Question 1

A particle moves along a straight line. Its displacement S varies with time t according to the law $s^{2}=a t^{2}+2 b t+c$ ( $\mathrm{a}, \mathrm{b}$ and c are constants). The acceleration of this particle varies as

Question 2

A ball A (mass $m_{1}$ ) moving with velocity v experiences an elastic collision with another stationary ball B (mass $m_{2}$ ). Each ball flies apart symmetrically relative to the initial direction of motion of ball A, at an angle $\theta$. Ratio of the masses of balls $\frac{m_{1}}{m_{2}}$ is

Question 3

A solid cylinder of mass $m$ is rolling without slipping on a rough horizontal surface, under the action of a horizontal force $F$ such that the line of action of $F$ passes through centre $C$ of the cylinder. Choose the correct alternative.

Question 4

A motor pump is used to deliver water at a certain rate $r$ from a given pipe. To obtain thrice as much water from the same pipe in the same time, the power of the motor has to be increased to

Question 5

Two small solid balls of masses $m$ and $8 m$ made up of same material are tried at the two ends of a thin weightless thread. They are dropped from a balloon in air. The tension T of thread during fall, after the motion of balls has reached steady state is

Question 6

Obtain the value of $\frac{e^{2}}{2 \varepsilon_{0} h c}$

Question 7

A sound source of fix frequency is in unison with an open end organ pipe of length 30.0 cm and a close end organ pipe of length 23.0 cm (both of same diameter). Both pipes are sounding their first overtone. If velocity of sound is $340 \mathrm{ ms}^{-1}$, frequency of sound source is nearly

Question 8

Solar constant for Earth is 2.0 cal per $\mathrm{cm}^{2}$ per minute. [ $1 \mathrm{cal}=4.2 \mathrm{ J}$ ]. Angular diameter of the Sun (as seen from the Earth) is $\frac{1}{2}$ (= half a degree). Treating Sun as a black body, its surface temperature is estimated to be nearly

Question 9

A concave mirror when placed in air has a focal length $f=20 \mathrm{ cm}$. The mirror is now placed horizontally and filled with a thin layer of water having refractive index $\frac{4}{3}$. The object is placed horizontally near the principal axis at a distance $d$ from the mirror such that a real, inverted image is formed at the same plane as the object, as shown in the figure. What is the value of $d$ ?

Question 10

When a sample of atoms is irradiated by neutrons, radioactive atoms are produced at a constant rate $R$, which decay with decay constant $\lambda$. The number of radioactive atoms accumulated after an irradiation time $t$ is given by

Question 11

Three uncharged capacitors $C_{1}=2 \mu F, C_{2}=3 \mu F$ and $C_{3}=5 \mu F$ are connected as shown in figure to one another at O and to points $\mathrm{A}, \mathrm{B}$ and D at potentials $V_{A}=300 \mathrm{ V}, V_{B}=200 \mathrm{ V}$ and $V_{D}=400 \mathrm{ V}$ respectively the potential $V_{o}$ at O is

Question 12

A cyclic process $1-2-3-4-1$ consisting of two isobars $2-3$ and $4-1$, an isochor $1-2$ and a process $3-4$ represented by straight line on a $\mathrm{P}-\mathrm{V}$ diagram, as shown in figure, involves n moles of an ideal gas. The gas temperatures at states $1,2,3$ and $4$ are $T_{1}, T_{2}, T_{3}$ and $T_{4}$ respectively. Also points 3 and 4 lie on the same isotherm. The work done by gas during the cycle is

Question 13

An insect of negligible mass is sitting on a block of mass M , tied with a spring of force constant K . The block performs simple harmonic motion vertically with amplitude A in front of a mirror which is include at $60^{\circ}$ with the vertical as shown. The maximum speed of insect relative to its image will be

Question 14

A concave lens of focal length 10 cm is placed between two convex lenses of focal length 10 cm and 20 cm at a separation of 5 cm between the first and second lens and 10 cm between the second and third lens. An object is placed at 30 cm in front of the first convex lens. The final image is formed beyond the third lens at a distance v from it. Then

Question 15

A point source $S$ of light is placed at a depth $d$ below the surface of water in a large and deep lake. Fraction of light that escapes in space above directly from water (refractive index $=\mu$ ) surface is given by

Question 16

A convex lens is held 45 cm above the bottom of an empty tank. The image of a point object at bottom of tank is formed $36 c m$ above the lens. Now a liquid is poured into the tank upto a height of $40 c m$ above the bottom. It is found that distance of image of same point object at the bottom of the tank is 60 cm above the lens. Refractive index of liquid is

Question 17

A potential of 5 V is applied across the face of a pure germanium plate of area $2 \times 10^{-4} \mathrm{ m}^{2}$ and of thickness $1.2 \times 10^{-3} \mathrm{ m}$. Concentration of carriers in germanium at room temperature is $1.6 \times 10^{6} \mathrm{ m}^{-3}$, Mobility of electrons and holes are $0.4 \mathrm{ m}^{2} \mathrm{ V}^{-1} \mathrm{ s}^{-1}$ and $0.2 \mathrm{ m}^{2} \mathrm{ V}^{-1} \mathrm{ S}^{-1}$ respectively. The current produced in germanium plate at room temperature,

Question 18

Fission of one nucleus of. ${ }^{235} U$ releases 200 MeV energy in average. Minimum amount of. ${ }^{235} U$ required to run 1000 MW reactor per year of continuous operation (assuming $30 %$ efficiency) is

Question 19

In a young's double slit experiment distance between slits is $d=1 \mathrm{ mm}$, Wavelength of light used is 600 nm and distance of screen from the plane of slits is $D=1 \mathrm{ m}$. the minimum distance between two points on the screen where intensity falls to $75 %$ of maximum intensity will be (Assume both sources of equal power).

Question 20

A ball is projected from horizontal ground. It attains a maximum height $H$ on its projectile path and there after strikes a stationary smooth vertical wall and falls on ground vertically below the point of maximum height. Assume the collision with wall to be perfectly elastic, the height of the point on the wall where the ball strikes is

Question 21

As shown in figure, a block of mass $m$ is projected from wall A with velocity $2 v_{0}$, on the rough surface with constant sliding friction to hit the wall B with velocity $v_{0}$. With what velocity same mass $m$ should be projected to hit the wall B with same velocity $v_{0}$ if the surface is now moving upward with an acceleration of $a=4 g$

Question 22

A sphere of radius $R$, is charged with volume charge density $\rho$ such that $\rho \propto r$ ( $r$ is distance from Centre). Variation of electric field $E$ with $r$ (For all values of $r: r \leq R$ and $r>R$ ) is best represented by

Question 23

A system of capacitors $C_{1}=4 \mu F, C_{2}=1 \mu F, C_{3}=2 \mu F$ and $C_{4}=3 \mu F$ connected across a battery of emf $\mathrm{E}=15$ V is shown in figure. The charge that will flow, through the switch K , when it is closed, is

Question 24

A simplification of a kind of interlock is shown in figure. All surfaces are smooth and frictionless. The body $m$ has a mass $\mathrm{m}=1 \mathrm{ kg}$ and the block $M=15 \mathrm{ kg}$. The time ' m ' takes to reach the base if it is released at height $\mathrm{h}=4$ meter above the base of M , is [ use $\mathrm{g}=10 \mathrm{ ms}^{-2}$ ]

Question 25

A number $n$ of identical balls, each of mass m and radius $r$ are stringed like beads at random and at rest along smooth, rigid horizontal rod of length $L$ mounted between immovable supports; $\frac{r}{L}$ is small but not negligible.

