NSEP 2021 Question Paper

Question 1

Consider the process of the melting of a spherical ball of ice originally at 0° Assuming that the heat is being absorbed uniformly through the surface and the rate of absorption is proportional to the instantaneous surface area. Which of the following is true for the radius (r) of the ice ball at any instant of time? Assume that the initial radius of the ice ball at $t=0$ is $r=R_{0}$ and that the shape of the ball always remains spherical during melting. Also assume that L and $\rho$ are respectively the latent heat and density of ice at $0^{\circ}$

(a) radius decreases exponentially with time as $r=R_{0} e^{-\frac{k t}{\rho L}}$. Here k is constant

(b) radius decreases exponentially with time as $r=R_{0} e^{-\frac{k \rho t}{2 L}}$

(d) radius of the ice ball decreases with time linearly with a slope $-\frac{k \rho}{2 L}$

Question 2

The work done by the three moles of an ideal gas in the cyclic process ABCD shown in the diagram is approximately. Given that $$ \begin{aligned} & \mathrm{T}_{1}=100 \mathrm{ K}, \mathrm{ T}_{2}=200 \mathrm{ K} \text { and } & \mathrm{T}_{3}=600 \mathrm{ K}, \mathrm{ T}_{4}=300 \mathrm{ K} \end{aligned} $$

(a) 7.5 kJ

(b) 5.0 kJ

(d) Zero

Question 3

The molar specific heat capacity of a certain gas is expressed as $C=C_{V}+\alpha \frac{P}{T}$. The equation of state for the process can be written as ( $\alpha$ and A are constant)

(a) $\mathrm{PV}=\mathrm{RT}$

(b) $V=\alpha T^{2}$

(d) $T=A e^{\frac{V}{\alpha}}$

Question 4

A metal bar of length $\ell$ moves with a velocity $v$ parallel to an infinitely long straight wire carrying a current I as shown in the figure. If the nearest end of the perpendicular bar always remains at a distance $2 \ell$ from the current carrying wire, the potential difference (in volt) between two ends of the moving bar is

(a) $\frac{\mu_{0} I v}{2 \pi}$

(b) $\frac{\mu_{0} I v}{6 \pi}$

(d) $\frac{\mu_{0} I v}{2 \pi} \ell$

Question 5

Two point charges +Q each are located at $(0,0)$ and $(\mathrm{L}, 0)$ at a distance L apart on the X - axis. The electric field (E) in the region $0<\mathrm{x}<\mathrm{L}$ is best represented by

(a) Fig. a

(b) Fig. b

(d) Fig. d

Question 6

A long straight wire AB of length $\mathrm{L}(\mathrm{L} \gg \mathrm{a}, \mathrm{L} \gg \mathrm{b})$ and resistance R is connected to a time varying source of emf $\mathrm{V}(\mathrm{t})$. The variation of applied emf $\mathrm{V}(\mathrm{t})$ with time is shown in Fig. B. A circular metallic loop of radius $\mathrm{r}=\mathrm{b}$ is placed coplanar with the current carrying wire with its centre at a distance 'a' from the axis of the wire as shown. The induced current in the loop is

(a) clockwise from 0 to T/2 and anticlockwise from T/2 to T

(b) anticlockwise from 0 to T/2 and clockwise from T/2 to T

(d) anticlockwise from 0 to T

Question 7

A simple circuit consists of a known resistance $R_{A}=2 M \Omega$ and an unknown resistance $\mathrm{R}_{\mathrm{B}}$ both in series with a battery of 9 volt and negligible internal resistance. When the voltmeter is connected across the resistance $\mathrm{R}_{\mathrm{A}}$, it measures 3 volt but when the same voltmeter is connected across $\mathrm{R}_{\mathrm{B}}$ it reads 4.5 volt. The voltmeter measures 9 V across the battery. Considering that the voltmeter has a finite resistance r , the correct option is

