NSEP 2024 Question Paper

Question 1

An air filled parallel plate capacitor, with plate area $A$ and plate separation $d$, is connected to a battery of emf $V$ volt having negligible internal resistance. One of the plates of the capacitor vibrates with amplitude ' $a$ ' ( $a \ll d$ ) and angular frequency $\omega$. If the instantaneous current in the circuit reaches a maximum value $I_{0}$, the amplitude of the vibrations is ' $a$ ' equal to

Question 2

A small ball of mass $m$ is attached to one end of a massless un-stretchable string of length $I$ and is held at the point $P$. The other end of the string is fixed to a support at $O$ such that $O P$ is horizontal. The minimum downward speed $u$, that should be imparted to the ball at the point $P$ so that the ball can complete the vertical circle without any slack in the string, is

Question 3

An illuminated point object is placed on the principal axis, in front of an equi-convex glass lens of focal length $f=40 \mathrm{ cm}$, at a distance of $u=1.20 \mathrm{ m}$ from the lens. A reflecting plane mirror has been placed behind the lens perpendicular to the principal axis and facing the lens. The nature of the final image and the distance of the plane mirror from the lens, so as to form the final image at the plane mirror itself, is

Question 4

A ball is kicked horizontally form the top $C$ of a hemispherical rock $A C B$ of radius $R$ on a horizontal ground, with a velocity $v$, so as not to hit the rock at any point during its flight. Choose the correct statement (see the figure)

Question 5

A uniform beam of light of intensity $60 \mathrm{ mW} / \mathrm{m}^{2}$ is incident on a totally absorbing sphere of radius $2.0 \mu \mathrm{ m}$. The density of the material of the sphere is $\rho=5.0 \times 10^{3} \mathrm{ kg} / \mathrm{m}^{3}$. The sphere is placed in a region of space where gravitational force can be neglected/ignored. The magnitude of acceleration of the sphere due to the incidence of the light is

Question 6

The ratio of lengths, radii and Young's moduli of steel to brass wires in the figure are $\alpha, \beta$ and $\gamma$, respectively. The corresponding ratio of the increase in their lengths is

Question 7

The fuse, in the upper branch of the circuit shown, is an ideal 4.0 A fuse. The fuse has zero resistance as long as current through it remains less than 4.0 A . The fuse blows out when the current reaches 4.0 A . Needless to say that the resistance becomes infinite thereafter. Switch $S$ is closed at time $t=0$. The fuse blows out at time

Question 8

The bent wire $P Q R$ shown in the figure lies in a uniform magnetic field $\vec{B}=3.0 \hat{i}+4.0 \hat{k} T$. The direction $\hat{k}$ being normal to the plane of the paper and directed towards the viewer. The two straight sections $P Q$ and $Q R$ of the wire each have length 2.0 m and the wire carries a current of 2.5 A . Net force on the wire due to magnetic field $\vec{B}$ is

Question 9

A charged spherical capacitor consists of two concentric spherical shells of radii $a$ and $b(b>a)$. Half of the stored electrical energy of this system lies within a spherical region of radius $r$ if

Question 10

Two stars of masses $M_{1}=M_{S}$ and $M_{2}=15 M_{S}$ (where $M_{S}=$ mass of the Sun) form a binary system. The stars are revolving round each other, always being at a separation of $d=4 \mathrm{AU}$ between them ( 1 AU is distance of the Earth from the Sun), move in circular orbits about their center of mass. The period of revolution of each star is

Question 11

The total energy released in $\alpha$ decay of a stationary Radium nucleus ${ }^{226} \mathrm{Ra}$ (mass 116 u ) is $Q=4.9 \mathrm{MeV}$ (mass of $\alpha$ particle $m_{\alpha}=4 \mathrm{u}$ ) then the

Question 12

A listener at rest (with respect to the air and the ground) hears a sound of frequency $f_{1}$ from a source moving towards him with a velocity of $15 \mathrm{ ms}^{-1}$, towards East. If the listener now moves towards the approaching source with a velocity of $25 \mathrm{ ms}^{-1}$, towards West, he hears a frequency $f_{2}$ that differs from $f_{1}$ by 40 Hz . The frequency $f$ of the sound produced by the source is (speed of sound in the air is $340 \mathrm{ ms}^{-1}$ )

Question 13

A uniform rod of length $\ell$ swings from a pivot as a physical pendulum. The position of the pivot can be varied along the length of the rod. The minimum time period with which the rod can oscillate with an appropriate position of the pivot is $T=2 \pi \sqrt{\frac{L}{g}}$ where $L$ is equal to

