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

(a) $\frac{1}{k} \ln (1+k u t)$

(b) $\frac{1}{k} \ln$ kut

(c) $k$ ln kut

(d) $\frac{1}{k} \ln (1-k u t)$

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

(a) a and $\frac{1}{\mathrm{~T}}$

(b) a and T

(c) a and $\mathrm{T}^{2}$

(d) T and a ${ }^{2}$

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

(a) linear acceleration of the center of the ring along the plane is $a=\frac{g}{2}$

(b) force of friction between the ring and the inclined plane is $\mathrm{F}_{\text {friction }}=\frac{\mathrm{mg}}{4}$

(c) the ring keeps rolling for all values of the coefficient of friction $\mu \geq \frac{1}{4}$

(d) linear acceleration of the center of the ring along the plane is $a=\frac{g}{3}$

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 .

(a) $\mathrm{S}^{\prime}=\mathrm{S}$

(b) $\mathrm{S}^{\prime}=\frac{\mathrm{m}}{\mathrm{M}} \mathrm{S}$

(c) $S^{\prime}=\frac{m}{m+M} S$

(d) $S^{\prime}=\frac{M}{M+m} S$

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

(a) 4.278

(b) 3.767

(c) 3.256

(d) 931.478

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

(a) at least $\frac{P+Q \sin \theta}{m g+Q \cos \theta}$

(b) at least $\frac{P+Q \cos \theta}{m g+Q \sin \theta}$

(c) equal to $\frac{\mathrm{P}+\mathrm{Q} \sin \theta}{\mathrm{mg}-\mathrm{Q} \cos \theta}$

(d) equal to $\frac{P+Q \cos \theta}{m g-Q \sin \theta}$

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

(a) mgh

(b) $\frac{4}{5} \mathrm{mgh}$

(c) $\frac{5}{6} \mathrm{mgh}$

(d) $\frac{6}{7} \mathrm{mgh}$

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

(a) $2.0 \mathrm{~ms}^{-2}$

(b) $2.5 \mathrm{~ms}^{-2}$

(c) $3.5 \mathrm{~ms}^{-2}$

(d) $1.5 \mathrm{~ms}^{-2}$

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

(a) $\mathrm{T}=\frac{\mathrm{h}_{1} \mathrm{t}_{2}^{2}-\mathrm{h}_{2} \mathrm{t}_{1}^{2}}{\mathrm{~h}_{1} \mathrm{t}_{2}-\mathrm{h}_{2} \mathrm{t}_{1}}$

(b) $\mathrm{T}=\frac{\mathrm{h}_{1} \mathrm{t}_{2}^{2}+\mathrm{h}_{2} \mathrm{t}_{1}^{2}}{\mathrm{~h}_{2} \mathrm{t}_{1}+\mathrm{h}_{1} \mathrm{t}_{2}}$

(c) $\mathrm{T}=\frac{\mathrm{h}_{1} \mathrm{t}_{1}^{2}-\mathrm{h}_{2} \mathrm{t}_{2}^{2}}{\mathrm{~h}_{1} \mathrm{t}_{1}-\mathrm{h}_{2} \mathrm{t}_{2}}$

(d) $\mathrm{T}=\frac{\mathrm{h}_{1} \mathrm{t}_{1}^{2}+\mathrm{h}_{2} \mathrm{t}_{2}^{2}}{\mathrm{~h}_{1} \mathrm{t}_{1}+\mathrm{h}_{2} \mathrm{t}_{2}}$

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

(a) $\frac{\mu m g}{M}$

(b) $\frac{\mu \mathrm{mg}}{\mathrm{m}+\mathrm{M}}$

(c) $\frac{2 \mu \mathrm{mg}}{\mathrm{M}}$

(d) $\frac{2 \mu m g}{m+M}$

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

(a) $30^{\circ}$

(b) $60^{\circ}$

(c) $90^{\circ}$

(d) $120^{\circ}$

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

(a) $\left[\frac{G m}{r}\left(\frac{1+2 \sqrt{2}}{4}\right)\right]^{\frac{1}{2}}$

(b) $3 \sqrt{\frac{G m}{r}}$

(c) $\sqrt{\frac{G m}{r}(1+2 \sqrt{2})}$

(d) $\left[\frac{1}{2} \frac{\mathrm{Gm}}{\mathrm{r}}\left(\frac{1+\sqrt{2}}{2}\right)\right]^{\frac{1}{2}}$

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

(a) $\frac{2}{3} \rho g$

(b) $\frac{4}{3} \rho \mathrm{~g}$

(c) $\frac{8}{3} \rho \mathrm{~g}$

(d) $\frac{1}{3} \rho \mathrm{~g}$

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 $13.6 \mathrm{~g} / \mathrm{cm}^{3}$ ]

(a) 9.6 mm of Hg

(b) 3.2 mm of Hg

(c) 1.5 mm of Hg

(d) 6.0 mm of Hg

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

(a) $x=1, y=1, z=1$

(b) $x=-1, y=1, z=1$

(c) $x=1, y=-1, z=1$

(d) $x=1, y=1, z=-1$

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

(a) $2 \pi \mathrm{~S}$

(b) $\frac{\mathrm{S}}{2 \pi}$

(c) $\frac{\mathrm{s}}{\sqrt{2 \pi}}$

(d) $\frac{\mathrm{s}}{2 \sqrt{\pi}}$

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

(a) $2 \pi \sqrt{\left(\frac{m+2 M}{m+3 M}\right) \times\left(\frac{3 L}{2 g}\right)}$