Collision between balls, or between balls and supports, are perfectly elastic. One of the balls is struct horizontally so as to acquire a speed $v$. Resulting outward force felt by supports, averaged over a long time, is

Question 26

A cylindrical tumbler of diameter $d$ has smooth sides and smooth edge. A thin of length $L$ is balanced on the edge of the tumbler as shown in figure. The angle $\alpha$ that the rod makes with horizontal for this trick to work is

Question 27

End A of a uniform thin rod of length $2 L$ is in boiling water $\left(100^{\circ} \mathrm{C}\right)$ and end B is in melting ice $\left(0^{\circ} \mathrm{C}\right) . \mathrm{P}$ and Q are two points at distance $\frac{L}{2}$ from A and B respectively. A similar bent rod of length $\frac{3 L}{2}$ of same material and equal cross section is joined to rod $A B$ between points $P$ and $Q$ as shown in figure. Then

Question 28

Two stars of masses $M$ and $m(M=2 m)$ separated by a distance $d=3$ astronomical unit, revolve in circular orbit about their centre of mass with a period of 2 years. If $M_{s}$ is mass of Sun then

Question 29

A thin uniform rod of mass $M$ is bent into four adjacent semicircles of radius of curvature $R$ lying in same plane. Moment of inertia of the bent rod about an axis through one end A and perpendicular to plane of the rod is

Question 30

Three point charges $+q,-2 q$ and $+q$ are placed on $x$-axis at $x=-d, x=0$ and $x=+d$ respectively. The value of electric field at a point P on x axis at $x=r(r \gg d)$ is given by $E=\frac{1}{4 \pi \varepsilon_{0}} \frac{a Q}{r^{n}}$ (Here $Q=2 q d^{2}$ ). Then

Question 31

The frequency of the transverse oscillations of a proton (mass M ) trapped in a cylindrical relativistic electron beam of circular cross section of radius R and current $/$ is given by [assume that speed v of relativistic electrons $\approx \mathrm{c}$ (the speed of light in vacuum) and ignore magnetic effect

Question 32

Current $I$ flows through a long thin-walled metallic cylinder of radius $R$ with a thin longitudinal slit of width $\xi(\xi \ll R)$ running parallel to the axis of the cylinder. The magnetic induction $B$ produced at any point on the axis of the cylinder is approximately

Question 33

The reading of the ammeter, used in the electrical network shown below, is $20 m A$, a long time after the key $K$ is closed

The reading of the same ammeter, immediately after the key was closed was

Question 34

At the Earth's surface, a projectile is launched straight up at a speed of $10.0 \mathrm{ km} / \mathrm{s}$. Height to which it will rise is $[g$ at surface of Earth $=9.8 \mathrm{ ms}^{-2}$ and radius of earth $\mathrm{R}=6400 \mathrm{ km}$ ]

Question 35

A small sphere of mass 2.00 g is released from rest in a large cylindrical vessel filled with oil. The resistive force due to viscosity of oil acting on sphere is proportional to its velocity. Sphere approaches a terminal speed of 5.00 $\mathrm{cm} / \mathrm{s}$. The time it takes the sphere to reach $90.0 %$ of its terminal speed is approximately.

Question 36

A static point charge $Q$ is located just above the centre $C(\delta 0)$ of a horizontal circle of radius $R$ on its geometric axis, as shown in figure. The magnitude of electric flux through this circle is

Question 37

Three small identical neutral metal balls are at the vertices of an equilateral triangle. The balls are in turn touched to an isolated large charged conducting sphere whose centre is on a line perpendicular to the plane of triangle and passing through its centre. As a result the first and second balls have acquired charges $q_{1}$ and $q_{2}$ respectively. The charge acquired by the third ball is [Assume that charge and potential of large spherical conductor change insignificantly in charging od the balls and that charges on balls are spherically symmetric]

Question 38

Voltage across the load L is controlled by using circuit as shown in figure. P is a potentiometer. Resistance $R_{L}$ of the load and $R_{P}$ of the potentiometer are equal to R . Load L is connected to the middle of potentiometer. Input voltage V is constant. If now $R_{L}$ is doubled, the voltage across load will change by a factor

Question 39

A small block A of mass 2 kg is attached to a spring of force constant $1200 \mathrm{Nm}^{-1}$, and rests on a smooth horizontal surface at $\mathrm{x}=0$ as shown in figure. A second block B of mass 1 kg slides along the surface towards A at $6 m s^{-1}$ and sticks to it. Assuming that the collision occurs at $t=0$, position $x$ (in meter) of block $A$ as a function of time $t$ is expressed as

Question 40

Two plane glass testing slides each of surface area A are stuck with each other by a small water drop squeezed between them as an extremely thin film of thickness $d$. If the surface tension of water be $T$ and the contact be zero, then the force required to pull apart the two glass plates will be

Question 41

The rate of flow of a certain liquid of viscosity $\eta$ through a horizontal capillary of length $l$ and radius $r$ is $Q$ when the pressure head at the inlet is just twice the atmospheric pressure. The rate of flow of the same liquid through another capillary of length $2 l$ and radius $2 r$ when the inlet pressure head is 4 times the atmospheric pressure will be (The outlet being open to atmosphere in each case)

Question 42

A uniform rod of the material of Young's modulus $Y$ is pushed over a smooth horizontal surface by a constant horizontal force F . The area of cross-section of the rod is A . The compressional strain in the rod is

Question 43

A total charge $Q$ is uniformly distributed over a non - conducting ring of radius $r$. There is a time varying magnetic field perpendicular to its plane and changing at the uniform rate of $\frac{d B}{d t}$. The magnitude of induced tangential electric field $E$ on ring is

Question 44

DC emf of 15 V is applied to a circuit containing 5 H inductance and $10 \Omega$ resistance in series at $\mathrm{t}=0$. The ratio of the currents in the circuit at $t=0.5 \mathrm{sec}$ and at $t=1.0 \mathrm{sec}$ is

Question 45

An insulating rod of length/ carries charge $q$ distributed uniformly II over its length. The rod is pivoted at its midpoint and is rotated at a frequency f (in Hz ) about an axis perpendicular to the rod passing through the point at the pivot. The magnetic moment of the system is

Question 46

A circular loop of radius $r$ is placed inside another circular loop of radius $R(R \gg r)$. The loops are coplanar and concentric. The manual inductance $(\mathrm{M})$ of the system is proportional to

Question 47

The amplitude of the electric and magnetic fields associated with a beam of light of intensity $477.9 \mathrm{ W} / \mathrm{m}^{2}$ are, respectively,

Question 48

Given that the critical angle of incidence for total internal reflection within a transparent material when placed in air is $45^{\circ}$. The Brewster's angle of incidence for light propagating from air to the transparent material will be

Question 49

A hydrogen atom is in ground state ( $n=1$ ). The magnetic field produced by revolving electron, at centre of atom is $B_{0}$. Atom is excited to state ${ }_{n=4}$. According to Bohr model, the correct alternative(s) is/are

