IIT JEE 2021 Physics Question Paper with Answer and Solution

The smallest division on the main scale of a Vernier calipers is $0.1 \text{ cm}$. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is (in $\text{ cm}$)

An ideal gas undergoes a four-step cycle as shown in the $P-V$ diagram below. During this cycle,in which step is heat absorbed by the gas?

$A$ projectile is thrown from a point $O$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \text{ m/s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,$0.5 \text{ s}$ after the splitting. The other part,$t$ seconds after the splitting,falls to the ground at a distance $x$ meters from the point $O$. The acceleration due to gravity $g = 10 \text{ m/s}^2$.
$(1)$ The value of $t$ is. . . . . .
$(2)$ The value of $x$ is. . . . .

$A$ horizontal force $F$ is applied at the center of mass of a cylindrical object of mass $m$ and radius $R$,perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration $a$. The acceleration due to gravity is $g$. Given that the object rolls without slipping,which of the following statement$(s)$ is(are) correct?
$(A)$ For the same $F$,the value of $a$ does not depend on whether the cylinder is solid or hollow
$(B)$ For a solid cylinder,the maximum possible value of $a$ is $2 \mu g$
$(C)$ The magnitude of the frictional force on the object due to the ground is always $\mu m g$
$(D)$ For a thin-walled hollow cylinder,$a = \frac{F}{2m}$

$A$ particle of mass $M=0.2 \ kg$ is initially at rest in the $xy$-plane at a point $(x=-l, y=-h)$,where $l=10 \ m$ and $h=1 \ m$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10 \ m/s^2$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin,in $SI$ units,are represented by $\vec{L}$ and $\vec{\tau}$,respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x, y$ and $z$-directions,respectively. If $\hat{k}=\hat{i} \times \hat{j}$,then which of the following statement$(s)$ is(are) correct?
$(A)$ The particle arrives at the point $(x=l, y=-h)$ at time $t=2 \ s$.
$(B)$ $\vec{\tau}=2 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$.
$(C)$ $\vec{L}=4 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$.
$(D)$ $\vec{\tau}=\hat{k}$ when the particle passes through the point $(x=0, y=-h)$.

$A$ cylindrical tube,with its base as shown in the figure,is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $\theta=45^{\circ}$. $P_1$ and $P_2$ are pressures at points $1$ and $2$,respectively,located at the base of the tube. Let $\beta=(P_1-P_2) / (\rho g d)$,where $\rho$ is the density of water,$d$ is the inner diameter of the tube,and $g$ is the acceleration due to gravity. Which of the following statement$(s)$ is(are) correct?
$(A)$ $\beta=0$ when $a=g / \sqrt{2}$
$(B)$ $\beta>0$ when $a=g / \sqrt{2}$
$(C)$ $\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a=g / 2$
$(D)$ $\beta=\frac{1}{\sqrt{2}}$ when $a=g / 2$

$A$ thin rod of mass $M$ and length $a$ is free to rotate in a horizontal plane about a fixed vertical axis passing through point $O$. $A$ thin circular disc of mass $M$ and radius $a/4$ is pivoted on this rod with its center at a distance $a/4$ from the free end so that it can rotate freely about its vertical axis,as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left(\frac{Ma^2\Omega}{48}\right) n$. The value of $n$ is. . . . .

$A$ small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at $0 \ K$. At time $t = 0$,the temperature of the object is $200 \ K$. The temperature of the object becomes $100 \ K$ at $t = t_1$ and $50 \ K$ at $t = t_2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $(t_2 / t_1)$ is:

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$,at a height $L$ above the hinge at point $O$. $A$ block of weight $\alpha W$ is attached at the point $P$ of the beam,as shown in the figure. The rope can sustain a maximum tension of $(2 \sqrt{2}) W$. Which of the following statement$(s)$ is(are) correct?