Question 14

The number density of conduction electrons in pure silicon at room temperature is about $10^{16} \mathrm{ m}^{-3}$. The number density of conduction electrons is increased by a factor of $10^{6}$ by doping the silicon lattice with phosphorus. Assume that at room temperature every phosphorus atom contributes one electron to the conduction band. The fraction of silicon atoms replaced by phosphorus atom is (Given that the density of silicon is $2.33 \mathrm{gm} / \mathrm{cm}^{3}$ and that the molar mass of silicon $M=28.1 \mathrm{gm}$ )

Question 15

A soap bubble 10 cm in radius, with a film thickness of $\frac{10}{3} \times 10^{-6} \mathrm{ cm}$, is charged to a potential of 80 V . The bubble bursts and converts into a single spherical drop. Assuming that the soap solution is a good conductor, the potential at the surface of the drop is

Question 16

Three resistances, each of $R=4 \Omega$ rated 16 W , are connected across $A$ and $B$ as shown. Potential difference of $V$ volt is applied between points $A$ and $B$.

Statement 1: Maximum potential difference $V$ that can be applied is 12 volt.

Statement 2: Maximum power that can be dissipated is 24 watt

Question 17

Water is filled in a vertical cylinder up to a certain height $h$. The cylinder is made to rotate with an angular velocity $\omega$ about a vertical axis coinciding with the axis of the cylinder. The water surface seen from top appears as a/an

Question 18

A uniformly charged non-conducting sphere with its center at $C$ carries positive charge with uniform charge density $+\rho$, except in a spherical cavity (inside the sphere) with center $O$. The electric field $E$ at any point inside the cavity is

Question 19

A cylindrical tank with base area $A=0.05 \mathrm{ m}^{2}$ is filled with water up to a height $H=50 \mathrm{ cm}$. There is a small hole of area $a=0.001 \mathrm{ m}^{2}(a \ll A)$ in the bottom of the tank. It takes times $t$ to empty the tank up to a height $\frac{H}{2}$ (i.e. to empty half of the water volume). The additional time required to empty the tank completely is

Question 20

$O A B C$ is a regular tetrahedron, each side of which is made of uniform wire of resistance $4 \Omega \mathrm{ m}$. The length of each side is 2 m . The point $M$ is the midpoint of the side $B C$. The resistance between $O$ and $M$ is

Question 21

A box weighing $W$ is placed on a rough horizontal floor. The coefficient of static friction between the box and the floor is $\mu$. To move the box, a pulling force $F$ is applied along the rope joined to the box at an angle $\theta$ with horizontal. By suitable choice of $\theta$, the minimum value of $F$ that can make the box move is

Question 22

The external wall of a room measuring $2 \mathrm{ m} \times 3 \mathrm{ m}$ consists of a layer of white pine of thickness $d_{\text {pine }}=2.0 \mathrm{ cm}$ and a layer of rock wool in succession. The external temperature is 36 K below the indoor temperature (Given the thermal conductivity coefficient of white pine $\mathrm{K}_{\mathrm{P}}=0.10 \mathrm{Wm}^{-1} \mathrm{kelvin}^{-1}$ and that of rock wool $\mathrm{K}_{\mathrm{w}}=0.04 \mathrm{Wm}^{-1}$ kelvin ${ }^{-1}$ ) The thickness of the layer of the rock wool, so that the thermal conduction rate $P_{\text {cond }}=\frac{d Q}{d t}$ across the wall does not exceed 120 watt (assuming no loss of heat during conduction and no other way of heat transfer other than conduction), is

Question 23

A 240 kg block is suspended from a fixed point $O$, at the end of a long ( $L=13 \mathrm{ m}$ ) massless rope. A horizontal force $F$ slowly pushes the block to move it a horizontal distance $d=5 \mathrm{ m}$ sideways, to a position $B$ where it remains stationary as shown in the figure.

Statement 1: Force $F$ at position $B$ is 980 N .