(b) $2 \pi \sqrt{\left(\frac{m+3 M}{m+2 M}\right) \times\left(\frac{2 L}{3 g}\right)}$

(c) $2 \pi \sqrt{\frac{3 \mathrm{~L}}{2 \mathrm{~g}}}$

(d) $2 \pi \sqrt{\left(\frac{2 m+M}{3 m+M}\right) \times\left(\frac{3 L}{2 g}\right)}$

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

(a) A and E are at rest for a moment while C and G have maximum speed

(b) B and D have upward velocity whereas F and H have downward

(c) D, E, F are moving downward at that moment

(d) B and H are moving downward at that moment

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

(a) $2.0 \mathrm{~ms}^{-1}$

(b) $1.5 \mathrm{~ms}^{-1}$

(c) $1.0 \mathrm{~ms}^{-1}$

(d) $0.5 \mathrm{~ms}^{-1}$

 

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

(a) $318 \mathrm{~ms}^{-1}$

(b) $332 \mathrm{~ms}^{-1}$

(c) $381 \mathrm{~ms}^{-1}$

(d) $401 \mathrm{~ms}^{-1}$

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

(a) $\frac{5 \mathrm{~T}}{2}$

(b) $\left(\frac{65}{2}\right)^{\frac{1}{4}} \mathrm{~T}$

(c) $\left(\frac{97}{2}\right)^{\frac{1}{4}} \mathrm{~T}$

(d) $\left(\frac{65}{4}\right)^{\frac{1}{4}} \mathrm{~T}$

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

(a) $-\frac{b^{2}}{a^{2}} c, 0$

(b) $-\frac{b^{2}}{a^{2}-c^{2}} c, 0$

(c) $-\frac{b^{2}}{a^{2}+b^{2}} c, 0$

(d) $-\frac{b^{2}}{a^{2}-b^{2}} c, 0$

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

(a) Value of $\mathrm{T}_{2}$ is $1740^{\circ} \mathrm{C}$

(b) Work done by the gas is 2160 R joule

(c) Internal energy of the gas increases by 1440 R joule

(d) $\frac{V_{2}}{V_{1}}=5.8$

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

(a) 3

(b) 5

(c) 7

(d) 9

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 $\mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1}$ )

(a) $\mathrm{E}_{\mathrm{rms}}=726.5 \mathrm{~V} / \mathrm{m}, \mathrm{B}_{\mathrm{rms}}=2.42 \mu \mathrm{~T}$

(b) $\mathrm{E}_{\mathrm{rms}}=7260 \mathrm{~V} / \mathrm{m}, \quad \mathrm{B}_{\mathrm{rms}}=242 \mathrm{nT}$

(c) $\mathrm{E}_{\text {rms }}=1030 \mathrm{~V} / \mathrm{m}, \mathrm{B}_{\text {rms }}=3.42 \mu \mathrm{~T}$

(d) $\mathrm{E}_{\mathrm{rms}}=10300 \mathrm{~V} / \mathrm{m}, \mathrm{B}_{\mathrm{rms}}=342 \mathrm{nT}$

 

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

(a) AND

(b) NAND

(c) OR

(d) XOR

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

(a) a straight line perpendicular to $x$ axis and passing through $(2,0)$

(b) only the point $(2,0)$

(c) a circle with center at ( $-2,0$ ) and radius 4

(d) an ellipse with foci at O and A

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

(a) 0.6 mA through $\mathrm{R}_{\mathrm{S}}$ and 5.0 mA through $\mathrm{R}_{\mathrm{L}}$

(b) 1.0 mA through $\mathrm{R}_{\mathrm{S}}$ and 1.0 mA through $\mathrm{R}_{\mathrm{L}}$

(c) 1.1 mA through $\mathrm{R}_{\mathrm{S}}$ and no current through $\mathrm{R}_{\mathrm{L}}$

(d) no current through $\mathrm{R}_{\mathrm{S}}$ and 11 mA through $\mathrm{R}_{\mathrm{L}}$

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

(a) the average current per pulse is 4 mA

(b) the peak value of power delivered by the electron beam is 50 MW

(c) the average power delivered by the electron beam is 200 W

(d) the average power delivered by the electron beam is 2 MW

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 $\mathrm{OQ}=3 \mathrm{~m}$ ]

(a) $\frac{\mu_{0} I}{2 \pi}\left(\hat{i}-\frac{3}{5} \hat{j}\right)$

(b) $\frac{\mu_{0} I}{5 \pi}\left(\hat{i}-\frac{7}{24} \hat{j}\right)$

(c) $\frac{\mu_{0} I}{5 \pi}\left(-\hat{i}+\frac{3}{8} \hat{j}\right)$

(d) $\frac{\mu_{0} I}{24 \pi}(2 \hat{i}+3 \hat{j})$

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

(a) $\pi f q \frac{R_{2}^{2}+R_{1}^{2}}{2}$

(b) $\pi f q \frac{R_{2}^{2}-R_{1}^{2}}{2}$

(c) $\pi \mathrm{fq} \frac{\mathrm{R}_{2}^{2}-\mathrm{R}_{1}^{2}}{4}$

(d) $2 \pi f q\left(R_{2}^{2}-R_{1}^{2}\right)$

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

(a) $\mathrm{f}=2 \pi \mathrm{RC}$

(b) $\mathrm{f}=\frac{1}{2 \pi \mathrm{RC}}$

(c) $\mathrm{f}=\frac{2 \pi}{\mathrm{RC}}$

(d) $f=\frac{1}{2 \pi \sqrt{R^{2}+C^{2}}}$

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

(a) 500 nm

(b) 550 nm

(c) 600 nm

(d) 750 nm

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

(a) Statement (1) is correct but statement (2) is wrong

(b) Statement (1) is wrong but statement (2) is correct

(c) Both statements (1) and (2) are wrong

(d) Both statements (1) and (2) are correct

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

(a) $2 r$

(b) 3 r

(c) 6 r

(d) 8 r

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

(a) 12 cm

(b) 24 cm

(c) 48 cm

(d) 96 cm

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]