Question 50

In an experimental set up to study the photoelectric effect a point source of light of power 3.2 mW is used. The source emits mono energetic photons of energy 5 eV and is located at a distance $\mathrm{d}=0.8 \mathrm{ m}$ from centre of a stationary metallic sphere of work function $\mathrm{W}=3.0 \mathrm{eV}$. The radius of the sphere is $\mathrm{R}=8 \mathrm{ mm}$. Assume that the sphere is isolated and photo electrons are instantly swept away after emission. Also assume that the efficiency off photoelectric emission is one for every $10^{6}$ photons. In the present set up

Question 51

Two identical Carnot (cycles) engines operate between maximum and minimum temperatures $T_{1}$ and $T_{2}$ and volume limits, $V_{a}, V_{b}, V_{c},$ and $V_{d}$ as shown in figure. Given that $\frac{V_{c}}{V_{a}}=e^{3}$ and $\frac{T_{1}}{T_{2}}=e$ (e is the base of natural logarithm). Engine 1 operates on mono atomic gas while engine 2 on diatomic gas. Choose correct alternatives

Question 52

In a certain machine two steel plates are separated by a hardened steel cylindrical roller (see fig). In operation, the plates move back and forth horizontally, perpendicular to the axis of roller, and the roller rolls freely between plates without slipping on either one. At a particular instant plate A is moving with a speed of $18 \mathrm{ cm} \mathrm{sec}{ }^{-1}$ to the right and an acceleration of $30 \mathrm{ cm} \mathrm{sec}^{-2}$ to the left, and the plate B is moving with a speed of $6 \mathrm{ cm} \mathrm{sec}{ }^{-1}$ to the right and an acceleration of $8 \mathrm{ cm} \mathrm{sec}^{-2}$ to the left. At that instant, for the roller

Question 53

Each of 9 sides of frame ACDEF b has resistance R (Nine in all) A current $I$ enters at A and leaves at B. Choose the correct alternatives.

Question 54

A long uniform rod of length $L$ and mass $M$ is pivoted vertically on a horizontal, friction less pivot at its lower end. The rod is released from rest in its vertical position OA (see figure). It falls off without slipping at O . At the instant the rod is horizontal,

Question 55

There are four layers of glass plates, placed on top of each other such that bottom one has thickness $a_{1}$ and refractive index $n_{1}=2.7$. Next one has thickness $a_{2}$ and refractive index $n_{2}=2.43$. The third one and the top one has thickness $a_{3}$ and $a_{4}$ and refractive indices $n_{3}$ and $n_{4}$ respectively. Three rays starting at the same moment from $A_{1}, A_{2}$ and $A_{3}$ reach pints $B_{2}, B_{3}, B_{4}$ at the same time, with their angles of incidence being critical angle. You are given $A_{1} B_{1}=A_{2} B_{2}=A_{3} B_{3}=A_{4} B_{4}=b=10 \mathrm{ mm}$. Choose correct statement (s).

Question 56

In an isolated asteroid of radius $R$ and uniform density $\rho$, a spherical cavity of diameter $A C=R$ is excavated,

where $C$ is centre of asteroid. Choose correct alternative(s)

Question 57

A small positively charged ball of mass $m$ is suspended by a long insulating thread of negligible mass. Other positively charged small ball is oved very slowly from a large distance (along horizontal direction) until it is at original position A of first ball. As a result, the first ball rises by $h$ to position B such that $h \ll l$. Choose the correct statement(s).

Question 58

A rope of mass $m$ and length $L$ is suspended vertically. A mass $M$ is suspended from bottom of the rope. A transverse wave is produced on the rope, which travels the length of rope in time $t$ choose the correct statement(s)

Question 59

A long solenoid having 1000 turns per meter carries a current of 1 A . It has a soft iron core of $\mu_{r}=1000$. The core is heated beyond the Curie temeperature ( $T_{c}$ ). Then

Question 60

A thin infinitely long metal sheet of appreciable finite width b carrying current $I$ (distributed uniformly through out of its cross section) parallel to its length is placed in an external magnetic field $B_{e}$ Parallel to its plane and perpendicular to the direction of current

Question 1

A point mass m moves in a straight line under a retardation $k v^{2}$ [where $k$ is a positive constant and $v$ is the instantaneous velocity]. The initial velocity of the point mass is $u$. The displacement of the point mass at time t is

Question 2

In the arrangement shown in figure, 'a' represents the magnitude of acceleration of small blocks A and B while ' T ' is the tension in the massless string passing over the frictionless and massless pulley. The sum of the masses of blocks A and B is constant. For this system, a linear relationship can be obtained between

Question 3

A thin uniform circular ring of mass m is rolling without slipping down an inclined plane of inclination $30^{\circ}$ with the horizontal. The coefficient of friction between the ring and the surface is $\mu$. The correct statement is

Question 4

A bullet of mass m can penetrate a target (a heavy block of mass M ) up to a distance S , when the target M is held stationary by a stopper P (shown in figure). Up to what distance $\mathrm{S}^{\prime}$ the bullet will penetrate if the block of mass M is free to move (i.e. when the stopper P is removed) on the frictionless surface T .

Question 5

Knowing that the atomic masses of Al and Mg are respectively ${ }_{13}^{25} \mathrm{Al}=24.990432 \mathrm{u}$ and ${ }_{12}^{25} \mathrm{Mg}=24.985839 \mathrm{u}$ while electron mass is often expressed as $\mathrm{m}_{\mathrm{e}}=0.511 \mathrm{MeV}$, the Q value (energy liberated) of the $\beta$ decay nuclear reaction ${ }^{25} \mathrm{Al} \rightarrow{ }^{25} \mathrm{Mg}+\mathrm{e}^{+}+v$ in MeV is

Question 6

A block of mass m , lying on a rough horizontal plane, is acted upon by a horizontal force P and simultaneously by another force Q acting at an angle $\theta$ from the vertical as shown. The block will remain in equilibrium if the coefficient of friction between the block and the surface $S$ is

Question 7

Knowing that the acceleration due to gravity on the Earth surface is g and the radius of the Earth is R , a small body of mass m falls on the Earth from a height $\mathrm{h}=\frac{\mathrm{R}}{5}$ above the Earth's surface. During the freefall, the potential energy of the falling body decreases by

Question 8

At some instant, a motor car is moving on a circular path of radius 600 m , with a speed $\mathrm{u}=30 \mathrm{ ms}^{-1}$. If its speed is increased at a rate of $2 \mathrm{ ms}^{-2}$, the magnitude of the acceleration of the car at that instant is

Question 9

A cricket ball, thrown across a field, is at heights of $\mathrm{h}_{1}$ and $\mathrm{h}_{2}$ above the point of projection, at time $\mathrm{t}_{1}$ and time $t_{2}$ after the throw, respectively. It is then caught by the wicket keeper at the same height as that from which it was thrown. The Time of Flight (T) of the ball is

Question 10

A plate of mass M is placed on a horizontal frictionless surface S . A block of mass m is placed on the plate. The coefficient of dynamic friction between the block and the plate is $\mu$. If a horizontal force $\mathrm{F}=2 \mu \mathrm{mg}$ is applied to the block (as shown), the acceleration of the plate will be

Question 11

A simple pendulum, with a bob of mass m , oscillates in a vertical plane, with an angular amplitude $\theta_{0}$. The tension in its string when it passes through the mean position is 2 mg . Neglecting the effect of air friction and the viscosity of air, the angular amplitude $\theta_{0}$ is