$A$ source,approaching with speed $u$ towards the open end of a stationary pipe of length $L$,is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination$(s)$ of $u$ and $f_s$,will the sound reaching the pipe lead to a resonance?
$(A)$ $u=0.8 v$ and $f_s=f_0$
$(B)$ $u=0.8 v$ and $f_s=2 f_0$
$(C)$ $u=0.8 v$ and $f_s=0.5 f_0$
$(D)$ $u=0.5 v$ and $f_s=1.5 f_0$

$A$ soft plastic bottle,filled with water of density $1 \text{ g/cc}$,contains an inverted glass test-tube with some air (ideal gas) trapped inside,as shown in the figure. The test-tube has a mass of $5 \text{ g}$,and it is made of thick glass with a density of $2.5 \text{ g/cc}$. Initially,the bottle is sealed at atmospheric pressure $P_0 = 10^5 \text{ Pa}$,such that the volume of the trapped air is $V_0 = 3.3 \text{ cc}$. When the bottle is squeezed from the outside at a constant temperature,the pressure inside increases and the volume of the trapped air decreases. It is observed that the test-tube begins to sink at a pressure $P_0 + \Delta P$ without changing its orientation. At this pressure,the volume of the trapped air is $V_0 - \Delta V$.
Let $\Delta V = X \text{ cc}$ and $\Delta P = Y \times 10^3 \text{ Pa}$.
$(1)$ The value of $X$ is
$(2)$ The value of $Y$ is

$A$ pendulum consists of a bob of mass $m=0.1 \ kg$ and a massless inextensible string of length $L=1.0 \ m$. It is suspended from a fixed point at height $H=0.9 \ m$ above a frictionless horizontal floor. Initially,the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. $A$ horizontal impulse $P=0.2 \ kg \cdot m/s$ is imparted to the bob at some instant. After the bob slides for some distance,the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J \ kg \cdot m^2/s$. The kinetic energy of the pendulum just after the lift-off is $K$ Joules. $(1)$ The value of $J$ is. . . . . . $(2)$ The value of $K$ is. . . . . Give the answers of the questions $(1)$ and $(2)$.

$A$ thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle,as shown in the figure below. On each side of the partition,there is one mole of an ideal gas,with specific heat at constant volume,$C_v = 2R$. Here,$R$ is the gas constant. Initially,each side has a volume $V_0$ and temperature $T_0$. The left side has an electric heater,which is turned on at very low power to transfer heat $Q$ to the gas on the left side. As a result,the partition moves slowly towards the right,reducing the right side volume to $V_0 / 2$. Consequently,the gas temperatures on the left and the right sides become $T_L$ and $T_R$,respectively. Ignore the changes in the temperatures of the cylinder,heater,and the partition.
$(1)$ The value of $\frac{T_R}{T_0}$ is
$(A)$ $\sqrt{2}$ $(B)$ $\sqrt{3}$ $(C)$ $2$ $(D)$ $3$
$(2)$ The value of $\frac{Q}{RT_0}$ is
$(A)$ $4(2\sqrt{2}+1)$ $(B)$ $4(2\sqrt{2}-1)$ $(C)$ $(5\sqrt{2}+1)$ $(D)$ $(5\sqrt{2}-1)$

The distance between two stars of masses $3 M_S$ and $6 M_S$ is $9 R$. Here $R$ is the mean distance between the centers of the Earth and the Sun,and $M_S$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period $n T$,where $T$ is the period of Earth's revolution around the Sun. The value of $n$ is. . . . .

An extended object is placed at point $O$,$10 \ cm$ in front of a convex lens $L_1$ and a concave lens $L_2$ is placed $10 \ cm$ behind it,as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are $20 \ cm$. The refractive index of both the lenses is $1.5$. The total magnification of this lens system is

$A$ heavy nucleus $Q$ of half-life $20 \text{ minutes}$ undergoes alpha-decay with a probability of $60 \%$ and beta-decay with a probability of $40 \%$. Initially,the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

In the circuit shown below,the switch $S$ is connected to position $P$ for a long time so that the charge on the capacitor becomes $q_1 \mu C$. Then $S$ is switched to position $Q$. After a long time,the charge on the capacitor is $q_2 \mu C$.
$(1)$ The magnitude of $q_1$ is
$(2)$ The magnitude of $q_2$ is
Give the answer of question $(1)$ and $(2)$

Two point charges $-Q$ and $+Q / \sqrt{3}$ are placed in the $xy$-plane at the origin $(0,0)$ and a point $(2,0)$,respectively,as shown in the figure. This results in an equipotential circle of radius $R$ and potential $V = 0$ in the $xy$-plane with its center at $(b, 0)$. All lengths are measured in meters.
$(1)$ The value of $R$ is. . . . meter.
$(2)$ The value of $b$ is. . . . . .meter.