(a) $R_{B}=3 M \Omega$ and $r=6.0 M \Omega$

(b) $R_{B}=2.5 M \Omega$ and $r=6.0 M \Omega$

(d) $R_{B}=4.5 M \Omega$ and $r=6.0 M \Omega$

Question 8

The optical powers of the objective and the eyepiece of a compound microscope are 100 D and 20 D respectively. The microscope magnification being equal to 50 when the final image is formed at $\mathrm{d}=25 \mathrm{ cm}$ i.e., the least distance of distinct vision. If the separation between the objective and the eyepiece is increased by 2 cm , the magnification of the microscope will be

(a) 62

(b) 50

(d) 25

Question 9

A hollow non-conducting cone of base radius $\mathrm{R}=50 \mathrm{ cm}$ and semi vertical angle of $15^{0}$ has been uniformly charged on its curved surface up to three-fourth of its slant length from base with a surface charge density $\sigma=2.5 \mu \mathrm{C} / \mathrm{m}^{2}$. The electric field produced at the location of the vertex of the cone is

(a) $\frac{\sigma \ln 2}{2 \varepsilon_{0}}$

(b) $\frac{\sigma \ln 2}{4 \varepsilon_{0}}$

(d) $\frac{\sigma \ln 2}{16 \varepsilon_{0}}$

Question 10

A freely falling spherical rain drop gathers moisture (maintaining its spherical shape all the way) from the atmosphere at a rate $\frac{d m}{d t}=k t^{2}$ where t is the time and m is the instantaneous mass of the drop, the constant $k=12 g m / s^{3}$. If the drop, of initial mass $m_{0}=2 g m$, starts falling from rest, the instantaneous velocity of the drop exactly after 5 second shall be (ignore air friction and air buoyancy)

(a) $12.4 \mathrm{ ms}^{-1}$

(b) $49.0 \mathrm{ m} \mathrm{ s}^{-1}$

(d) data insufficient

Question 11

Two planets, each of mass M and radius R are positioned (at rest) in space, with their centres a distance 4R apart. You wish to fire a projectile from the surface of one planet to the other. The minimum initial speed for which this may be possible is

(a) $\sqrt{\frac{2 G M}{5 R}}$

(b) $\sqrt{\frac{2 G M}{3 R}}$

(d) $\sqrt{\frac{3 G M}{2 R}}$

Question 12

A thin uniform metallic rod of length L and radius R rotates with an angular velocity $\omega$ in a horizontal plane about a vertical axis passing through one of its ends. The density and the Young's modulus of the material of the rod are $\rho$ and Y respectively. The elongation in its length is

(a) $\frac{\rho \omega^{2} L^{3}}{6 Y}$

(b) $\frac{\rho \omega^{2} L^{3}}{3 Y}$

(d) $\frac{\rho \omega^{2} L^{3}}{2 Y}$

Question 13

Consider a particle of mass m with a total energy E moving in a one dimensional potential field. The potential V(x) is plotted against x in the figure beside. The plot of momentum - position graph of this particle is qualitatively best represented by

All plots are symmetrical about x - axis.

(a) Fig. a

(b) Fig. b

(d) Fig. d

Question 14

Knowing that the parallel currents attract, the inward pressure on the curved surface of a thin walled, long hollow metallic cylinder of radius $\mathrm{R}=50 \mathrm{ cm}$ carrying a current of $\mathrm{i}=2 \mathrm {amp}$ parallel to its axis distributed uniformly over the entire circumference, is

(a) $2.05 \times 10^{-1} \mathrm{Nm}^{-2}$

(b) $2.55 \times 10^{-3} N m^{-2}$

(d) $2.55 \times 10^{-7} \mathrm{Nm}^{-2}$

Question 15

Two masses move on a collision path as shown. Before the collision the object with mass 2M moves with a speed v making an angle $\theta=\sin ^{-1} \frac{3}{5}$ to the x-axis while the object with mass M moves with a speed $\frac{3}{2} v$ making an angle $\phi=\sin ^{-1} \frac{4}{5}$ with the x-axis. After the collision the object of mass 2M is observed to be moving to the right along the x-axis with a speed of $\frac{4}{5} v$. There are no external forces acting during the collision. The correct option is

(a) The velocity of mass M, after the collision, is zero.