Question 12

Because of their mutual gravitational attraction, four identical planets each of mass m are orbiting in a circular path of radius r in the same sense (angular direction). The magnitude of the velocity of each planet is

Question 13

A rigid square sheet of size $2 \mathrm{ m} \times 2 \mathrm{ m}$ is hinged at the middle of the vertical edges to serve as a door which can turn about the horizontal axis $\mathrm{OO}^{\prime}$. A fluid of density $\rho$ fills the space to the left of the sheet up to its top. The horizontal force F required (to be applied at the lower edge) to hold the sheet vertical is

Question 14

A major artery in human body, with radius 0.4 cm , carries blood at a flow rate of 5.0 cubic centimeters per second. The pressure difference of blood per meter length of the artery is nearly [Given that the coefficient of viscosity ( $\eta$ ) of blood at body temperature is $4.0 \times 10^{-3} \mathrm{ Pa}$.s and the density of mercury is $13.6 \mathrm{ g} / \mathrm{cm}^{3}$ ]

Question 15

If P represents radiation pressure, E represents radiation energy striking per unit area per unit time and c represents speed of light then the possible values of non-zero integers $x, y$ and $z$ such that $\mathrm{P}^{x} \mathrm{E}^{y} \mathrm{c}^{z}$ is dimensionless, may be

Question 16

A large tank, open at the top, has two small holes in the vertical wall. One is a square hole of side 's' at a depth $h$ below the top and the other is a circular hole of radius r at a depth 4 h below the top (given that $\mathrm{s} \ll \mathrm{h} ; \mathrm{r} \ll \mathrm{h})$. When the tank is completely filled up to the brim with water, the quantity of water flowing out per second from each hole is the same, then r is equal to

Question 17

A pendulum consists of a heavy but very small bob of mass M suspended at the end of a rigid rod of mass m and length L . The time period of small oscillations in the vertical plane, about a horizontal axis through the upper end of the rod is

Question 18

A transverse wave is travelling along a long stretched string from left to right (along + ve $x$ direction). The snapshot of a small part of the string at any moment $t$ is shown in the figure. At this particular instant

Question 19

Two tuning forks, with natural frequency 700 Hz each, move relative to a stationary observer. Fork (1) moves towards the observer while the fork (2) moves away from the observer. Both the forks move with same velocity $v$ on the same line. The observer, standing between the two forks, hears 4 beats per sec. Using the speed of sound in air as $v_{s}=350 \mathrm{ ms}^{-1}$, the speed of each tuning fork is

Question 20

The speed of sound in a mixture of 1 mole of Helium (molar mass $=4 \mathrm{ g}$ ) and 2 moles of oxygen (molar mass $=32 \mathrm{ g}$ ) at $27^{\circ} \mathrm{C}$ is nearly

Question 21

Three identical large metal plates are kept parallel and close to each other. Each plate can be treated as an ideal black body and has very high thermal conductivity. The first and third plates are maintained at high temperature $\mathrm{T}_{1}=3 \mathrm{ T}$ and $\mathrm{T}_{3}=2 \mathrm{ T}$. The temperature $\mathrm{T}_{2}$ of the middle (i.e. second) plate under steady state condition is

Question 22

A thin uniform circular disc of radius ' $a$ ' is placed in XY plane with its center at origin ( 0,0 ). A small circular disc of radius $b$ with center at ( $c, 0$ ) is cut and taken out to create a hole. The center of mass of the remaining disc is at

Question 23

One mole of an ideal monoatomic gas, contained in a cylinder fitted with movable piston, is originally at $\mathrm{P}_{1}, \mathrm{ V}_{1}$ and $\mathrm{T}_{1}=27^{\circ} \mathrm{C}$. The gas is slowly heated. Initially 8.31 watt-hour of energy is added to it; at the same time it is allowed to expand at constant pressure to a new state $\mathrm{P}_{1}, \mathrm{ V}_{2}$ and $\mathrm{T}_{2}$. The correct option is

Question 24

A non-conducting solid sphere, of radius R , with its center at A , has a spherical cavity of diameter R with center at B as shown. There is no charge in the cavity while the solid part has a uniform volume charge density $\rho$. Electric potential at the center of the sphere (at point A) is $V=\frac{k \rho R^{2}}{12 \epsilon_{0}}$ (in SI units) where the value of $k$ is

Question 25

Energy from the Sun falls on the Earth surface at the rate of $1400 \mathrm{ W} / \mathrm{m}^{2}$, which is known as solar constant. The respective rms values $\mathrm{E}_{\mathrm{rms}}$ and $\mathrm{B}_{\mathrm{rms}}$ of electric and magnetic fields in the sunlight (electromagnetic radiation) reaching Earth surface are (Take speed of light $\mathrm{c}=3 \times 10^{8} \mathrm{ ms}^{-1}$ )

Question 26

The figure below depicts the voltage wave forms of binary input signals A and B and the output signal C of a certain logic gate.

The logic gate is

Question 27

Two electric charges, $+q$ at the origin $\mathrm{O}(0,0)$ and $-2 q$ at the point $\mathrm{A}(6,0)$ are placed on $x$ axis. The locus of the point P in $x-y$ plane where the potential vanishes $(\mathrm{V}=0)$ is

Question 28

In the circuit shown, the Zener diode is an ideal one with breakdown voltage of 5.0 volt. The values of the resistances are $\mathrm{R}_{\mathrm{S}}=10 \mathrm{k} \Omega$ and $\mathrm{R}_{\mathrm{L}}=1 \mathrm{k} \Omega$. The current through the resistances, when the supply voltage is 11.0 V , is

Question 29

In an accelerator the electrons are accelerated up to an energy of 50 MeV . The electrons do not emerge continuously from the accelerator rather they come in pulses at time interval of 5.0 milliseconds. Each pulse has a much shorter duration of 200 nanoseconds. Electron current during the pulse is 100 mA , while the current is zero between the two successive pulses (see figure), then

Question 30

Two infinitely long straight parallel wires perpendicular to the plane of the paper are 5 m apart. One of the wires, P carries current I out of the plane of the paper and the other, Q carries the current I into the plane of paper. The magnetic field B at the origin O of the coordinate system with $x$ and y axes as perpendicular and parallel to PQ , respectively, is [Given $\mathrm{OP}=4 \mathrm{ m}$ and $\mathrm{OQ}=3 \mathrm{ m}$ ]

Question 31

Charge q is uniformly distributed over the surface of a thin non-conducting annular disc of inner radius $\mathrm{R}_{1}$ and outer radius $\mathrm{R}_{2}$. The disc is made to rotate with constant frequency f , about an axis passing through the center of the annular disc and perpendicular to its plane. The magnetic moment of the disc is

Question 32

For a resistance R and capacitance C in series, the impedance is twice that of a parallel combination of the same elements when used with an AC voltage of frequency $f$. The frequency $f$ of the applied emf is

Question 33

In an experiment on photoelectric effect on a metal surface, one finds a stopping potential of 1.8 V for the wavelength of 300 nm and a stopping potential of 0.9 V for the wavelength of 400 nm . The cutoff wavelength $\lambda_{0}$ (the maximum wavelength that can produce photoelectric effect) for the metal is

Question 34

Given that the power dissipated in $5 \Omega$ resistance is 7.2 W in the circuit shown. Statement (1): Power dissipated in $6 \Omega$ resistance is 6 W . Statement (2): Potential difference $\mathrm{V}_{\mathrm{AB}}$ between A and B is $\mathrm{V}_{\mathrm{AB}}=12.4 \mathrm{ V}$ Then

Question 35

A transparent and homogeneous sphere of glass of radius r is immersed in water (refractive indices of glass and water being ${ }_{a} \mu_{g}=\frac{3}{2}$ and ${ }_{a} \mu_{w}=\frac{4}{3}$ ). The image of a point object O , located at distance d on its axis in front of the sphere, is formed at point I at the same distance d from the sphere on the opposite side as shown.