(a) $\frac{v}{7 R}$

(b) $\frac{3 v}{7 R}$

(c) $\frac{6 v}{7 R}$

(d) $\frac{v}{R}$

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

(a) $B=\frac{\mu_{0} i}{4 \pi} \frac{2 \sqrt{2} \ell^{2}}{\left(4 x^{2}+\ell^{2}\right) \sqrt{2 x^{2}+\ell^{2}}}$

(b) $B=\frac{\mu_{0} i}{4 \pi} \frac{4 \sqrt{2} \ell x}{\left(x^{2}+\ell^{2}\right) \sqrt{2 x^{2}+\ell^{2}}}$

(c) $B=\frac{\mu_{0} i}{4 \pi} \frac{4 \times 2 \sqrt{2} \ell^{2}}{\left(4 x^{2}+\ell^{2}\right) \sqrt{2 x^{2}+\ell^{2}}}$

(d) $B=\frac{\mu_{0} i}{4 \pi} \frac{4 \sqrt{2} \ell x}{\left(4 x^{2}+\ell^{2}\right) \sqrt{x^{2}+\ell^{2}}}$

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

(a) is proportional to $\frac{\lambda}{y^{2}}$ irrespective of whether y is positive or negative.

(b) is always directed away and perpendicular to the line of charge.

(c) has a vanishing component parallel to the line of charge.

(d) is directed along a straight line of slope $m=-1$ if y is positive but along a line of slope $m=+1$ if y is negative.

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

(a) $\vec{E}=\frac{\sigma}{2 \epsilon_{0}}(-\hat{i}+\hat{k})$

(b) $\vec{E}=\frac{\sigma}{2 \epsilon_{0}} \hat{j}$

(c) $\vec{E}=\frac{\sigma}{2 \epsilon_{0}}\left[-\frac{x}{|x|} \hat{i}+\frac{z}{|z|} \hat{k}\right]$

(d) $\vec{E}=\frac{\sigma}{\epsilon_{0}}\left[\frac{x}{|x|} \hat{i}-\frac{z}{|z|} \hat{k}\right]$

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 $r \geq R$,

(a) $\mathrm{E}=0$ : both at $\mathrm{r}=0$ and $\mathrm{r}=\mathrm{R}$

(b) $\mathrm{E} \propto \mathrm{r}$ for $\mathrm{r}<\mathrm{R}$ and $E \propto \frac{1}{r^{2}}$ for $r \geq R$

(c) the magnitude of E increases with r and reaches its maximum at $r=\frac{2 R}{3}$

(d) the maximum electric field produced by the given charge distribution is $E_{\max }=\frac{\rho_{0} R}{3 \in_{0}}$

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

(a) $R_{\text {effective }}=R_{1}+\sqrt{R_{1}^{2}+2 R_{1} R_{2}}$

(b) $R_{\text {effective }}=R_{2}+\sqrt{R_{1}^{2}+2 R_{1} R_{2}}$

(c) $\mathrm{R}_{\text {effective }}=\mathrm{R}_{1}+\sqrt{3 \mathrm{R}_{1} \mathrm{R}_{2}}$

(d) $R_{\text {effective }}=R_{1}+\sqrt{R_{2}^{2}+2 R_{1} R_{2}}$

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

(a) $\mathrm{B}=\frac{3}{8} \frac{\mu_{0} \mathrm{~J}}{\mathrm{a}}$

(b) $\mathrm{B}=\frac{3}{4} \mu_{0} \mathrm{Ja}$

(c) $\mathrm{B}=\frac{3}{8} \mu_{0} \mathrm{Ja}$

(d) $\mathrm{B}=\frac{5}{12} \mu_{0} \mathrm{Ja}$

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

(a) Statement (1) is correct but statement (2) is wrong

(b) Statement (1) is wrong but statement (2) is correct

(c) Both statements (1) and (2) are wrong

(d) Both statements (1) and (2) are correct

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

(a) 22.1

(b) 5.52

(c) 55.2

(d) 2.76

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

(a) $\frac{2}{9}$

(b) $\frac{3}{9}$

(c) $\frac{4}{9}$

(d) $\frac{5}{9}$

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

(a) $\frac{q_{1}-q_{2}}{C}$

(b) $\frac{q_{1}-q_{2}}{2 C}$

(c) $\frac{q_{1}-q_{2}}{4 C}$

(d) $\frac{q_{1}+q_{2}}{2 C}$

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

(a) $\frac{3}{2}$

(b) $\frac{4}{3}$

(c) $\frac{5}{4}$

(d) $\frac{6}{5}$

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

(a) tension at midpoint of the rod is $\mathrm{T}=1200 \mathrm{~N}$.

(b) tension in the rod varies with distance r from the axis of rotation as $\mathrm{T}=1600 \mathrm{r}^{2} \mathrm{~N}$

(c) stress in the rod at $\mathrm{r}=0.5 \mathrm{~m}$ is $3.0 \times 10^{8} \mathrm{Nm}^{-2}$

(d) elongation of the rod is $\frac{8}{3} \mathrm{~mm}$

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

(a) Wavelength of the transverse wave is 20 cm .

