JEE Main 2026 January 21, Shift 2 Question Paper with Solutions

A large drum having radius R is spinning around its axis with angular velocity $\omega$ as shown in the figure. The minimum value of $\omega$ so that body of mass M remains stuck to the inner wall of the drum taking the coefficient of friction between drum surface and mass M as $\mu$ is

As shown in the diagram, when the incident ray is parallel to base of the prism the emergent ray grazes along the second surface. If refractive index of the material of prism is $\sqrt{2}$, the angle $\theta$ of prism is

Keeping significant figures in view physical quantities 52.01 m, 153.2 m and 0.123 m is

The r.m.s. speed of oxygen molecules at $47^{\circ }\text{C}$ is equal to that of hydrogen molecules kept at ______ ${}^{\circ }\text{C}$. (mass of oxygen molecule/mass of hydrogen molecule = 32/2)

A capacitor C is first charged fully with potential difference of $V_0$ and disconnected from battery. The charged capacitor is connected across an inductor having inductance L. In t sec 25% of initial energy in the capacitor is transferred to inductor. The value of t is _______ sec.

Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centres have an initial separation of 4 R. Both spheres are given and initial speed u towards each other. The minimum value of u, so that they can just touch each other is (Take $k=\frac{1}{4\pi {\in }_0}$ and assume $k{Q}^2>G{m}^2$ where G is the gravitational constant)

Two cars A and B each of mass $10^{3}kg$ are moving on parallel tracks separate by distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is _______ Js.

An infinitely long straight wire carrying current $I$ is bent in a planer shape as shown in the diagram. The radius of the circular part is $r$. The magnetic field at the centre $O$ of the circular loop is

The charge stored by the capacitor C in the given circuit in the steady state is ______ $\mu C$.

The kinetic energy of simple harmonic oscillator is oscillating with angular frequency of 176 rad/sec. The frequency of the simple harmonic oscillator is ______Hz. $(\pi =22/7)$

A river of width 200m flowing from west to east with a speed 18 km/h. A boat moving with speed of 36 km/h in still water, is made to travel 1 round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also displacement along the river bank are ____ and _____ respectively.

A spherical body of radius r and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $v_0$. Estimated maximum error in the quantity $\eta$ is (ignore errors associated with $\sigma ,\rho$ and g, gravitational acceleration)

Two known resistance of $R\Omega$ and $2R\Omega$ and one unknown resistance $X\Omega$ are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is $X\Omega$, then value of X is ______ $\Omega$.

The pulley shown in the figure is made using a thin rim and two rods of equal length to the diameter of rim. The rim and each rod have a mass of M. Two blocks of mass M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the block is _____ (Assume no slipping of string on pulley)

Given below are two statements:
Statement 1: In Young's double slit experiment, the angular separation of fringes will increase as the screen is moved away from the plane of the slits.
Statement II: In a Young's double slit experiment, the angular separation of fringes will increase when monochromatic source is replaced by another monochromatic source of higher wavelength.
In the light of the above statements, choose the correct answer from the options given below:

A battery with EMF $E$ and internal resistance $r$ is connected across a resistance $R$. The power consumption in $R$ will be maximum when:

A body of mass 2 kg is moving along x direction such that its displacement as function of time is given by $x(t)=a{t}^2+\beta t+\gamma m,$ where $\alpha =1m/s^2$, $\beta =1m/s$ and $\gamma =1m$. The work done on the body during the time interval $t=2s$ to $t=3s$, is _______ J.

The total length of potentiometer wire AB is $50cm$ in the arrangement as shown in the figure. If P is the point where the galvanometer shows Zero reading then length of AP is ______ cm.

Surface tension of two liquids (having same densities). $T_1$ and $T_2$ are measured using capillary rise method utilising two tubes of inner radius $r_1$ and $r_2$ where $r_1>r_2$. The measure liquid heights in these tubes are $h_1$ and $h_2$ respectively. (ignore weight of liquid about the lowest point of meniscus). The heights $h_1$ and $h_2$ and surface tension $T_1$ and $T_2$ satisfy the relation.

The energy of an electron in an orbit of Bohr's atom is $-0.04E_{0}eV$ where $E_0$ is the ground state energy. If L is the angular momentum of electron in this orbit and h is planks constant then $\frac{2\pi L}{h}$ is ______.

A diatomic gas $(\gamma =1.4)$ does $100J$ of work when it is expanded isobarically. Then the heat given to the gas is ________ J.

