NSEP 2024 24th Nov Question Paper

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 $$n^{\text{th}}$$ orbit is

An infinite number of conducting rings having increasing radii $$r_0, r_1, r_2, r_3$$ and so on, such that $$r_0 = r$$, $$r_1 = 2r$$, $$r_2 = 2^2r$$, $$r_3 = 2^3r$$ and so on up to infinity, 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

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 $$C = 2R$$. The volume of the gas gets tripled at constant pressure during the process. The final temperature attained by the gas is

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, assuming the spring obeys Hooke's law,

Two small positively charged spherical balls are suspended from a common point at the ceiling by non-conducting massless strings of equal length $$\ell$$. 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

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 $$a$$, where $$a\ll d$$, 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

A small ball of mass $$m$$ is attached to one end of a massless un-stretchable string of length $$\ell$$ and is held at the point $$P$$. The other end of the string is fixed to a support at $$O$$ such that $$OP$$ 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

An illuminated point object is placed on the principal axis, in front of an equi-convex glass lens of focal length $$f=40\ \text{cm}$$, at a distance of $$u=1.20\ \text{m}$$ 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

A ball is kicked horizontally from the top $$C$$ of a hemispherical rock $$ACB$$ 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 from the figure.

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

A uniform beam of light of intensity $$60\ \text{mW m}^{-2}$$ is incident on a totally absorbing sphere of radius $$2.0\ \mu\text{m}$$. The density of the material of the sphere is $$\rho=5.0\times10^3\ \text{kg m}^{-3}$$. The sphere is placed in a region of space where gravitational force can be neglected or ignored. The magnitude of acceleration of the sphere due to the incidence of the light is

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

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

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

Two stars of masses $$M_1=M_S$$ and $$M_2=15M_S$$, where $$M_S$$ is the mass of the Sun, form a binary system. The stars are revolving round each other, always being at a separation of $$d=4\ \text{AU}$$ 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

The total energy released in $$\alpha$$ decay of a stationary Radium nucleus $$^{226}\mathrm{Ra}$$, printed with mass $$116\ \mathrm{u}$$, is $$Q=4.9\ \text{MeV}$$. The mass of the $$\alpha$$ particle is $$m_\alpha=4\ \mathrm{u}$$. Then the correct statement is

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 $$15\ \text{m s}^{-1}$$ towards East. If the listener now moves towards the approaching source with a velocity of $$25\ \text{m s}^{-1}$$ towards West, he hears a frequency $$f_2$$ that differs from $$f_1$$ by $$40\ \text{Hz}$$. The frequency $$f$$ of the sound produced by the source is, taking the speed of sound in air as $$340\ \text{m s}^{-1}$$,

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

The number density of conduction electrons in pure silicon at room temperature is about $$10^{16}\ \text{m}^{-3}$$. 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 atoms is, given the density of silicon as $$2.33\ \text{g cm}^{-3}$$ and the molar mass of silicon as $$28.1\ \text{g mol}^{-1}$$,

A soap bubble $$10\ \text{cm}$$ in radius, with a film thickness of $$\frac{10}{3}\times10^{-6}\ \text{cm}$$, is charged to a potential of $$80\ \text{V}$$. 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

Three resistances, each of $$R=4\ \Omega$$ rated $$16\ \text{W}$$, 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 $$12\ \text{V}$$.
Statement 2 - Maximum power that can be dissipated is $$24\ \text{W}$$.