(b) Maximum magnitude of transverse acceleration of any point on the rope is nearly $800 \mathrm{~ms}^{-2}$

(c) If the oscillator produces maximum negative displacement at $x=0$ at time $\mathrm{t}=0$, the equation of the wave can be expressed as $\mathrm{y}(x, \mathrm{t})=-0.02 \sin [10 \pi x-100 \pi \mathrm{t}]$ in SI units.

(d) Tension in the given rope remaining unchanged, if a harmonic oscillator of frequency 200 Hz is used (instead of earlier frequency 100 Hz ), the wavelength will be 10 cm .

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

(a) number of nuclei $\mathrm{N}(\mathrm{t})$, at time t , is $\mathrm{N}(\mathrm{t})=\frac{1}{\lambda}\left[\left(\alpha-\lambda \mathrm{N}_{0}\right) \mathrm{e}^{-\lambda \mathrm{t}}\right]$

(b) if $\alpha=\lambda \mathrm{N}_{0}$, the number of nuclei $\mathrm{N}(\mathrm{t})$ at any time t will remain constant

(c) if $\alpha=2 \lambda \mathrm{~N}_{0}$ then $\mathrm{N}(\mathrm{t})=2 \mathrm{~N}_{0}$ as $\mathrm{t} \rightarrow \infty$

(d) if $\alpha=2 \lambda N_{0}$, the number of nuclei $N(t)$ after one half-life of $A$ is $N\left(\frac{T}{2}\right)=\frac{3}{2} N_{0}$

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)

(a) The refractive index of the film is 1.6

(b) The fringe width near O is 2 mm

(c) On the screen, $\mathrm{O}^{\prime}$ is above O

(d) The distance D of the screen from the double slit is nearly 1.15 m

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

(a) Electric field strength at distance $r(r<a)$ from the center is $\vec{E}=\frac{1}{4 \pi \epsilon_{0}} \frac{4 Q}{a^{3}} \vec{r}$

(b) Charge on the inner surface of the conducting spherical shell is +2 Q

(c) Charge on the outer surface of the conducting spherical shell is +2 Q

(d) Electrical energy stored in region $0<\mathrm{r}<\mathrm{a}$ [i.e. in the inner sphere] is $\frac{2 \mathrm{Q}^{2}}{5 \pi \in_{0} \mathrm{a}}$

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

(a) the value of Z is 5

(b) the energy required to excite the electron from the $3^{\text {rd }}$ orbit to the $4^{\text {th }}$ orbit is 16.53 eV (nearly)

(c) the wavelength of electromagnetic radiation required to liberate the electron completely when in the first Bohr orbit is $36.56 \AA$

(d) the angular momentum of an electron in the second Bohr orbit is $1.056 \times 10^{-33} \mathrm{Js}$

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

(a) Work done on the gas in going from A to B is $\mathrm{W}_{\mathrm{AB}}=\frac{\mathrm{MP}_{0}}{\rho_{0}} \ell \mathrm{n} 2$

(b) Work done by the gas in the process BC is $\mathrm{W}_{\mathrm{BC}}=\frac{\mathrm{MP}_{0}}{2 \rho_{0}}$

(c) Efficiency $(\eta)$ of the complete cycle ABCA is $\eta=\frac{2}{5}(1-\ell \operatorname{n} 2)$

(d) Heat rejected by the gas in the complete cycle $A B C A$ is $Q_{A B C A}=\frac{{M P_{0}}^{\rho_{0}}}{\rho_{0}}(1-\ell \operatorname{n} 2)$

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

(a) I downwards in $\mathrm{R}_{1}$

(b) I downwards in $\mathrm{R}_{2}$

(c) 2 I upwards in $\mathrm{R}_{3}$

(d) zero in each $\mathrm{R}_{1}, \mathrm{R}_{2}$ and $\mathrm{R}_{3}$

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

(a) $\mathrm{U}(x)$ can be expressed as $U(x)=\frac{\alpha}{x_{0}^{2}}\left[\left(\frac{x_{0}}{x}\right)^{2}-\frac{x_{0}}{x}\right]$

(b) velocity of the particle $v(x)$ as a function of $x$ can be expressed as $v(x)=\left[\frac{2 \alpha}{m x_{0}^{2}}\left\{\frac{x_{0}}{x}-\left(\frac{x_{0}}{x}\right)^{2}\right\}\right]^{\frac{1}{2}}$

(c) the maximum speed of the particle is $v_{\text {max }}=\sqrt{\frac{\alpha}{2 m x_{0}^{2}}}$

(d) the total energy of the particle $\mathrm{KE}(x)+\mathrm{U}(x)$ is zero

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

(a) the spring will be compressed to a maximum when at a length of $v \sqrt{\frac{M}{3 K}}$

(b) the kinetic energy of A and B together, when the spring is compressed to the maximum, is $\frac{M v^{2}}{6}$

(c) the blocks A and B stop for a moment when the spring is at the maximum compression

(d) the time required to reach the maximum compression from the normal length is $\frac{\pi}{2} \sqrt{\frac{2 \mathrm{M}}{3 \mathrm{~K}}}$

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

(a) the tension $\mathrm{T}=\mathrm{T}_{0} \frac{\mathrm{r}_{0}^{4}}{\mathrm{r}^{4}}$

(b) the kinetic energy $\mathrm{K}=\mathrm{K}_{0} \frac{\mathrm{r}_{0}^{2}}{\mathrm{r}^{2}}$

(c) the radius r of the circular path just when the string breaks is 0.40 m

(d) the work done by the tension in the string in reducing the radius of circle from $\mathrm{r}_{0}$ to $\frac{\mathrm{r}_{0}}{2}$ is $4 \mathrm{~K}_{0}$