The distance d is equal to

Question 36

A certain substance, with a dielectric constant $\mathrm{k}=2.5$ and the dielectric strength $\mathrm{E}=1.8 \times 10^{7} \mathrm{ N} / \mathrm{C}$, completely fills the space between the plates of a parallel plate capacitor (with circular plates) of capacitance $\mathrm{C}=72.0 \mathrm{nF}$. The minimum diameter of the circular plates, to ensure that the capacitor can withstand a potential difference of $\mathrm{V}=4.0 \mathrm{kV}$, is

Question 37

A uniform solid sphere of radius R rolls without slipping on a rough horizontal surface with a forward velocity $v$ of its center. On its way, it suddenly encounters a small step of height 0.2 R as shown. The angular velocity of the sphere just after the impact is [given that the sphere does not bounce back, rather it goes ahead up the step]

Question 38

The magnetic field (B) produced by the current $i$ flowing through the sides of a square loop of side $\ell$, at a point P at distance $x$ from the center of the square, on the axis perpendicular to the plane of the square loop and passing through its center, is

Question 39

A linear positive charge distribution, with linear charge density $\lambda$ coulomb per meter, extends along $+x$ - axis from $x=0$ to $x=\infty$.

The electric field $\vec{E}$ at any point $\mathrm{P}(0, \mathrm{y})$ on the y - axis

Question 40

Imagine a situation, in which an infinite sheet with positive charge $+\sigma$ per unit area lies in the xy-plane and a second infinite sheet with negative charge $-\sigma$ per unit area lies in the yz-plane. The net electric field E at any point $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ [that does not lie on either of these planes xy or yz ] can be expressed as

Question 41

For the electric field E , in a region of space where a non-uniform, but spherically symmetric distribution of charge has a charge density $\rho(\mathrm{r})$ as $\quad \rho(r)=\rho_{0}\left(1-\frac{r}{R}\right)$ for $r \leq R, \quad$ one can say that $\rho(r)=0 \quad$ for $r \geq R$,

Question 42

A typical network of resistances $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ shown below extends to infinity towards the right. The total resistance $R_{\text {effective }}$ of this network between points $A$ and $B$ is

Question 43

A cylindrical cavity of diameter 'a' exists inside a long solid cylinder of diameter '2a' as shown in figure. Both the cylinder and the cavity are taken to be infinitely long. The axis of the cavity is parallel to the axis of the cylinder and is at a distance $\frac{\mathrm{a}}{2}$ from it. A uniform current of current density $\mathrm{J}\left(\mathrm{Am}^{-2}\right)$ flows through the cylinder along its length and not through the cavity. The magnitude of the magnetic field at a point P on the surface of the cylinder lying farthest from the axis of the cavity, is

Question 44

A thin uniform rod, of length $\ell=0.200 \mathrm{ m}$ with negligible mass, is attached to the floor by a frictionless hinge at a fixed point P . A horizontal spring connects the other end of the rod to a vertical wall. The rod is in a uniform magnetic field $\mathrm{B}=0.500$ tesla directed into the plane of paper. There is a current $\mathrm{i}=10.0 \mathrm{ A}$ in the rod in the direction shown. Force constant of the spring is $5.00 \mathrm{ N} / \mathrm{m}$. The rod is in equilibrium at $\theta=\tan ^{-1} \frac{4}{3}$ Statement (1) Torque on the rod due to magnetic force is 0.1 Nm clockwise Statement (2) In equilibrium the energy stored in the spring is 0.039 J

Question 45

The electric flux through a certain area of a dielectric medium is $\phi=\left(8.00 \times 10^{3}\right) t^{4}$ in SI units. The displacement current through that area is 12.5 pA at a time $\mathrm{t}=20.0 \mathrm{ ms}$. The dielectric constant of the dielectric medium is

Question 46

A thin equi-convex lens of flint glass (refractive index $\mu_{1}$ ) is kept coaxially in contact with another thin equi-concave lens of crown glass (refractive index $\mu_{2}$ ). The system is completely immersed in water $\left({ }_{a} \mu_{w}=\frac{4}{3}\right)$.

Parallel rays of light incident parallel to the principal axis in water are focused by this system at a distance of 24 cm beyond the system. The thickness of the system is negligible. If the radius of curvature of each surface is $\mathrm{R}=20 \mathrm{ cm}$, the difference ( $\mu_{1}-\mu_{2}$ ) is

Question 47

Two identical large thin metal plates carrying charges $+\mathrm{q}_{1}$ and $+\mathrm{q}_{2}\left(\mathrm{q}_{1}>\mathrm{q}_{2}\right)$, respectively, are kept close at a distance d apart and parallel to each other to form a parallel plate capacitor of capacitance C . The potential difference between the plates is

Question 48

In a certain electrical network, the three nodes $\mathrm{A}, \mathrm{B}$ and C are each at a potential of 1.0 volt while the node D is at a potential 2.0 volt. The potential at the Node O in volt is

Question 49

A thin uniform metallic rod, of length $\ell=1.0 \mathrm{ m}$ and area of cross section $\mathrm{A}=2 \mathrm{ mm}^{2}$, is made to rotate with angular velocity $\omega=400 \mathrm{rad} / \mathrm{s}$ in a horizontal plane about a vertical axis through one of its ends. The density and the Young's modulus of the material of the rod are $\rho=10^{4} \mathrm{ kg} \mathrm{ m}^{-3}$ and $\mathrm{Y}=2.0 \times 10^{11} \mathrm{Nm}^{-2}$. Taking r as the distance of a point on the rod from the axis of rotation, the

Question 50

One end of a long and thin rope, stretched horizontally with a tension $\mathrm{T}=8 \mathrm{ N}$, along $x$ axis, is supporting a weight after passing over a pulley fixed on a vertical pole (see figure). At the other end, a simple harmonic oscillator (a clamped iron rod along the axis of a solenoid fed with AC voltage and oscillating between north and south poles) at $x=0$, generates a transverse wave of frequency 100 Hz and an amplitude of 2 cm , in the rope. The wave propagates along the rope. The mass per unit length of the rope is $20 \mathrm{ g} / \mathrm{m}$. Ignoring the effect of gravity (on the rope), the correct option(s) is /are

Question 51

Nuclei of a radioactive element A are being produced at a constant rate $\alpha$. The element A has a decay constant $\lambda$. If there are $\mathrm{N}_{0}$ nuclei at $\mathrm{t}=0$, then

Question 52

In Young's double slit experiment, a fine beam of coherent monochromatic light of wavelength $\lambda=600 \mathrm{ nm}$ is incident on identical slits $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ at separation d. The intensity at the central maximum formed at O is $\mathrm{I}_{\text {max }}$ and the angular fringe width is $\beta=0.1^{\circ}$. When a thin transparent film is placed in front of the slit $\mathrm{S}_{2}$, the intensity at O changes. It is found that the smallest thickness of the film, for which the intensity at O becomes half the maximum intensity ( i.e. $\frac{\mathrm{I}_{\text {max }}}{2}$ ), is 250 nm . Neglecting the absorption of light by the film, the zero order fringe earlier at O now forms at $\mathrm{O}^{\prime}$ where $\mathrm{OO}^{\prime}=0.5 \mathrm{ mm}$. Choose correct option(s)