(b) The centre of mass is moving along x-axis before the collision.

(d) The total linear momentum of the system before the collision along x - axis is $\frac{5}{6} M v$

Question 16

A large hemispherical water tank of radius R is filled with water initially upto a height $h=\frac{R}{2}$. The water starts dripping out through a small orifice of cross section area 'a' at its spherical bottom. The time taken to get the tank completely empty (neglect viscosity) is

(a) $t=\frac{19 \pi R^{2}}{60 a} \sqrt{\frac{R}{g}}$

(b) $t=\frac{3 \pi R^{2}}{10 a} \sqrt{\frac{R}{g}}$

(d) $t=\frac{\pi R^{2}}{4 a} \sqrt{\frac{R}{g}}$

Question 17

If Pascal (Pa), the unit of pressure volt (V), the unit of potential and meter (L), the unit of length are taken as fundamental units, the dimensional formula for the permittivity $\varepsilon_{0}$ of free space is expressed as

(a) $\mathrm{Pa}^{-1} \mathrm{ V}^{2} \mathrm{ L}^{-2}$

(b) $\mathrm{Pa}^{1} \mathrm{ V}^{-2} \mathrm{ L}^{2}$

(d) $\mathrm{Pa}^{-1} \mathrm{ V}^{-2} \mathrm{ L}^{2}$

Question 18

A cycle wheel of mass M and radius R fitted with a siren at a point on its circumference, is mounted with its plane vertical on a horizontal axle at about 3 feet above the ground. An observer stands in the vertical plane of the wheel at 100 m away from the axle of the wheel on a horizontal platform. The siren emits a sound of frequency 1000 Hz and the wheel rotates clockwise with a uniform angular speed $\omega=\pi \mathrm{rad} / \mathrm{sec}$. Initially at $\mathrm{t}=0 \mathrm{sec}$ the siren is nearest to the observer and moves downwards. The observer records the highest pitch of sound for the first time after (speed of sound in air is $330 \mathrm{ ms}^{-1}$ )

(a) 0.30 s

(b) 1.8 s

(d) 9.8 s

Question 19

On a right angled transparent triangular prism ABC, when a ray of light is incident on face AB , parallel to the hypotenuse BC , it emerges out of the prism grazing along the surface AC . If instead the ray is made incident on face AC, parallel to the hypotenuse CB it gets totally reflected on face AB . The refractive index $\mu$ of the material of the prism is

(a) $\mu>\sqrt{2}$

(b) $\sqrt{2}>\mu>\sqrt{\frac{3}{2}}$

(d) $\mu<\sqrt{\frac{3}{2}}$

Question 20

A circular disc of radius $\mathrm{R}=10 \mathrm{ cm}$ is uniformly rolling on a horizontal surface with a velocity $\mathrm{v}=4 \mathrm{ ms}^{-1}$ of centre of mass without slipping, the time taken by the disc to have the speed of point A (which lies on the circumference) equal to the present speed of point B (point B lies midway between centre and the point A) is

(a) $\mathrm{t}=0.025 \mathrm{ s}$

(b) $\mathrm{t}=0.036 \mathrm{ s}$

(d) $\mathrm{t}=0.064 \mathrm{ s}$

Question 21

As shown in the figure, a particle of mass $m=10^{-10} k g$, moving with velocity $\mathrm{v}_{0}=10^{5} \mathrm{ m} / \mathrm{s}$ approaches a stationary fixed target with impact parameter $b$ from a large distance. If the fixed rigid target has a core with repulsive central force $F(r)=\frac{K}{r^{3}}$ where constant $\mathrm{K}>0$ and the particle scatters elastically. The closest distance of approach (if numerically $\mathrm{K}=\mathrm{b}^{2}$ ) is

(a) b

(b) $b \sqrt{2}$

(d) 2b

Question 22

If the specific activity of $\mathrm{C}^{14}$ nuclide in a certain ancient wooden toy is known to be $\frac{3}{5}$ of that in a recently fallen tree of the same class, the age of the ancient wooden toy is (The half life of $\mathrm{C}^{14}$ is 5570 years)

(a) 5570 years

(b) 4105 years

(d) 2785 years

In questions 23 and 24 mark your answer as

(a) If statement I is true and statement II is true and also if the statement II is a correct explanation of statement I

(b) If statement I is true and statement II is true but the statement II is a not a correct explanation of statement I

(d) If statement I is false but statement II is true

Question 23

Statement I: Work done in bringing a charge q from infinity to the center of a uniformly charged non - conducting solid sphere of radius R (with a total charge Q) is zero.