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

(a) the time period of the oscillating rectangular coil is $\mathrm{T}=0.19 \mathrm{~s}$

(b) the net change in flux through the circular coil wrapped on the iron cylinder is 4.0 weber

(c) the induced charge in the circular coil wrapped on the iron cylinder is $\mathrm{q}_{\text {ind }}=240 \mu \mathrm{C}$

(d) total resistance of the circuit containing the circular coil is $\mathrm{R}=33.3 \mathrm{k} \Omega$

1. In the Bohr model of hydrogen atom, the force between the nucleus and the electron is modified as $F=\frac{e^2}{4 \pi \epsilon_0}\left(\frac{1}{r^2}+\frac{\delta}{r^3}\right)$ where $\delta$ is a small constant. Using the Bohr radius $a_0=\frac{\epsilon_0 h^2}{\pi m e^2}$, the radius of $n^{\text {th }}$ orbit is

(a) $a_0 n^2-\delta$
(b) $a_0 n^2+\delta$
(c) $a_0(n-\delta)^2$
(d) $a_0(n+\delta)^2$

2. 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

(a) Zero
(b) $\frac{\mu_0 i}{4 r}$
(c) $\frac{\mu_0 i}{3 r}$
(d) $\frac{\mu_0 i}{2 r}$

3. 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 $\mathrm{C}=2 \mathrm{R}$. The volume of the gas gets tripled (at constant pressure) during the process. The final temperature attained by the gas is
(a) 3 T
(b) $3^2 \mathrm{~T}$
(c) $\frac{1}{3} \mathrm{~T}$
(d) $3^3 \mathrm{~T}$

4. 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).
(a) $2 \xi$
(b) $3 \xi$
(c) $4 \xi$
(d) $5 \xi$

5. 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

(a) $\frac{\sin \theta_1}{\sin \theta_2}=\frac{q_2}{q_1}$

(b) $\frac{\sin \theta_1}{\sin \theta_2}=\frac{m_2}{m_1}$

(c) $\frac{\tan \theta_1}{\tan \theta_2}=\frac{q_1}{q_2} \times \frac{m_2}{m_1}$

(d) $\frac{\sin \theta_1}{\sin \theta_2}=1$

6. 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 << d) and angular frequency $\omega$. If the instantaneous current in the circuit reaches a maximum value $\mathrm{I}_{0}$, the amplitude of the vibrations is ' a ' equal to\\

(a) $\frac{I_{0} d^{2}}{\epsilon_{0} \omega V A}$

(b) $\frac{2 \mathrm{I}_{0} \mathrm{~d}}{\epsilon_{0} \omega \mathrm{VA}}$

(c) $\frac{2 \mathrm{I}_{0} \mathrm{~d}^{2}}{\epsilon_{0} \omega \mathrm{VA}}$

(d) $\frac{I_{0} d^{2}}{2 \in_{0} \omega V A}$

7. 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

(a) $\sqrt{2 g \ell}$

(b) $\sqrt{3 g \ell}$

(c) $\sqrt{4 g \ell}$

(d) $\sqrt{5 g \ell}$

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

(a) virtual image, 40 cm

(b) real image, 40 cm

(c) virtual image, 50 cm

(d) real image, 20 cm

9. 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 (see the figure)

(a) The ball will just strike at B

(b) The ball will strike at D where $\mathrm{d}=\mathrm{BD}=(\sqrt{2}-1) \mathrm{R}$

(c) The ball will strike at D such that $\mathrm{d}=\mathrm{BD}=(2-\sqrt{2}) \mathrm{R}$

\includegraphics[max width=\textwidth, alt={}, center]{4e5549a2-6dc0-4de7-8b54-53954cd4753c-04_313_538_1398_1311}

(d) The speed $v$ of the ball should always be greater than the critical speed $v_{0}$ where $v_{0}$ is $\sqrt{\frac{g R}{2}}$

10. 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) $\frac{2 \alpha}{3 \beta^{2} \gamma}$

(b) $\frac{5 \alpha}{3 \beta^{2} \gamma}$

(c) $\frac{2 \alpha \beta^{2}}{3 \gamma}$

(d) $\frac{5 \alpha \beta^{2}}{3 \gamma}$

11. 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

(a) $1.5 \mathrm{~ms}^{-2}$

(b) $3.0 \mathrm{~ms}^{-2}$

(c) $7.5 \mathrm{~ms}^{-2}$

(d) Zero

12. 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

(a) $t=2$ sec

(b) $t=4$ sec

(c) $t=8$ sec

(d) It won't blow

13. The bent wire PQR 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 PQ and QR 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

(a) $40 \hat{\mathrm{j}} \mathrm{N}$

(b) $-20 \hat{\mathrm{j}} \mathrm{N}$

(c) $-40 \hat{\mathrm{j} ~ \mathrm{~N}}$

(d) $-20 \sqrt{3} \hat{\mathrm{j}} \mathrm{N}$

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

(a) $r=\sqrt{a b}$

(b) $r=\frac{a+b}{2}$

(c) $r=\frac{2 a b}{a+b}$

(d) $r=\frac{a^{2}+b^{2}}{a+b}$

15. 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 $\mathrm{d}=4 \mathrm{AU}$ 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

(a) 1 year

(b) 2 years

(c) 4 years

(d) 8 years

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

(a) energy of the recoiled daughter nucleus is nearly 8.7 keV

(b) energy of the recoiled daughter nucleus is nearly 4.81 MeV

(c) recoil speed of the daughter nucleus is $2.74 \times 10^{5} \mathrm{~ms}^{-1}$