A particle having electric charge $3\times 10^{-19}$ C and mass $6\times 10^{-27}$ kg is accelerated by applying an electric potential of $1.21V$. Wave length of the matter wave associated with the particle is $\alpha \times 10^{-12}m$. The value of $\alpha$ is ______. (Take Planck's constant $=6.6\times 10^{-34}J.s$)

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid 20 cm/s. The terminal velocity of another ball of same material and having radius 3mm in same fluid will be ______ cm/sec.

In a Young's double slit experiment setup, the two slits are kept 0.4 mm apart and screen is placed at 1m from slits. If a thin transparent sheet of thickness $20\mu m$ is introduced in front of one of the slits then center bright fringe shifts by 20 mm on the screen. The refractive index of transparent sheet is given by $\frac{\alpha }{10}$, where $\alpha$ is ______.

An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, $\sigma =10\text{mho}/m$. The ratio of maximum conduction current density to maximum displacement current density is _______.
$\left[\text{Take}\frac{1}{4\pi {\in }_0}=9\times 10^{9}N{m}^2/C^2\right]$

Aqueous $\text{HCl}$ reacts with ${\text{MnO}}_2(s)$ to form ${\text{MnCl}}_2(aq),{\text{Cl}}_2(g)$ and ${\text{H}}_2\text{O}(l)$. What is the weight (in g) of ${\text{Cl}}_2$ liberated when 8.7 g of ${\text{MnO}}_2(s)$ reacted with excess aqueous $\text{HCl}$ solution? (Given Molar mass in $g{\text{mol}}^{-1}$ $\text{Mn}=55,\text{Cl}=35.5,\text{O}=16,\text{H}=1$)

Consider the following data: ${\Delta }_f{H}^{\ominus }$ (methane, g) $=-XkJ{\text{mol}}^{-1}$. Enthalpy of sublimation of graphite $=YkJ{\text{mol}}^{-1}$. Dissociation enthalpy of ${\text{H}}_2=ZkJmo{l}^{-1}$. The bond enthalpy of $\text{C}-\text{H}$ bond is given by

Consider the following spectral lines for atomic hydrogen: A) First line of Paschen series B) Second line of Balmer series C) Third line of Paschen series D) Fourth line of Bracket series. The correct arrangement of the above lines in ascending order of energy is:

For a closed circuit Daniell cell, which of the following plots is the accurate one at a given temperature ?

Decomposition of A is a first order reaction at T(K) and is given by $\text{A}(g)\to \text{B}(g)+\text{C}(g)$. In a closed 1 L vessel, 1 bar A(g) is allowed to decompose at T(K). After 100 minutes, the total pressure was 1.5 bar. What is the rate constant $(in{\min }^{-1})$ of the reaction ? $(\log 2=0.3)$

Given below are two statements
Statement -I: The correct order in terms of atomic/ ionic radii is $\text{Al}>\text{Mg}>{\text{Mg}}^{2+}>{\text{Al}}^{3+}$
Statement -II : The correct order in terms of the magnitude of electron gain enthalpy is $\text{Cl}>\text{Br}>\text{S}>\text{O}$. In the light of the above statements, choose the correct answer from the options given below:

Given below are two statements
Statement I : The correct order in terms of bond dissociation energy order is : ${\text{Cl}}_2>{\text{Br}}_2>{\text{F}}_2>{\text{I}}_2$
Statement II: The correct trend in the covalent character of the metal halides is $[{\text{SnCl}}_4>{\text{SnCl}}_2]$, $[{\text{PbCl}}_4>{\text{PbCl}}_2]$ and $[{\text{UF}}_4>{\text{UF}}_6]$. In the light of the above statements , choose the correct answer from the options given below :

On heating a mixture of common salt and ${\text{K}}_2{\text{Cr}}_2{\text{O}}_7$ in equal amount along with concentrated ${\text{H}}_2{\text{SO}}_4$ in a test tube, a gas is evolved . Formula of the gas evolved and oxidation state of the central metal atom in the gas respectively are :

The correct increasing order of $\text{C}-\text{H}(A),\text{C}-\text{O}(B),\text{C}=\text{O}(C)\&\text{C}\equiv \text{N}(D)$ bonds in terms of covalent bond length is :

Given below are some of the statements about Mn and ${\text{Mn}}_2{\text{O}}_7$. Identify the correct statements.
A) Mn forms the oxide ${\text{Mn}}_2{\text{O}}_7$, in which Mn is in its highest oxidation state.
B) Oxygen stabilizes the Mn in higher oxidation states by forming multiple bonds with Mn.
C) ${\text{Mn}}_2{\text{O}}_7$ is an ionic oxide.
D) The structure of ${\text{Mn}}_2{\text{O}}_7$ consists of one bridged oxygen.
Choose the correct answer from the options given below :