$A$ wide slab consisting of two media of refractive indices $n_1$ and $n_2$ is placed in air as shown in the figure. $A$ ray of light is incident from medium $n_1$ to $n_2$ at an angle $\theta$,where $\sin \theta$ is slightly larger than $1/n_1$. Take the refractive index of air as $1$. Which of the following statement$(s)$ is(are) correct?
$(A)$ The light ray enters air if $n_2 = n_1$
$(B)$ The light ray is finally reflected back into the medium of refractive index $n_1$ if $n_2 < n_1$
$(C)$ The light ray is finally reflected back into the medium of refractive index $n_1$ if $n_2 > n_1$
$(D)$ The light ray is reflected back into the medium of refractive index $n_1$ if $n_2 = 1$

Which of the following statement$(s)$ is(are) correct about the spectrum of hydrogen atom?
$(A)$ The ratio of the longest wavelength to the shortest wavelength in Balmer series is $9/5$.
$(B)$ There is an overlap between the wavelength ranges of Balmer and Paschen series.
$(C)$ The wavelengths of Lyman series are given by $\lambda = \frac{\lambda_0}{1 - 1/m^2}$,where $\lambda_0$ is the shortest wavelength of Lyman series and $m$ is an integer.
$(D)$ The wavelength ranges of Lyman and Balmer series do not overlap.

$A$ long straight wire carries a current,$I = 2 \text{ A}$. $A$ semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire,the rod,and the rails lie in the same horizontal plane,as shown in the figure. Two ends of the semi-circular rod are at distances $1 \text{ cm}$ and $4 \text{ cm}$ from the wire. At time $t = 0$,the rod starts moving on the rails with a speed $v = 3.0 \text{ m/s}$. $A$ resistor $R = 1.4 \text{ } \Omega$ and a capacitor $C_0 = 5.0 \text{ } \mu\text{F}$ are connected in series between the rails. At time $t = 0$,$C_0$ is uncharged. Which of the following statement$(s)$ is(are) correct? $\left[\mu_0 = 4\pi \times 10^{-7} \text{ SI units}, \ln 2 = 0.7\right]$
$(A)$ Maximum current through $R$ is $1.2 \times 10^{-6} \text{ A}$
$(B)$ Maximum current through $R$ is $3.8 \times 10^{-6} \text{ A}$
$(C)$ Maximum charge on capacitor $C_0$ is $8.4 \times 10^{-12} \text{ C}$
$(D)$ Maximum charge on capacitor $C_0$ is $2.4 \times 10^{-12} \text{ C}$

An $\alpha$-particle (mass $4 \text{ amu}$) and a singly charged sulfur ion (mass $32 \text{ amu}$) are initially at rest. They are accelerated through a potential $V$ and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region,the $\alpha$-particle and the sulfur ion move in circular orbits of radii $r_\alpha$ and $r_s$,respectively. The ratio $(r_s / r_\alpha)$ is:

For a prism of prism angle $\theta=60^{\circ}$,the refractive indices of the left half and the right half are,respectively,$n_1$ and $n_2$ $(n_2 \geq n_1)$ as shown in the figure. The angle of incidence $i$ is chosen such that the incident light rays will have minimum deviation if $n_1=n_2=n=1.5$. For the case of unequal refractive indices,$n_1=n$ and $n_2=n+\Delta n$ (where $\Delta n \ll n$),the angle of emergence $e=i+\Delta e$. Which of the following statement$(s)$ is (are) correct?
$(A)$ The value of $\Delta e$ (in radians) is greater than that of $\Delta n$
$(B)$ $\Delta e$ is proportional to $\Delta n$
$(C)$ $\Delta e$ lies between $2.0$ and $3.0$ milliradians,if $\Delta n=2.8 \times 10^{-3}$
$(D)$ $\Delta e$ lies between $1.0$ and $1.6$ milliradians,if $\Delta n=2.8 \times 10^{-3}$

$A$ physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_0$,where $\vec{E}$ is the electric field,$\vec{B}$ is the magnetic field,and $\mu_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity(ies)?
$(A)$ $\frac{\text{Energy}}{\text{charge} \times \text{current}}$
$(B)$ $\frac{\text{Force}}{\text{Length} \times \text{Time}}$
$(C)$ $\frac{\text{Energy}}{\text{Volume}}$
$(D)$ $\frac{\text{Power}}{\text{Area}}$