(d) speed of emitted $\alpha$ particle is nearly $1.5 \times 10^{6} \mathrm{~ms}^{-1}$

17. 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 $340 \mathrm{~ms}^{-1}$ )

(a) 520 Hz

(b) 450 Hz

(c) 480 Hz

(d) 550 Hz

18. 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

(a) $\frac{\ell}{2}$

(b) $\frac{\ell}{\sqrt{2}}$

(c) $\frac{\ell}{\sqrt{3}}$

(d) $\frac{\ell}{2 \sqrt{3}}$

19. 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 atoms is (Given that the density of silicon is $2.33 \mathrm{gm} / \mathrm{cm}^{3}$ and that the molar mass of silicon $\mathrm{M}=28.1 \mathrm{gm}$ )

(a) $1.0 \times 10^{-7}$

(b) $2.0 \times 10^{-7}$

(c) $4.0 \times 10^{-7}$

(d) $5.0 \times 10^{-7}$

20. 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

(a) 2 kV

(b) 4 kV

(c) 6 kV

(d) 8 kV

21. Three resistances, each of $\mathrm{R}=4 \Omega$ 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.

(a) Statements 1 and 2 both are wrong

(b) Statement 1 is correct but statement 2 is wrong

(c) Statement 1 is wrong but statement 2 is correct

(d) Statements 1 and 2 both are correct

22. 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

(a) ellipsoid

(b) hemisphere

(c) paraboloid

(d) hyperboloid

23. 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 $\mathbf{E}$ at any point inside the cavity is

(a) zero

(b) uniform

(c) directed radially outward

(d) directed radially inward

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

(a) t

(b) $t \sqrt{2}$

(c) $\mathrm{t}(\sqrt{2}-1)$

(d) $\mathrm{t}(\sqrt{2}+1)$

25. OABC 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 BC . The resistance between O and M is

(a) $5 \Omega$

(b) $4 \Omega$

(c) $10 \Omega$

(d) $15 \Omega$

26. 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

(a) $\mu \mathrm{W}$

(b) $\frac{\mathrm{W}}{\sqrt{1+\mu^{2}}}$

(c) $\frac{\mu \mathrm{W}}{\sqrt{1+\mu^{2}}}$

(d) $\frac{W \sqrt{1-\mu^{2}}}{\mu}$

27. The external wall of a room measuring $2 \mathrm{~m} \times 3 \mathrm{~m}$ consists of a layer of white pine of thickness $\mathrm{d}_{\text {pine }}=2.0 \mathrm{~cm}$ 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 $\mathrm{P}_{\text {cond }}=\frac{\mathrm{dQ}}{\mathrm{dt}}$ 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) 7.2 cm

(b) 6.4 cm

(c) 4.8 cm

(d) 0.8 cm

28. A 240 kg block is suspended from a fixed point O , at the end of a long ( $\mathrm{L}=13 \mathrm{~m}$ ) massless rope. A horizontal force F slowly pushes the block to move it a horizontal distance $\mathrm{d}=5 \mathrm{~m}$ 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

(a) Only statement 1 is correct

(b) Only statement 2 is correct

(c) Both statements 1 and 2 are wrong

(d) Both statements 1 and 2 are correct

29. The number of radioactive nuclei N of a radioactive sample is experimentally measured as a function of time t . At $\mathrm{t}=0, \mathrm{~N}(\mathrm{t}=0)=50,000$ and at $\mathrm{t}=10 \mathrm{~s}, \mathrm{~N}(\mathrm{t}=10 \mathrm{~s})=5000 \pm 100$. 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 $x, L t x \rightarrow 0, \ln (1+x) \approx x$ ]

(a) 0.26 s

(b) 0.15 s

(c) 0.05 s

(d) 0.10 s

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

(a) mono atomic

(b) diatomic

(c) tri-atomic

(d) mixture of monoatomic and diatomic

31. 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

(a) $\tan \alpha=\frac{\mathrm{h}}{\mathrm{L}}-\tan \theta$

(b) $\tan \alpha=\frac{2 \mathrm{~h}}{\mathrm{~L}}+\tan \theta$

(c) $\tan \alpha=\tan \theta-\frac{2 \mathrm{~h}}{\mathrm{~L}}$

(d) $\tan \alpha=\frac{2 \mathrm{~h}}{\mathrm{~L}}-\tan \theta$

32. 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

(a) $v=\sqrt{g\left(L-\frac{y_{0}^{2}}{L}\right)}$

(b) $v=\sqrt{2 g\left(L-\frac{y_{0}^{2}}{L}\right)}$

(c) $v=\sqrt{g\left(L-y_{0}\right)}$

(d) $v=\sqrt{2 g\left(L-y_{0}\right)}$

33. 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

(a) $\frac{\mathrm{D}}{\mathrm{d}}-1$

(b) $\frac{\mathrm{D}}{\mathrm{d}}+1$

(c) $\frac{\mathrm{D}+\mathrm{d}}{\mathrm{D}-\mathrm{d}}$

(d) $\frac{\mathrm{D}}{\mathrm{d}}$

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

(a) 1.5 kHz

(b) 500 Hz

(c) 326 Hz

(d) 126 Hz

35. 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?

(a) A

(b) B

(c) neither A nor B

(d) both A and B

36. 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)

(a) $\frac{15 \mathrm{~L}}{16}$

(b) $\frac{8 \mathrm{~L}}{9}$

(c) $\frac{\mathrm{L}}{2}$

(d) $\frac{3 \mathrm{~L}}{4}$

37. 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 $\left.=m_{e}=0.00055 u\right]$. The disintegration energy (i.e. the Q value) is approximately

(a) 1.45 MeV

(b) 0.94 MeV

(c) 0.43 MeV

(d) -1.45 MeV

38. 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 displacements $\mathrm{y}_{1}$ and $\mathrm{y}_{2}$ are in meter)

(a) 32.7 cm

(b) 16.35 cm

(c) 6.54 cm

(d) 5.45 cm

39. 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

(a) $\mathrm{R}=10 \mathrm{r}$

(b) $R=20 r$

(c) $R=40 r$

(d) $r=20 R$

40. 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 $U=\alpha \times \frac{Q^{2}}{4 \pi \epsilon_{0} R}$. The value of $\alpha$ is

(a) Zero

(b) $\frac{3}{5}$

(c) $\frac{1}{2}$

(d) $\frac{1}{10}$

41. 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]

(a) $\frac{\mu_{0}}{4 \pi^{2}} \frac{I^{2}}{n R^{3}}$ outward

(b) $\frac{\mu_{0}}{2 \pi} \frac{I^{2}}{n R^{3}}$ inward

(c) $\frac{\mu_{0}}{2 \pi^{2}} \frac{I^{2}}{n R^{3}}$ inward

(d) $\frac{\mu_{0}}{2 \pi^{2}} \frac{I^{2}}{n R^{3}}$ outward

42. In Young's double slit experiment, a bright fringe is observed at $\mathrm{y}=1.5 \mathrm{~cm}$ 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 $\mathrm{y}=1.5 \mathrm{~cm}$ is

(a) 13

(b) 8

(c) 7

(d) 6

43. An illuminated point object is placed in front of an equi-convex lens of refractive index $\mu=1.5$ and focal length $\mathrm{f}=40 \mathrm{~cm}$, at a distance of $\mathrm{u}=1.20 \mathrm{~m}$ 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

(a) 60 cm , virtual

(b) 80 cm , virtual

(c) 110 cm , real

(d) 130 cm , real

44. A paramagnetic gas at room temperature $27^{\circ} \mathrm{C}$ is placed in an external uniform magnetic field of magnitude $\mathrm{B}=1.5$ tesla . The atoms of the gas have magnetic dipole moment $\mu=1.0 \mu_{\mathrm{B}}$ where $1 \mu_{\mathrm{B}}=\frac{\mathrm{eh}}{4 \pi \mathrm{~m}}$ is Bohr magneton, and m is the mass of electron. The energy difference $\Delta \mathrm{U}_{\mathrm{B}}$ 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

(a) 17.4

(b) 1.74

(c) 34.8

(d) 8.7

45. 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 $\mathrm{h}=25 \mathrm{~mm}$. Surface Tension of water is $\mathrm{T}=0.073 \mathrm{Nm}^{-1}$. Since the water wets the glass completely, it may be concluded that the

(a) length of capillary above the water surface is insufficient so the water will flow out as a fountain

(b) water will rise up to the brim forming a meniscus of radius 1.2 mm at the top

(c) water will rise up to the brim forming a meniscus of radius 0.6 mm at the top

(d) water will rise up to the brim forming a meniscus of radius equal to radius of the capillary

46. 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

(a) 200

(b) 300

(c) 320

(d) 180

47. A thin plastic disc, which is a quarter of a circle of radius $\mathrm{R}=0.6 \mathrm{~m}$, lies in the first quadrant of $\mathrm{x}-\mathrm{y}$ 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 $\mathrm{P}(0,0,0.8 \mathrm{~m})$ is

(a) $\frac{\sigma}{8 \epsilon_{0}}$

(b) $\frac{\sigma}{8 \pi \epsilon_{0}}$

(c) $\frac{\sigma}{20 \epsilon_{0}}$

(d) $\frac{\sigma}{40 \epsilon_{0}}$

48. The charge on a small point object at A produces an electric potential $\mathrm{V}=3$ volt at a point P . Because of an unavoidable situation, leakage of the charge from the object at A starts at $\mathrm{t}=0$ 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 OD. The resulting magnetic field $\mathbf{B}$, at this instant, at the location O (such that $\mathrm{OD}=1.0 \mathrm{~m}$ ) is

(a) $8.1 n T$

(b) $\frac{1}{9} \mu T$

(c) $9.9 \mu \mathrm{~T}$

(d) $9.0 n T$

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 $\mathrm{t}=0$. 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 I follows curve (2). The following conclusion(s) may be drawn

(a) E and R are unchanged but L has increased

(b) $\frac{E}{R}$ is unchanged but $\frac{L}{R}$ has increased

(c) $\frac{E}{R}$ is unchanged but $\frac{L}{R}$ has decreased

(d) Stored magnetic energy has increased

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. 2M. The correct alternative(s) is/are [Assume that the Sun and the Earth are point masses]

(a) The period of revolution of the Earth around the Sun now becomes $\frac{1}{\sqrt{2}}$ year

(b) There is no change in angular momentum of the Earth around the Sun

(c) Minimum distance of the Earth from Sun during revolution is now $\frac{r}{3}$

(d) Maximum speed of the Earth during revolution around the Sun is $3 \sqrt{\frac{G M}{r}}$

51. An optical fibre consists of a glass core (refractive index $\mathrm{n}_{1}$ ) 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)

(a) Maximum value of $\theta$ for which a ray can travel down the fibre is $\theta=\sin ^{-1} \sqrt{n_{1}^{2}-n_{2}^{2}}$