Statement II: The potential difference between the Centre and the surface of the uniformly charged non - conducting solid sphere of radius R (with a total charge Q) is $\frac{1}{4 \pi \varepsilon_{0}} \times \frac{Q}{2 R}$.

Question 24

Statement I: The current flowing through a p-n junction is more in forward bias than that in the reverse bias.

Statement II: The diffusion current, dominant in forward bias, is more than the drift current, dominant in the reverse bias.

In the following questions, any number of options 4,3,2, or 1 may be correct. Marks will be awarded only if all the correct options are bubbled.

Question 25

A simple pendulum consisting of a small bob of mass m attached to a massless inextensible string of length $\ell$, hanging vertically from the ceiling, is oscillating in a vertical plane with an angular amplitude $\theta_{m}$ such that the maximum tension in its string is three times the minimum tension in the string i.e., $\mathrm{T}_{\text {max }}=3 \mathrm{ T}_{\text {min }}$. The correct option(s) is/are

(a) The maximum tension in the string is $\mathrm{T}_{\text {max }}=\mathrm{mg}\left(3-2 \cos \theta_{\mathrm{m}}\right)$

(b) The maximum tension in the string is $T_{\text {max }}=\frac{9}{5} m g$

(d) The angular amplitude $\theta_{\mathrm{m}}$ lies in the range $\frac{\pi}{4}<\theta_{m}<\frac{\pi}{3}$

Question 26

Two small masses m and M lie on a large horizontal frictionless circular track of radius R. The two masses are free to slide on the track but constrained to move along a circle. Initially the two masses are tied by a thread with a compressed spring between them (spring of negligible length being attached with none of the two masses). The compressed spring stores a potential energy $\mathrm{U}_{0}$. At a certain time $\mathrm{t}=0$ the thread is burnt and the two masses are released to run opposite to each other leaving the spring behind. The total mechanical energy remaining conserved. On the circular track the two masses make a head on perfectly elastic collision. Take $\mathrm{M}=2 \mathrm{ m}$ for all calculations. Which of the following option(s) is / are correct?

(a) The angle turned by mass m before the collision is $\theta=4 \frac{\pi}{3}$

(b) The velocity of mass m on the track is $u=\sqrt{\frac{4 U_{0}}{3 m}}$

(d) The time taken for the second collision is $t_{2}=2 \pi R \sqrt{\frac{2 m}{3 U_{0}}}$

Question 27

The electric field component of an electromagnetic wave is expressed as $E=(3 j+b k) \times 10^{-3} \sin \left[10^{7}(x+2 y+3 z-\beta t)\right]$ in SI units. Taking $c=3 \times 10^{8} m \mathrm{ s}^{-1}$ as the speed of electromagnetic wave in vacuum, choose the correct option(s)

(a) The value of constant beta is $\beta=3 \times 10^{8} \times \sqrt{14}$

(b) The value of constant b is $\mathrm{b}=2$.

(d) The amplitude of magnetic field is $B=1.20 \times 10^{-11}$ Tesla

Question 28

A parallel beam of light is made incident (as shown) on the flat diametric plane of a transparent semi-circular thin sheet of thickness $\mathrm{t}(\mathrm{t} \ll \mathrm{R})$ of refractive index $\mu=\sqrt{2}$ at an angle of $45^{0}$. As a result of refraction, the light enters the semi-circular sheet and comes out at its curved surface.

(a) Light rays come out at the curved surface for values of $\theta$ in the range $75^{0} \leq \theta \leq 165^{0}$.