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 or an

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

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

$$OABC$$ is a regular tetrahedron, each side of which is made of uniform wire of resistance $$4\ \Omega\text{ m}^{-1}$$. The length of each side is $$2\ \text{m}$$. The point $$M$$ is the midpoint of the side $$BC$$. The resistance between $$O$$ and $$M$$ is

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

The external wall of a room measuring $$2\ \text{m}\times3\ \text{m}$$ consists of a layer of white pine of thickness $$d_{pine}=2.0\ \text{cm}$$ and a layer of rock wool in succession. The external temperature is $$36\ \text{K}$$ below the indoor temperature. The thermal conductivity coefficients are $$K_P=0.10\ \text{W m}^{-1}\text{K}^{-1}$$ for white pine and $$K_W=0.04\ \text{W m}^{-1}\text{K}^{-1}$$ for rock wool. The thickness of the layer of the rock wool, so that the thermal conduction rate $$P_{cond} =\frac{dQ}{dt}$$ 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

A $$240\ \text{kg}$$ block is suspended from a fixed point $$O$$, at the end of a long $$L=13\ \text{m}$$ massless rope. A horizontal force $$F$$ slowly pushes the block to move it a horizontal distance $$d=5\ \text{m}$$ sideways, to a position $$B$$ where it remains stationary as shown in the figure.
Statement 1 - Force $$F$$ at position $$B$$ is $$980\ \text{N}$$.
Statement 2 - Work done by the force to bring the box from $$A$$ to $$B$$ is $$2352\ \text{J}$$.

The number of radioactive nuclei $$N$$ of a radioactive sample is experimentally measured as a function of time $$t$$. At $$t=0$$, $$N=50000$$ and at $$t=10\ \text{s}$$, $$N=5000\pm100$$. The half-life of the sample is estimated from these measurements. The error in the estimation of the half-life is approximately. For small $$x$$, use $$\ln(1+x)\approx x$$.

An amount of heat equal to $$10.61\ \text{J}$$ is given to an ideal gas at constant pressure of one atmosphere, $$1.01\times10^5\ \text{Pa}$$. As a result the volume of the gas increases by $$30.0\ \text{cm}^3$$. The gas is

A ball is projected from point $$O$$ on the ground with a certain velocity $$u$$ 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 $$v$$ with horizontal at this point $$P$$ is expressed as

A flexible chain, of length $$L$$ and uniform mass per unit length $$\lambda$$, slides off the edge of a frictionless table as shown. 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, that is when the chain is just to leave the edge, is

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

In the circuit shown below, the resistance $$R=\sqrt3\times10^3\ \Omega$$, the inductance $$L=2\ \text{H}$$ and the capacitance $$C=1\ \mu\text{F}$$ have been connected to an AC supply of peak voltage $$V_{\max}=5\ \text{V}$$ at a frequency $$\omega$$. Either switch $$S_1$$ or switch $$S_2$$ is closed at a time. In either case, the same maximum current $$i_{\max}$$ is recorded in the circuit. The frequency of the AC source is nearly

There are two given physical quantities $$A=\frac{F}{\rho v^2D}$$ and $$B=\frac{\rho vD}{\eta}$$ 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 or are dimensionless?

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 from the top of the string, that is from the support, equal to

In a positron decay, the radionuclide $$^{11}\mathrm{C}$$ decays according to $$^{11}\mathrm{C}\rightarrow{}^{11}\mathrm{B}+e^++\nu$$. The respective atomic masses are $$^{11}\mathrm{C}=11.01142\ \mathrm{u}$$, $$^{11}\mathrm{B}=11.00931\ \mathrm{u}$$ and the mass of positron equals the mass of electron, $$m_e=0.00055\ \mathrm{u}$$. The disintegration energy, that is the $$Q$$ value, is approximately

Two mechanical waves, given by $$y_1=A\sin(8x-50t)$$ and $$y_2=A\sin\left(8x+50t+\frac{\pi}{3}\right)$$, travelling in opposite directions along the $$x$$ axis, superpose. The position of the node for $$x>0$$ nearest to the origin is, with the displacements in metre,

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

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 $$U=\alpha\times\frac{Q^2}{4\pi\varepsilon_0R}$$. The value of $$\alpha$$ is

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, assuming all the conduction electrons are moving with drift velocity,