Question 53

An insulated non-conducting solid sphere of radius 'a', carrying a positive charge +4 Q uniformly distributed throughout its volume, is surrounded by a concentric thick conducting spherical shell of inner radius $b$ and outer radius $c$. This thick shell carries a negative charge - 2Q (see figure). The correct option(s) is/are

Question 54

A single electron orbits around a stationary nucleus of charge +Ze in a hydrogen-like atom, where Z is the atomic number and e is the magnitude of the charge on an electron. It requires 47.25 eV to excite the electron from second Bohr orbit to the third Bohr orbit. Ionization energy of hydrogen atom is 13.6 eV . Then

Question 55

One mole of an ideal monoatomic gas of molecular mass M undergoes a cyclic process (ABCA) shown in the figure as a density ( $\rho$ ) versus pressure (P) curve. The correct option(s) is/are

Question 56

Two ideal inductors $\mathrm{L}_{1}=\mathrm{L}_{2}=\mathrm{L}$ and three identical resistors $\mathrm{R}_{1}=\mathrm{R}_{2}=\mathrm{R}_{3}=\mathrm{R}$ have been connected to a DC source of emf E as shown in the circuit. When the key K is kept pressed (closed) for a long time, the current through the resistance $\mathrm{R}_{1}$ on the extreme right is measured to be I . Immediately after releasing (switching off) the key, the current through the resistors is

Question 57

A particle of mass $m$ moves along $x$ axis with its potential energy as $U(x)=\frac{\alpha}{x^{2}}-\frac{\beta}{x}$ where $\alpha$ and $\beta$ are positive constants. The particle is released from rest at $x_{0}=\frac{\alpha}{\beta}$. Then

Question 58

Two blocks A and B , of masses M and 2 M , respectively, are connected by a massless spring of natural length $\mathrm{L}_{0}$ and spring constant K . The blocks are initially at rest on a smooth horizontal floor with spring at its natural length $\mathrm{L}_{0}$. A third block C of mass M , identical to that of block A , moves on the floor with speed $v$ along the line joining A and B and collides with A elastically. In the subsequent motion

Question 59

A small block B of mass $\mathrm{m}=0.25 \mathrm{ kg}$, lying on a frictionless horizontal table, is attached to a massless cord (breaking strength 40 N ) passing through a narrow hole C at the center of the table. Initially when the block is revolving in a circle of radius $\mathrm{r}_{0}=0.80 \mathrm{ m}$ about a vertical axis through the hole, with a tangential speed of $\mathrm{v}_{0}=4.00 \mathrm{ m} / \mathrm{s}$; the tension in the string is $\mathrm{T}_{0}$ and the kinetic energy of the block is $\mathrm{K}_{0}$. The string is then pulled down slowly from below, decreasing the radius of circular path from $\mathrm{r}_{0}$ to r so that the kinetic energy of the block is now K and the tension in the string is T . As a result

Question 60

A circular coil of thin insulated copper wire ( $\mathrm{N}=2000$ turns), wrapped around an iron cylinder of cross-section area $\Delta \mathrm{S}=0.001 \mathrm{ m}^{2}$, is connected to a suspended type moving coil ballistic galvanometer. The suspended rectangular coil of the galvanometer is of mass $m=80 \mathrm{ g}$, length $\ell=5 \mathrm{ cm}$, breadth $\mathrm{b}=3$ cm and has $\mathrm{n}=100$ turns of fine copper wire wound on a non-metallic frame of ivory. This rectangular coil of the galvanometer is free to execute torsional oscillations in a radial magnetic field $\mathrm{B}=0.1$ tesla. The galvanometer is being used to measure the charge by employing the formula $\mathrm{q}=\frac{\mathrm{T}}{2 \pi} \frac{\mathrm{c}}{\mathrm{nAB}} \theta$. [Given that the moment of inertia of the oscillating coil about the vertical axis is $I=2.7 \times 10^{-6} \mathrm{ kg} \mathrm{ m}^{2}$ and the torsional constant (torsional rigidity) of the suspension fiber is $\mathrm{c}=3.0 \times 10^{-3} \mathrm{Nm} /$ radian : $\mathrm{A}=\ell \times b$ is the area of the coil] When the magnetic induction of 1.0 weber per meter ${ }^{2}$, perpendicular to the plane of the circular coil, is reversed (in opposite direction), a deflection of 40 mm is observed on a scale placed 1.0 meter away in front of the reflecting mirror attached with the suspension fiber of the rectangular coil. The correct statement(s) is/are

Question 1

A target of ${ }^{7} \mathrm{Li}$ is bombarded with a proton beam of current $10^{-4}$ ampere for 1 hour to produce ${ }^{7} \mathrm{Be}$ of activity $1.8 \times 10^{8}$ disintegrations per second. Assuming that bombarding of 1000 protons produces one ${ }^{7} \mathrm{Be}$ radioactive nucleus, the half-life of ${ }^{7} \mathrm{Be}$ is estimated to be approximately

Question 2

A long straight wire carrying a current $i=10 A$ and a rectangular metallic loop of dimensions $b \times c$ lie in the same plane as shown in the figure. The parameters are $a=10 \mathrm{ cm}, b=30 \mathrm{ cm}$ and $c=50 \mathrm{ cm}$. The mutual inductance of the system is nearly

Question 3

Impedance of a given series LCR circuit, fed with alternating current, is the same for two frequencies $f_{1}$ and $f_{2}$. The resonance frequency $f_{R}$ of the circuit is

Question 4

A lawn roller is a solid cylinder of mass $M$ and radius $R$. As shown in the figure, it is pulled at its center by a horizontal force $F$ and rolls without slipping on a horizontal surface. Then the

Question 5

A hydrogen atom ( $M_{\mathrm{H}}=1.67 \times 10^{-27} \mathrm{ kg}$ ), initially at rest, emits a photon and goes from the excited state $n=5$ to the ground state. The recoil speed of the atom is nearly

Question 6

Two nuclides $A$ and $B$ are isotopes. The nuclides $B$ and $C$ are isobars. All the three nuclides $A, B$ and $C$ are radioactive. You may then conclude that

Question 7

Numerical aperture of an optical fibre is a measure of

Question 8

Heavy stable nuclei have more neutrons than protons. This is because of the fact that

Question 9

An equi-concave lens of radii of curvature of the two surfaces numerically equal to 7 cm and refractive index $\mu=1.5$ has a small silver dot on the rear surface. As a result of this, a ray of light incident parallel to the principal axis gets reflected from its rear surface and then reflected also from the inner front surface. The ray after the second reflection emerges out of the thin lens and appears to focus at a point $P$ on the principal axis. The point $P$ lies

Question 10

Light emerges out uniformly from a point source placed at the focus of a concave mirror to give out a spherical wave front. As a result of reflection of the paraxial rays from the concave mirror, according to Huygen's theory the reflected light is in the form of a