(b) The range of angle $\theta$ is independent of the angle of incidence.

(d) All the emergent rays of light shall cross the line OP which is a refracted ray at $\theta$=120$^{\circ}$

Here $\theta$ is the angle between the vertical diameter AB and the concerned radius of the
semicircular sheet of radius R.

Question 29

A certain rod of uniform area of cross section $\mathrm{A}\left(\mathrm{A}=1.0 \mathrm{ cm}^{2}\right)$ with its length = 2 m is thermally insulated on its lateral surface. The thermal conductivity $(K)$ of the material of the rod varies with temperature $T$ as $K=\frac{\alpha}{T}$ where $\alpha$ is a constant. The two ends of the rod are maintained at temperature of $T_{1}=90^{\circ}$ - and $T_{2}=10^{\circ}$.. The correct option(s) is /are

(a) The temperature at 50 cm from the colder end is 17.32°

(b) The temperature at 50 cm from the hotter end is $51.96^{\circ}$ -

(d) The temperature gradient is numerically higher near the hot end compare to that near the cold end.

Question 30

Positronium is a short-lived $\left(\approx 10^{-9} \mathrm{ s}\right)$ bound state of an electron and a positron (a positively charged particle with mass and charge equal (in magnitude) to an electron) revolving round their common centre of mass. If $\mathrm{E}_{0}, \mathrm{v}_{0}$ and $\mathrm{a}_{0}$ are respectively the ground state energy, the orbital speed of electron in first orbit and the radius of the first ( $\mathrm{n}=1$ ) Bohr orbit for Hydrogen atom, the corresponding quantities E, v and a for the positronium are

(a) $E=\frac{E_{0}}{2}$

(b) $a=a_{0}$

(d) $E=E_{0}, v=v_{0}, a=a_{0}$

Question 31

A thin double convex lens of radii of curvature $\mathrm{R}_{1}=20 \mathrm{ cm}$ and $\mathrm{R}_{2}=60 \mathrm{ cm}$ is made-up of a transparent material of refractive index $\mu=1.5$. Choose the correct option(s)

(a) The focal length of the lens is $\mathrm{f}=30 \mathrm{ cm}$ when in air.

(b) The lens behaves as a concave mirror of focal length $f_{M}=10 c m$ when silvered on the surface of radius $\mathrm{R}_{2}=60 \mathrm{ cm}$

(c) The lens behave as a concave lens (diverging lens) if the image space beyond $\mathrm{R}_{2}=60 \mathrm{ cm}$ radius surface is filled with a transparent liquid of refractive index $\mu=\frac{5}{3}$. The object space prior to the surface of radius $\mathrm{R}_{1}=20 \mathrm{ cm}$ is air.

(d) A beam of rays incident parallel to principal axis focuses at 48 cm behind the lens if water $\left(\mu=\frac{4}{3}\right)$ fills the entire space behind the surface of radius $\mathrm{R}_{2}=60 \mathrm{ cm}$. The object space prior to the surface of radius $\mathrm{R}_{1}=20 \mathrm{ cm}$ is air.

Question 32

A thick hollow cylinder of height h and inner and outer radii a and $\mathrm{b}(\mathrm{b}>\mathrm{a})$ made up of a poorly conducting material of resistivity $\rho$ lies coaxially inside a long solenoid at its middle. The radius of the solenoid is larger than b . Throughout the interior of the solenoid, a uniform time varying magnetic field $B=\beta t$ is produced parallel to solenoid axis. Here $\beta$ is a constant. In this time varying magnetic field

(a) the emf induced at a certain radius $\mathrm{r}(\mathrm{a}<\mathrm{r}<\mathrm{b})$ in the hollow cylinder is $\pi r^{2} \beta$

(b) the induced current circulating in the thick hollow cylinder between radii a and b is $i=\frac{\beta h}{4 \rho}\left(b^{2}-a^{2}\right)$

(c) the resistance offered to the circulation of current by the thick hollow cylinder is $R=\frac{2 \pi \rho}{h \times \ln \frac{b}{a}}$

(d) no electric field is detectable outside the solenoid.