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

An illuminated point object is placed in front of an equi-convex lens of refractive index $$\mu=1.5$$ and focal length $$f=40\ \text{cm}$$, at a distance of $$u=1.20\ \text{m}$$ in front of the lens on its principal axis. Water with $$\mu=\frac43$$ fills the space behind the lens up to a distance of $$40\ \text{cm}$$ 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

A paramagnetic gas at room temperature $$27\ ^\circ\text{C}$$ is placed in an external uniform magnetic field of magnitude $$B=1.5\ \text{T}$$. The atoms of the gas have magnetic dipole moment $$\mu=1.0\mu_B$$ where $$\mu_B=\frac{eh}{4\pi m}$$ 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 is $$x\times10^{-4}\ \text{eV}$$. The value of $$x$$ is

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

The container $$A$$ in the figure holds an ideal gas at a pressure of $$5.0\times10^5\ \text{Pa}$$ at $$27\ ^\circ\text{C}$$. 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 $$1.0\times10^5\ \text{Pa}$$ at $$127\ ^\circ\text{C}$$. 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 $$\text{kPa}$$ is

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

The charge on a small point object at $$A$$ produces an electric potential $$V=3\ \text{V}$$ 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 $$3\ \mu\text{C s}^{-1}$$. To maintain the potential of $$3\ \text{V}$$ 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 $$10\ \mu\text{C}$$ and the direction of its velocity is perpendicular to $$OD$$. The resulting magnetic field $$B$$ at this instant at the location $$O$$, such that $$OD=1.0\ \text{m}$$, is

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 a second time and the growth of current $$I$$ follows curve 2. The following conclusion or conclusions may be drawn.

The Earth is revolving around the Sun of mass $$M$$ in a circular orbit of radius $$r$$ with a period of revolution $$T=1\ \text{year}$$. Suppose that during the revolution of the Earth, at any instant, the mass of the Sun instantaneously becomes double, that is $$2M$$. The correct alternative or alternatives are, assuming that the Sun and the Earth are point masses,

An optical fibre consists of a glass core of refractive index $$n_1$$ surrounded by a cladding of 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 or options.

A series $$LCR$$ circuit fed with AC has resonant angular frequency $$\omega_0=2.0\times10^4\ \text{rad s}^{-1}$$. When the same circuit is driven at an angular frequency of $$2.5\times10^4\ \text{rad s}^{-1}$$, it has an impedance of $$1.0\ \text{k}\Omega$$ and phase constant of $$\phi=45^\circ$$. The values of $$L$$, $$C$$ and $$R$$ for this circuit may be

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

In the given combination of capacitors $$C_1=4\ \mu\text{F}$$, $$C_2=2\ \mu\text{F}$$, $$C_3=2\ \mu\text{F}$$, $$C_4=4\ \mu\text{F}$$ and $$C_5=3\ \mu\text{F}$$, a source of $$12\ \text{V}$$ is connected across the points $$A$$ and $$B$$. Then the

One mole of an ideal monoatomic gas, initially at temperature $$T$$, is compressed to $$\frac18$$ 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

A solid ball of mass $$M$$ and radius $$R$$ 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 $$h=\alpha R$$, where $$0\le\alpha\le1$$. Choose the correct option or options.

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 $$n=2$$,

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

A uniform rod $$OA$$ of length $$L$$ is freely pivoted at one end $$O$$. Let $$C$$ be the midpoint, that is $$OC=CA=\frac L2$$. Choose the correct statement or statements.

A boy stands on a stationary ice boat on a frictionless horizontal flat iceberg. The boy and the boat have a combined mass of $$M=60\ \text{kg}$$. Two balls of masses $$m_1=10\ \text{kg}$$ and $$m_2=20\ \text{kg}$$ 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 $$v_{rel}=6\ \text{m s}^{-1}$$ relative to the boat just when the ball is being thrown. The resulting speed of the system of the boat and the boy is