Statement 2 Work done by the force to bring the box from $A$ to $B$ is 2352 J

Question 24

The number of radioactive nuclei $N$ of a radioactive sample is experimentally measured as a function of time $t$. At $t=0, N(t=0)=50,000$ and at $t=10 \mathrm{ s}, N(t=10 \mathrm{ s})=5000 \pm 100$. The half-life of the sample is estimated from these measurements. The error in the estimation of the half-life is approximately [note that for small values of $x$, $L t x \rightarrow 0, \ln (1+x) \approx x]$

Question 25

An amount of heat equal to 10.61 J is given to an ideal gas at constant pressure of one $\mathrm{atm}\left(1.01 \times 10^{5} \mathrm{ Pa}\right)$. As a result the volume of the gas increases by $30.0 \mathrm{ cm}^{3}$. The gas is

Question 26

A ball is projected from point $O$ on the ground with a certain velocity $\vec{u}$ at angle $\theta$ from horizontal. When it reaches point $P$ located at a horizontal distance $L$ from $O$ and is at a height $h$ above the ground, the angle $\alpha$ subtended by the velocity vector $\vec{v}$ with horizontal at this point $P$ is expressed as

Question 27

A flexible chain, of length $L$ and uniform mass per unit length $\lambda$, slides off the edge of a frictionless table (see figure). Initially a length $y=y_{0}$ of the chain hangs over the edge, with the chain held at rest. Now the chain is let free. The velocity of the chain when the chain becomes completely vertical (i.e. when the chain is just to leave the edge) is

Question 28

A Keplerian telescope is adjusted in its normal setting for parallel rays, mounting of the objective has diameter $D$ and the diameter of the image of that mounting formed by the eye piece of the telescope is $d$. Magnifying power of the telescope is

Question 29

In the circuit shown below the resistance $R=\sqrt{3} \times 10^{3} \Omega$, the inductance $L=2 \mathrm{H}$ and the capacitance $C=1 \mu \mathrm{ F}$ have been connected to an AC supply of the peak voltage of $\mathrm{V}_{\max }=5$ volt at a frequency $\omega$. Either the switch $S_{1}$ or the switch $S_{2}$ is closed at a time. In either case, same maximum current ( $i_{\max }$ ) is recorded in the circuit. The frequency of the AC source is nearly

Question 30

There are two given physical quantities $A=\frac{F}{\rho v^{2} D}$ and $B=\frac{\rho v D}{\eta}$ where $F$ is force, $\rho$ is density, $D$ is diameter, $v$ is velocity and $\eta$ is the coefficient of viscosity. Which of the two $A$ and $B$ is/are dimensionless?

Question 31

A uniform inextensible string of length $L$ and mass $M$ is suspended vertically from a rigid support. A transverse pulse is allowed to propagate down through the string from the support. At the same time, a ball of mass $m$ is dropped from the rigid support. The ball will pass the pulse at a distance of (from the top of the string i.e. from the support)

Question 32

In a positron decay, the radionuclide ${ }^{11} C$ decays according to ${ }^{11} C \longrightarrow{ }^{11} B+e^{+}+v$ [the respective atomic masses are ${ }^{11} C=11.01142 u,{ }^{11} B=11.00931 u$ and the mass of positron $=$ the mass of electron $=m_{e}=0.00055$ u]. The disintegration energy (i.e. the $Q$ value) is approximately

Question 33

Two mechanical waves, given by $y_{1}=A \sin (8 x-50 t)$ and $y_{2}=A \sin \left(8 x+50 t+\frac{\pi}{3}\right)$ travelling in opposite directions along $x$-axis, superpose. The position of the node (for $x>0$ ) nearest to the origin is (the displacement $y_{1}$ and $y_{2}$ are in meter)

Question 34

At a certain location on the Earth surface, the intensity of sunlight is $1.00 \mathrm{ kW} / \mathrm{m}^{2}$. A perfectly reflecting concave mirror, of radius of curvature $R$ and aperture radius $r$, is facing the Sun to produce the light intensity of $100 \mathrm{ kW} / \mathrm{m}^{2}$ at the image. Knowing that the disc of the Sun subtends an angle of 0.01 radian at the Earth surface, the relation between $R$ and $r$ is

Question 35

A charge $Q$ is uniformly distributed throughout the volume of a non-conducting sphere of radius $R$. The total electrostatic energy of electric field inside the sphere is $U=\alpha \times \frac{Q^{2}}{4 \pi \varepsilon_{0} R}$. The value of $\alpha$ is

Question 36

A long straight vertical wire of circular cross section of radius $R$ contains $n$ conduction electrons per unit volume. A current $I$ flows upward in the wire. The expression for magnetic force on an electron at the surface of the wire is [Assume all the conduction electrons are moving with drift velocity]

Question 37

In Young's double slit experiment, a bright fringe is observed at $y=1.5 \mathrm{ cm}$ from the center of the fringe pattern when monochromatic light of wavelength 612 nm is used. The screen is at 1.4 m from the plane of the two slits, whose separation is 0.4 mm . The number of dark fringes between the center and the said bright fringe at $y=1.5 \mathrm{ cm}$ is

Question 38

An illuminated point object is placed in front of an equi-convex lens of refractive index $\mu=1.5$ and focal length $f=40 \mathrm{ cm}$, at a distance of $u=1.20 \mathrm{ m}$ in front of the lens on its principal axis. Water ( $\mu=4 / 3$ ) fills the space behind the lens up to a distance of 40 cm from the lens. The final image is formed on the principal axis beyond the lens at a distance $v$ from the lens. The value of $v$ and the nature of image are

Question 39

A paramagnetic gas at room temperature $27^{\circ} \mathrm{C}$ is placed in an external uniform magnetic field of magnitude $B=$ 1.5 tesla. The atoms of the gas have magnetic dipole moment $\mu=1.0 \mu_{B}$ where $1 \mu_{B}=\frac{e h}{4 \pi m}$ is Bohr magneton, and $m$ is the mass of electron. The energy difference $\Delta U_{B}$ between the parallel alignment and the antiparallel alignment of the atom's magnetic dipole moment, with respect to the external field $B$, is $x \times 10^{-4} \mathrm{eV}$ where the value of $x$ is

Question 40

A glass capillary with inner diameter of 0.40 mm is vertically submerged in water so that the length of its part protruding above the water surface is $h=25 \mathrm{ mm}$. Surface Tension of water is $T=0.073 \mathrm{Nm}^{-1}$. Since the water wets the glass completely, it may be concluded that the

Question 41

The container $A$ in the figure holds an ideal gas at a pressure of $5.0 \times 10^{5} \mathrm{ Pa}$ at $27^{\circ} \mathrm{C}$. It is connected to the container $B$ by a thin tube fitted with a closed valve. Container $B$ with volume four times the volume of $A$ holds the same ideal gas at a pressure $1.0 \times 10^{5} \mathrm{ Pa}$ at $127^{\circ} \mathrm{C}$. The valve is now opened to allow the pressure to equalize, but the temperature of each container is maintained as before. The common pressure in the two containers (in kPa) is

Question 42

A thin plastic disc, which is a quarter of a circle of radius $R=0.6 \mathrm{ m}$, lies in the first quadrant of $x-y$ plane, with the center of curvature at the origin $O$ as shown. It is charged uniformly on one side (one face) with surface charge density $\sigma$. Electric potential at point $P(0,0,0.8 \mathrm{ m})$ is

Question 43

The charge on a small point object at $A$ produces an electric potential $V=3$ volt at a point $P$. Because of an unavoidable situation, leakage of the charge from the object at $A$ starts at $t=0$ at a constant rate of $3 \mu \mathrm{Cs}^{-1}$. To maintain the potential of 3 volt at the point $P$, the object is made to move towards $P$ with a certain velocity $v$. When the point object crosses the point $D$ shown in the figure, it is found to have a charge of $10 \mu \mathrm{C}$ and the direction of its velocity is perpendicular to $O D$. The resulting magnetic field $B$, at this instant, at the location $O$ (such that $O D=1.0 \mathrm{ m}$ ) is $D$

Question 44

In the Bohr model of hydrogen atom, the force between the nucleus and the electron is modified as $F=\frac{e^{2}}{4 \pi \varepsilon_{0}}\left(\frac{1}{r^{2}}+\frac{\delta}{r^{3}}\right)$ where $\delta$ is a small constant. Using the Bohr radius $a_{0}=\frac{\varepsilon_{0} h^{2}}{\pi m e^{2}}$, the radius of $\mathrm{n}^{\text {th }}$ orbit is

Question 45

An infinite number of conducting rings having increasing radii $r_{0}, r_{1}, r_{2}, r_{3}$ and so on, such that $r_{0}=r$, $r_{1}=2 r$, $r_{2}=2^{2} r, r_{3}=2^{3} r$, and so on ... upto $\infty$, have been placed concentrically on a plane. All the rings carry the same current $i$ but the current in consecutive rings is in opposite direction as shown. The magnetic field produced at the common center of the rings is

Question 46

One mole of an ideal monoatomic gas, initially at temperature $T$, is heated in such a way that its molar heat capacity during the process of heating is $C=2 R$. The volume of the gas gets tripled (at constant pressure) during the process. The final temperature attained by the gas is

Question 47

A spring is compressed under the action of a constant force $F$. The compression of the spring is $\xi$. Suppose the direction of the force is reversed suddenly as well as its magnitude is doubled. The maximum extension of the spring beyond its natural length will now be (the spring obeys Hook's law).

Question 48

Two small positively charged spherical balls are suspended from a common point at the ceiling by nonconducting massless strings of equal length $I$. The first ball has mass $m_{1}$ and charge $q_{1}$ while the second ball has mass $m_{2}$ and charge $q_{2}$. If the two strings subtend angles $\theta_{1}$ and $\theta_{2}$, with the vertical as shown, then

Question 49

An electric circuit consists of a battery of emf $E$, an inductance $L$ and a resistance $R$ in series. The switch $S$ is closed at $t=0$. The current in the circuit grows exponentially with time as depicted by curve (1). The values of the circuit parameters ( $E, L$ or $R$ ) are now somehow changed. The circuit is closed second time, the growth of current / follows curve (2). The following conclusion(s) may be drawn

Question 50

The Earth is revolving around the Sun (mass $M$ ) in a circular orbit of radius $r$ with a period of revolution $T=1$ year. Suppose that during the revolution of the Earth, at any instant, the mass of the Sun instantaneously becomes double i.e. $2 M$. The correct alternative(s) is/are [Assume that the Sun and the Earth are point masses]

Question 51

An optical fibre consists of a glass core (refractive index $n_{1}$ ) surrounded by a cladding (refractive index $n_{2}<n_{1}$ ). A beam of light enters one end of the fibre at $P$, from air at angle $\theta$, with the axis of fibre as shown in the figure. Choose the correct option (s)

Question 52

According to Bohr theory, in the ground state of hydrogen atom, an electron revolves in circular orbit of radius $r$ with velocity $v$ and circulation frequency $f$. The magnetic dipole moment of the electronic orbit is $p_{m}$. The magnetic field produced by the circulating electron at the center of the atom is $B$. Then for a $\mathrm{He}^{+}$ion in the state $n=2$ (electron in the $2^{\text {nd }}$ orbit)

Question 53

A body of mass $m=0.25 \mathrm{ kg}$ is moving along $x$ axis under the action of a conservative force. Its potential energy as a function of position $x$ is given by $U(x)=-\frac{100 x}{x^{2}+4} \mathrm{ J}(x$ in m$)$. Then

Question 54

A uniform rod $O A$ of length $L$ is freely pivoted at one end $O$. Let $C$ be the midpoint (i.e. $O C=\frac{L}{2}=C A$ ). Choose the correct statement(s)

Question 55

A boy stands on a stationary ice boat on a frictionless horizontal flat iceberg. The boy and the boat have a combined mass of $M=60 \mathrm{ kg}$. Two balls masses $m_{1}=10 \mathrm{ kg}$ and $m_{2}=20 \mathrm{ kg}$ are placed on the boat. In order to get the boat moving, the boy throws the balls backward horizontally either in succession or both together. In each case the balls are thrown backward with a certain speed $v_{\text {rel }}=6 \mathrm{ ms}^{-1}$ relative to the boat just when the ball is being thrown. The resulting speed of the system of the boat and the boy is

Question 56

A series LCR circuit fed with AC has resonant angular frequency of $\omega=2.0 \times 10^{4} \mathrm{rad} / \mathrm{s}$. When the same circuit is driven at an angular frequency of $2.5 \times 10^{4} \mathrm{rad} / \mathrm{s}$, it has impedance of $1.0 \mathrm{k} \Omega$ and phase constant of $\phi=45^{\circ}$. The values of $L, C$ and $R$ for this circuit may be

Question 57

A long straight wire, having a radius greater than 4.0 mm , carries a current that is uniformly distributed over its cross-section. The magnitude of magnetic field $B$ due to the current is 0.28 mT at $r=4.0 \mathrm{ mm}$ and 0.20 mT at $r=$ 10.0 mm , respectively, from the axis of the wire then the

Question 58

In the given combination of capacitors $C_{1}=4 \mu \mathrm{ F}, C_{2}=2 \mu \mathrm{ F}, C_{3}=2 \mu \mathrm{ F}, C_{4}=4 \mu \mathrm{ F}$ and $C_{5}=3 \mu \mathrm{ F}$, a source of 12 volt is connected across the points $A$ and $B$. Then the

Question 59

One mole of an ideal monoatomic gas, initially at temperature $T$, is compressed to $\frac{1}{8}$ of its volume by a piston in a cylinder such that the heat dissipated into the environment is equal to the change in the internal energy of the gas. Then the

Question 60

A solid ball of mass $M$ and radius $R$ (see figure) is lying on a horizontal table. The ball experiences a short horizontal impulse which imparts a momentum $p$ to the ball. The height of point of impact above the center line is $h=\alpha R(0 \leq \alpha \leq 1)$. Choose the correct option/option(s).