(b) Maximum value of $\theta$ for which a ray can travel down the fibre is $\theta=\cos ^{-1} \sqrt{n_{1}^{2}-n_{2}^{2}}$

(c) If $\theta=30^{\circ}$ (in air) and $n_{1}=1.50$, then for reflection just at the critical angle, the value of $n_{2}$ is $\sqrt{2}$

(d) A ray entering at $\theta=0$ travels a distance L from P to Q , inside the fibre directly along the fibre axis. Another ray travelling through fibre is repeatedly reflected at the critical angle, at interface of glass core and surrounding layer/cladding. Both the rays travel from point P to point Q on the axis. If the two rays started from point P at the same time, the difference $\Delta t$ in the time taken to reach the point Q by the two rays is $\Delta t=\frac{n_{1}}{n_{2}} \times \frac{L}{c}\left(n_{1}-n_{2}\right)$ (here c is the speed of light in vacuum)

52. 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 an impedance of $1.0 \mathrm{k} \Omega$ and phase constant of $\phi=45^{\circ}$. The values of $\mathrm{L}, \mathrm{C}$ and R for this circuit may be

(a) $\mathrm{R}=707 \Omega$

(b) $\mathrm{L}=78.6 \mathrm{mH}$

(c) $\mathrm{C}=31.8 \mathrm{nf}$

(d) $\mathrm{R}=506 \Omega$

53. 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 $\mathbf{B}$ due to the current is 0.28 mT at $\mathrm{r}=4.0 \mathrm{~mm}$ and 0.20 mT at $\mathrm{r}=10.0 \mathrm{~mm}$, respectively, from the axis of the wire then the

(a) magnitude of the magnetic field $\mathbf{B}$ at a distance $\mathrm{r}=2.0 \mathrm{~mm}$ from axis is 0.14 mT

(b) magnitude of the magnetic field $\mathbf{B}$ at $\mathrm{r}=5.0 \mathrm{~mm}$ is greater than that at $\mathrm{r}=6.0 \mathrm{~mm}$

(c) current flowing in the wire is $\mathrm{I}=20 \mathrm{~A}$

(d) current density in the wire is $\mathrm{J} \approx 1.1 \times 10^{5} \mathrm{~A} / \mathrm{m}^{2}$

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

(a) equivalent capacity between A and B is $\frac{8}{3} \mu \mathrm{~F}$

(b) stored electrical energy of the system is $204 \mu \mathrm{~J}$

(c) potential difference between X and Y is $\mathrm{V}_{\mathrm{XY}}=2$ volt

(d) potential difference across the capacitor $\mathrm{C}_{2}$ is 8 volt

55. 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

(a) molar heat capacity of the gas is $\mathrm{C}=\frac{3 \mathrm{R}}{2}$

(b) final temperature of the gas is 2 T

(c) work done on the gas is 3 RT

(d) equation of state of the process is $\mathrm{PV}^{4 / 3}=$ constant

56. 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 $\mathrm{h}=\alpha \mathrm{R}(0 \leq \alpha \leq 1)$. Choose the correct option/option(s).

(a) Translational energy of the ball is $\frac{p^{2}}{2 M}$

(b) Energy of pure rotational motion of the ball is $\frac{5}{4} \frac{\mathrm{p}^{2} \alpha^{2}}{\mathrm{M}}$

(c) For $\alpha=\frac{2}{5}$, the ball rolls without sliding

(d) For $\alpha=0$, pure rotational motion will be observed

57. 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 $\mathbf{B}$. Then for a $\mathrm{He}^{+}$ion in the state $\mathrm{n}=2$ (electron in the $2^{\text {nd }}$ orbit)

(a) the radius of the orbit is 2 r

(b) the magnetic dipole moment of the orbit is $2 p_{m}$

(c) the frequency of circulation is $\frac{f}{2}$

(d) the magnetic field at the center is $\frac{\mathrm{B}}{4}$

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

(a) force $\mathrm{F}(x)$ acting on the body at $x=0$ is 25 N

(b) there is stable equilibrium at $x=2 \mathrm{~m}$

(c) there is unstable equilibrium at $X=-2 \mathrm{~m}$

(d) the body executes (small) oscillations with angular frequency $\omega=0.5 \mathrm{rads}^{-1}$

 59. A uniform rod OA of length L is freely pivoted at one end O . Let C be the midpoint (i.e. $\mathrm{OC}=\frac{\mathrm{L}}{2}=\mathrm{CA}$ ).

Choose the correct statement(s)

(a) The rod can perform small angular oscillations with time period $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{~L}}{2 \mathrm{~g}}}$

(b)The rod is brought to the horizontal position and then released from rest. The angular velocity of the rod at the instant when it is vertical is $\omega=\sqrt{\frac{3 g}{\mathrm{~L}}}$

(c) When the oscillating rod comes to the vertical position, it breaks at midpoint C without generating any impulsive force, the largest angle from vertical reached by the upper part OC of the rod is $60^{\circ}$.

(d)After breaking, the lower part just falls vertically rotating about its center.

60. 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 of 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

(a) $V=2.00 \mathrm{~ms}^{-1}$ when both the balls $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ are thrown together.

(b) $\mathrm{V}=3.00 \mathrm{~ms}^{-1}$ when both the balls $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ are thrown together.

(c) $V=2.17 \mathrm{~ms}^{-1}$ if the balls are thrown one after the other, first $m_{1}$ and then $m_{2}$.

(d) $V=2.19 \mathrm{~ms}^{-1}$ if the balls are thrown one after the other, first $\mathrm{m}_{2}$ and then $\mathrm{m}_{1}$.