$A$ heavy nucleus $N$,at rest,undergoes fission $N \rightarrow P+Q$,where $P$ and $Q$ are two lighter nuclei. Let $\delta=M_N-M_P-M_Q$,where $M_P, M_Q$ and $M_N$ are the masses of $P, Q$ and $N$,respectively. $E_P$ and $E_Q$ are the kinetic energies of $P$ and $Q$,respectively. The speeds of $P$ and $Q$ are $v_P$ and $v_Q$,respectively. If $c$ is the speed of light,which of the following statement$(s)$ is(are) correct?
$(A)$ $E_P+E_Q=c^2 \delta$
$(B)$ $E_P=\left(\frac{M_P}{M_P+M_Q}\right) c^2 \delta$
$(C)$ $\frac{v_P}{v_Q}=\frac{M_Q}{M_P}$
$(D)$ The magnitude of momentum for $P$ as well as $Q$ is $c \sqrt{2 \mu \delta}$,where $\mu=\frac{M_P M_Q}{M_P+M_Q}$

Two concentric circular loops,one of radius $R$ and the other of radius $2R$,lie in the $xy$-plane with the origin as their common center,as shown in the figure. The smaller loop carries current $I_1$ in the anti-clockwise direction and the larger loop carries current $I_2$ in the clockwise direction,with $I_2 > 2I_1$. $\vec{B}(x, y)$ denotes the magnetic field at a point $(x, y)$ in the $xy$-plane. Which of the following statement$(s)$ is(are) correct?
$(A)$ $\vec{B}(x, y)$ is perpendicular to the $xy$-plane at any point in the plane.
$(B)$ $|\vec{B}(x, y)|$ depends on $x$ and $y$ only through the radial distance $r = \sqrt{x^2 + y^2}$.
$(C)$ $|\vec{B}(x, y)|$ is non-zero at all points for $r$.
$(D)$ $\vec{B}(x, y)$ points normally outward from the $xy$-plane for all the points between the two loops.

In a circuit,a metal filament lamp is connected in series with a capacitor of capacitance $C \mu F$ across a $200 V, 50 Hz$ supply. The power consumed by the lamp is $500 W$ while the voltage drop across it is $100 V$. Assume that there is no inductive load in the circuit. Take $rms$ values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $\varphi$.
Assume,$\pi \sqrt{3} \approx 5$.
$(1)$ The value of $C$ is . . . . . .
$(2)$ The value of $\varphi$ is
Give the answers of the questions $(1)$ and $(2)$:

$A$ special metal $S$ conducts electricity without any resistance. $A$ closed wire loop,made of $S$,does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop,of radius $a$,with its center at the origin. $A$ magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r \gg a$ from the center of the loop with its north pole always facing the loop,as shown in the figure.
The magnitude of the magnetic field of a dipole $m$,at a point on its axis at distance $r$,is $\frac{\mu_0}{2 \pi} \frac{m}{r^3}$,where $\mu_0$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments $m_1$ and $m_2$,separated by a distance $r$ on the common axis,with their north poles facing each other,is $\frac{k m_1 m_2}{r^4}$,where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.
$(1)$ When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure),the current induced in the loop will be proportional to
$(A) \frac{m}{r^3} \quad (B) \frac{m^2}{r^2} \quad (C) \frac{m}{r^2} \quad (D) \frac{m^2}{r}$
$(2)$ The work done in bringing the dipole from infinity to a distance $r$ from the center of the loop by the given process is proportional to
$(A) \frac{m}{r^5} \quad (B) \frac{m^2}{r^5} \quad (C) \frac{m^2}{r^6} \quad (D) \frac{m^2}{r^7}$

In order to measure the internal resistance $r_1$ of a cell of emf $E$,a meter bridge of wire resistance $R_0=50 \Omega$,a resistance $R_0/2$,another cell of emf $E/2$ (internal resistance $r$) and a galvanometer $G$ are used in a circuit,as shown in the figure. If the null point is found at $l=72 \text{ cm}$,then the value of $r_1$ is . . . . $\Omega$.

In a photoemission experiment,the maximum kinetic energies of photoelectrons from metals $P, Q$ and $R$ are $E_P, E_Q$ and $E_R$,respectively,and they are related by $E_P = 2E_Q = 2E_R$. In this experiment,the same source of monochromatic light is used for metals $P$ and $Q$,while a different source of monochromatic light is used for metal $R$. The work functions for metals $P, Q$ and $R$ are $4.0 \ eV$,$4.5 \ eV$ and $5.5 \ eV$,respectively. The energy of the incident photon used for metal $R$,in $eV$,is: