=0,\mathrm{y}\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
with …

Q11.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 con …

Q12.Let ${\alpha }_{\theta }$ and ${\beta }_{\theta }$ be the distinct roots of $2{\mathrm{x}}^2+(\cos \theta )\mathrm{x}-1=0,\theta \in (0,2\pi )$. 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 :

Q13.Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $\mathrm{f}:\mathrm{A}\to \mathrm{B}$ such that $1\in \mathrm{f}(\mathrm{A})$ is equal to :

Q14.If the system of linear equations :
$x+y+2z=6$,
$2\mathrm{x}+3\mathrm{y}+\mathrm{az}=\mathrm{a}+1$,
$-\mathrm{x}-3\mathrm{y}+\mathrm{bz}=2\mathrm{b}$,
where $a,b\in \mathbf{R}$, has infinitely many solutions, then $7\mathrm{a}+3\mathrm{b}$ is equal to :

Q15.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 :

Q16.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 …

Q17.If A and B are two events such that $\mathrm{P}(\mathrm{A}\cap \mathrm{B})=0.1$, and $\mathrm{P}(\mathrm{A}\mid \mathrm{B})$ and $\mathrm{P}(\mathrm{B}\mid \mathrm{A})$ are the roots of the equation $12{\mathrm{x}}^2-7\mathrm{x}+1=0$, then the value of …

Q18.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

Q19.Let the curve $\mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i})=4,\mathrm{z}\in \mathrm{C}$, divide the region $|z-3|\le 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha -\beta |$ equals :

Q20.The area of the region enclosed by the curves $y=x^2-4x+4$ and $y^2=16-8x$ is :

Q21.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 $6{\int }_{-1/2}^{1/2}f(x)dx=2\pi -\alpha$ then ${\alpha }^2$ is equal to ____.

Q22.Let $\mathrm{A}(6,8),\mathrm{B}(10\cos \alpha ,-10\sin \alpha )$ and $\mathrm{C}(-10\sin \alpha ,10\cos \alpha )$, be the vertices of a triangle. If $\mathrm{L}(\mathrm{a},9)$ and $\mathrm{G}(\mathrm{h},\mathrm{k})$ be its orthocenter and centroid respective …

Q23.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 ___ …

Q24.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
____ .

Q25.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