Question 11

An equi-convex lens of focal length ' $f$ ' is cut along a diameter, in two halves (pieces). The two identical pieces of the lens are now arranged as shown in the figure on a common axis at a separation $f$ between the two. The image of an object $A B$ placed at $x=0$ cannot be formed at the distance $x=\xi$ from the object along the axis, for the value of $\xi$ as

Question 12

During the processes of annihilation of a stationary electron of mass $m_{0}$ with a stationary positron of equal mass, a radiation is emitted. The wavelength of the resulting radiation is

Question 13

The convex surface of a concavo-convex lens of refractive index 1.5 and radii of curvature $R_{1}=20 \mathrm{ cm}$ and $R_{2}=40 \mathrm{ cm}$ has been silvered so as to make it reflecting. The distance of a luminous object from the reflecting system when placed in front of it on its principal axis, so that the image coincides with the object is

Question 14

Two balls are projected from the top of a cliff with equal initial speed $u$. One starts at angle $\theta$ above the horizontal while the other starts at angle $\theta$ below. Difference in their ranges on ground is

Question 15

A solid block of mass 3 kg is suspended from the bottom of a 5 kg block with the help of a rope $A B$ of mass 2 kg as shown in the figure. When pulled by an upward force $F$, the whole system experiences an upward acceleration $a=2.19 \mathrm{ ms}^{-2}$. Choose the correct option

Question 16

A block $P$ of mass 0.4 kg is attached to a vertical rotating spindle by two strings $A P$ and $B P$ of equal length 1.0 m as shown in the figure. The period of rotation is 1.2 s . Tensions $T_{1}$ and $T_{2}$ in string $A P$ and $B P$ are

Question 17

A particle of mass $m$ moves in a straight line under the influence of a certain force such that the power ( $P$ ) delivered to it remains constant. Starting from rest, the straight-line distance travelled by the moving particle in time $t$ is

Question 18

A bullet is fired vertically up with half the escape speed from the surface of the Earth. The maximum altitude reached by it (ignore the effect of rotation of the Earth) in terms of radius of Earth $R$ is

Question 19

A can is a hollow cylinder of radius $R$ and height $h$. Its ends are sealed with circular sheets of the same material. The can is made of thin sheet metal of areal mass density $\sigma\left(\mathrm{kg} / \mathrm{m}^{2}\right)$. Moment of inertia of this closed can about its vertical axis of symmetry is

Question 20

A particle of mass $m$ is revolving in a horizontal circle on a frictionless horizontal table with the help of a string tied to it and passing through a hole at the center of the table. Two equal masses $M$ are attached to the other end of the string as shown. If one of the hanging masses $M$ is removed gently, the radius of the circular motion of $m$

Question 21

Three stars of equal mass $M$ rotate in a circular path of radius $r$ about their center of mass such that the stars always remain equidistant from each other. The common angular speed ( $\omega$ ) of rotation of the stars can be expressed as

Question 22

The density of a liquid is $\rho$ at the surface. The bulk modulus of the liquid is $B$. The increase $\Delta \rho$ in the density of the liquid at a depth $h$ from the surface is (with $\Delta \rho \ll \rho$ )

Question 23

Water flows at $1.2 \mathrm{ m} / \mathrm{s}$ through a hose of diameter 1.59 cm . The time required to fill a cylindrical container of radius 2 m to a height of $h=1.25 \mathrm{ m}$ will be nearly

Question 24

A police car, moving at speed of $108 \mathrm{ km} / \mathrm{hour}$, approaches a truck moving at $72 \mathrm{ km} /$ hour in opposite direction. The natural frequency of the siren of the car is 800 Hz and the surrounding temperature is $27^{\circ} \mathrm{C}$. The frequency heard by the truck driver as the car passes him

Question 25

A rope of mass $M$ and length $L$ hangs vertically. Time needed for a transverse pulse to travel from its bottom end to the support is

Question 26

The figure shows a smooth tunnel $A B$ (length $=2 \ell$ ) in a uniform density planet (say Earth) of mass $M$ and radius $R$. A small ball of mass $m$ is released from rest at the end $A$ of the tunnel. Acceleration due to gravity at surface of the planet is $g$. Time taken by the ball to reach the end $B$ is

Question 27

When the speaker $S_{1}$ is switched ON , the sound intensity at a point $P$ in a room is 80 dB . But when the speaker $S_{2}$ is switched ON ( $S_{1}$ is switched OFF ), the sound intensity at the same point $P$ in the room is 85 dB . The sound intensity level (in dB ) at the same point $P$ in the room, if the two speakers $S_{1}$ and $S_{2}$ are simultaneously switched ON , is (consider the speakers to be incoherent)

Question 28

A block $B$ of mass 0.5 kg moving, on a horizontal frictionless table at $2.0 \mathrm{ ms}^{-1}$, collides with a massless pan $P$ (at origin $O$ ) and sticks to it. The pan is connected at the end of a horizontal un-stretched (relaxed) spring of force constant $K=32 \mathrm{Nm}^{-1}$ as shown in figure. After the block collides, the displacement $x(t)$ of the block as a function of time $t$ is given by

Question 29

Which of the following functions does not represent a traveling wave?

Question 30

Two Carnot heat engines are connected in series such that the sink of the first engine is heat source of the second. Efficiency of the engines are $\eta_{1}$ and $\eta_{2}$ respectively. Net efficiency $\eta$ of the combination is given by

Question 31

An air bubble of radius 2 mm at a depth 12 m below the surface of water at temperature of $8^{\circ} \mathrm{C}$, rises to the surface where the temperature is $16^{\circ} \mathrm{C}$. Neglecting the effect of Surface Tension, the radius of the bubble at the surface is estimated to be

Question 32

Two soap bubbles of radii $a$ and $b$ coalesce to form a single bubble of radius $c$ under isothermal conditions. If the external pressure is $P_{A}$, then the Surface Tension ( $T$ ) of the soap solution is

Question 33

An open-end organ pipe 30 cm in length and a closed-end organ pipe 23 cm in length, both of equal diameter, are each sounding their first overtone and both are in unison at 1100 Hz . The speed of sound in air, is estimated to be nearly

Question 34

The figure shows a lagged bar $X Y$ of non-uniform cross section. One end $X$ of the bar is maintained at $100^{\circ} \mathrm{C}$ and the other end $Y$ at $0^{\circ} \mathrm{C}$. The variation of temperature along its length from $X$ to $Y$ in steady state is best represented by the curve.

Question 35

An ideal gas ( $n$ moles) is initially at pressure $P$ and temperature $T$. It is cooled isochorically to a pressure $\frac{P}{4}$. The gas is then expanded at a constant pressure so as to attain back its initial temperature $T$. Work done by gas during the entire process is

Question 36

Assuming the Sun to be a spherical body (radius $R_{S}$ ) of surface temperature $T$, the total radiation power received by Earth (radius $R_{E}$ ) at a distance $r$ from Sun is

Question 37

The figure shows five point-charges on a straight line. Separation between successive charges is 10 cm . For what values of $q_{1}$ and $q_{2}$ would the net force on each of the other three charges be zero?

Question 38

Two equal blocks, each of mass $M$, hang on either side of a frictionless light pulley with a light string. A rider of mass $m$ is placed on one of the blocks (as shown). When the system is released, the block with rider descends a distance $H$ till the rider is caught by a ring that allows the block to pass through. The system moves a further distance $D$ taking time $t$. In such a situation, the acceleration due to gravity is

Question 39

A very small electric dipole of dipole moment $\vec{p}$ lies along the $x$ axis (i.e. $\vec{p}=p \hat{i}$ ) in a non-uniform electric field $\vec{E}=\frac{c}{x} \hat{i}$ (where $c$ is a constant). The force on the dipole is

Question 40

A conducting thick spherical shell of radii $a$ and $b(b>a)$ has been charged with uniform surface charge density $-\sigma \mathrm{C} / \mathrm{m}^{2}$ on the inner and $+\sigma \mathrm{C} / \mathrm{m}^{2}$ on the outer surface. Then

Question 41

A spherical conductor is charged up to a potential of 450 V . The potential outside, at a distance 15 cm from the surface, is 300 V . Then

Question 42

Capacitors $C_{1}=3 \mu \mathrm{ F}, C_{2}=6 \mu \mathrm{ F}, C_{3}=4 \mu \mathrm{ F}$ and $C_{4}=1 \mu \mathrm{ F}$ are connected in a circuit as shown to a battery of 60 V . Now if key $K$ is closed, the charge that will flow through $K$ is

Question 43

The electrical conductivity of a sample of semiconductor is found to increase when the electromagnetic radiation of wave length just shorter than 2480 nm is incident normally on its surface. The band gap of the semiconductor is

Question 44

A U-shaped conducting wire of mass $m=10 \mathrm{ g}$, having length of its horizontal section as $\ell=20 \mathrm{ cm}$, is free to move vertically up and down. The two ends of the wire are immersed in mercury for proper electrical contact. The wire is in a homogenous field of magnetic induction $B=0.1 \mathrm{ T}$ as shown. The wire jumps up to a height $h=3 \mathrm{ m}$ when a charge $q$, in the form of a current pulse, is sent through the wire. Considering that the duration of the current pulse is very small compared to the time of flight, the charge $q$ passed through the wire is estimated to be nearly

Question 45

A direct vision spectroscope has been designed to obtain dispersion without deviation by arranging alternate inverted thin prisms of crown glass (refractive index $\mu_{1}=\sqrt{2}$ ) and flint glass ( $\mu_{2}=\sqrt{3}$ ) with refracting angle $\theta_{\text {flint }}=3^{\circ}$. The refracting angle $\theta_{\text {crown }}$ of the crown glass prism is

Question 46

Continuous and characteristic X-rays are produced when electron beam accelerated by a high potential difference of V volt (say) is made to hit the metallic target such as Molybdenum in a modern Coolidge tube. Let $\lambda_{\text {min }}$ be the smallest possible wavelength of continuous X-rays and $\lambda_{L \alpha}$ be the maximum wavelength of the characteristic X-rays. Then

Question 47

While performing an experiment for determining the focal length of a concave mirror by u-v method, a student recorded the given sets of the positions (in cm ) of the object and the corresponding image on the bench as (12, $51),(18,54),(30,50),(48,34),(42,42)$ and $(78,98)$. She used an optical bench of length 1.5 m and the mirror is fixed at the 90 cm mark on the bench. The maximum acceptable error in the location of the image is 0.2 cm . The reading (observation) that cannot be obtained from experimental measurement and has been incorrectly recorded, for a mirror of focal length $=24 \mathrm{ cm}$, is

Question 48

A parallel beam, of 6.0 mW radiation of wavelength 200 nm and of area of cross-section $1.0 \mathrm{ mm}^{2}$, falls normally on a plane metallic surface. If the radiations are completely reflected, the pressure exerted by the radiations on the metallic surface is estimated to be

IN THE FOLLOWING QUESTIONS ANY NUMBER OF OPTIONS 4, 3, 2 OR 1 MAY BE CORRECT

Question 49

A metal rod of mass $m$ and length $\ell$ slides on frictionless parallel metal rails of negligible resistance. A resistance $R$ is connected between the rails at their ends as shown in the figure. A uniform magnetic field $B$ is directed into the plane of paper perpendicular to the plane of rails throughout the space. The rod is given an initial velocity vo (towards right). No other force acts on the rod. Then

Question 50

The magnetic field $\vec{B}=2 \times 10^{-5} \sin \left\{\pi\left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right)\right\} \hat{j} T$ represents a plane electromagnetic wave travelling in space with $x$ in meter and $t$ in second. The correct statement(s) are

Question 51

The force $F(x)$ acting on a body of mass m changes with position $x$ (in meter) as shown. It is given that the potential energy $U(x)=0$ at $x=0$ Choose correct option(s).

Question 52

A deuteron of mass $M$ moving at speed $v$ collides elastically with an $\alpha$-particle of mass $2 M$, initially at rest. The deuteron is scattered through $90^{\circ}$ from initial direction of its motion with speed $V_{d}$ while the $\alpha$-particle is scattered with speed $V_{\alpha}$ at an angle $\theta$ from the initial direction of motion of deuteron. Then

Question 53

Two plane progressive waves travelling on a string as

$Y_{1}=2.5 \times 10^{-3} \sin (30 x-420 t)$

$Y_{2}=2.5 \times 10^{-3} \sin (30 x+420 t)$

Superimpose to produce a standing wave. The variables $x$ and $y$ are in meter and $t$ is in second. Then

Question 54

Two moles of nitrogen in a container, of negligible thermal capacity, are initially at $17^{\circ} \mathrm{C}$. The gas is compressed adiabatically from an initial volume of 120 liter to 80 liter. The correct option(s) is/are

Question 55

A small dipole is placed at the origin with its dipole moment $\vec{P}=p \hat{i}$ oriented along $x$ axis. $E$ and $V$, are respectively, the electric field and potential at point $A(x, y)$. The observations at the Point $A(x, y)$ which is at a large distance $r$ from the origin, show that

Question 56

Two equal positive charges $+Q$ each lie on $y$ axis at ( $0, a$ ) and ( $0,-a$ ). The electric field strength $E$ at a point $(x, 0)$ satisfies:

Question 57

In the circuit shown, the current in the $8 \Omega$ resistance across $G$ and $H$ is $i=0.5$ ampere. The ammeter is ideal. The internal resistance of the cell is $0.8 \Omega$. Choose the correct option(s).

Question 58

In an experiment with Lummer Gehrcke plate, the two coherent beams of light, caused by multiple reflections inside the transparent plate of refractive index $\mu=1.54$, reach the points $P$ and $Q$ on the screen. The net path difference between the two beams reaching either at $P$ or $Q$ is $\Delta \mathrm{x}=5000 \mathrm{ nm}$. Which of the wavelengths in the visible range ( $\lambda=390 \mathrm{ nm}$ to $\lambda=780 \mathrm{ nm}$ ) is/are most likely to produce a constructive interference (a maximum) at the point $P$ as well as at $Q$ on the screen?

Question 59

Two identical transparent solid cylinders, each of radius 10 cm and refractive index $\mu=\sqrt{3}$, lie horizontally parallel to each other on a horizontal plane mirror with a separation $x$ between their horizontal axes. A ray of light is incident horizontally on the cylinder A at a height $h$ above the plane mirror so as to emerge from this cylinder at a height $h_{1}=0.1 \mathrm{ m}$ above the plane mirror. The ray emerging out from the first cylinder $A$ is reflected from the horizontal plane mirror to enter the second parallel cylinder $B$ at a height $h_{2}$ and then this ray emerges out of the second cylinder, parallel and in-line with the original incident ray. The correct statement(s) is/are:

Question 60

In the working of a $p-n$ junction

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).