A tuning fork of known frequency 256 Hz makes 5 beats per second with the vibrating string of a piano. The beat frequency decreases to 2 beats per second when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was
(1) $256 + 2 \mathrm{~Hz}$
(2) $256 - 2 \mathrm{~Hz}$
(3) $256 - 5 \mathrm{~Hz}$
(4) $256 + 5 \mathrm{~Hz}$

The work done in placing a charge of $8 \times 10^{-18}$ coulomb on a condenser of capacity 100 micro-farad is
(1) $16 \times 10^{-32}$ joule
(2) $3.1 \times 10^{-26}$ joule
(3) $4 \times 10^{-10}$ joule
(4) $32 \times 10^{-32}$ joule

An ammeter reads up to 1 ampere. Its internal resistance is 0.81 ohm. To increase the range to 10 A the value of the required shunt is
(1) $0.03\,\Omega$
(2) $0.3\,\Omega$
(3) $0.9\,\Omega$
(4) $0.09\,\Omega$

A 220 volt, 1000 watt bulb is connected across a 110 volt mains supply. The power consumed will be
(1) 750 watt
(2) 500 watt
(3) 250 watt
(4) 1000 watt

The length of a given cylindrical wire is increased by $100\%$. Due to the consequent decrease in diameter the change in the resistance of the wire will be
(1) $200\%$
(2) $100\%$
(3) $50\%$
(4) $300\%$

A thin rectangular magnet suspended freely has a period of oscillation equal to $T$. Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely in the same field. If its period of oscillation is $T'$, the ratio $\frac{T'}{T}$ is
(1) $\frac{1}{2\sqrt{2}}$
(2) $\frac{1}{2}$
(3) 2
(4) $\frac{1}{4}$

A magnetic needle lying parallel to a magnetic field requires W units of work to turn it through $60^{\circ}$. The torque needed to maintain the needle in this position will be
(1) $\sqrt{3}\mathrm{~W}$
(2) W
(3) $\frac{\sqrt{3}}{2}\mathrm{~W}$
(4) 2W

When the current changes from $+2$ A to $-2$ A in 0.05 second, an e.m.f. of 8 V is induced in a coil. The coefficient of self-induction of the coil is
(1) 0.2 H
(2) 0.4 H
(3) 0.8 H
(4) 0.1 H

In an oscillating LC circuit the maximum charge on the capacitor is Q. The charge on the capacitor when the energy is stored equally between the electric and magnetic field is
(1) $\frac{Q}{2}$
(2) $\frac{Q}{\sqrt{3}}$
(3) $\frac{Q}{\sqrt{2}}$
(4) $Q$

A metal wire of linear mass density of $9.8 \mathrm{~g/m}$ is stretched with a tension of 10 kg-wt between two rigid supports 1 metre apart. The wire passes at its middle point between the poles of a permanent magnet, and it vibrates in resonance when carrying an alternating current of frequency $n$. The frequency $n$ of the alternating source is
(1) 50 Hz
(2) 100 Hz
(3) 200 Hz
(4) 25 Hz

Two identical photocathodes receive light of frequencies $f_{1}$ and $f_{2}$. If the velocities of the photo electrons (of mass $m$) coming out are respectively $\mathrm{v}_{1}$ and $\mathrm{v}_{2}$, then
(1) $v_{1}^{2} - v_{2}^{2} = \frac{2h}{m}\left(f_{1} - f_{2}\right)$
(2) $v_{1} + v_{2} = \left[\frac{2h}{m}\left(f_{1} + f_{2}\right)\right]^{1/2}$
(3) $v_{1}^{2} + v_{2}^{2} = \frac{2h}{m}\left(f_{1} + f_{2}\right)$
(4) $\mathrm{v}_{1} - \mathrm{v}_{2} = \left[\frac{2h}{\mathrm{~m}}\left(\mathrm{f}_{1} - \mathrm{f}_{2}\right)\right]^{1/2}$

A body of mass $m$, accelerates uniformly from rest to $v _ { 1 }$ in time $t _ { 1 }$. The instantaneous power delivered to the body as a function of time $t$ is
(1) $\frac { m v _ { 1 } t } { t _ { 1 } }$
(2) $\frac { m v _ { 1 } ^ { 2 } t } { t _ { 1 } ^ { 2 } }$
(3) $\frac { m v _ { 1 } t ^ { 2 } } { t _ { 1 } }$
(4) $\frac { m v _ { 1 } ^ { 2 } t } { t _ { 1 } }$

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?
(1) moment of inertia
(2) angular momentum
(3) angular velocity
(4) rotational kinetic energy.

One solid sphere $A$ and another hollow sphere $B$ are of same mass and same outer radii. Their moment of inertia about their diameters are respectively $\mathrm { I } _ { \mathrm { A } }$ and $\mathrm { I } _ { \mathrm { B } }$ such that
(1) $I _ { A } = I _ { B }$
(2) $I _ { A } > I _ { B }$
(3) $I _ { A } < I _ { B }$
(4) $I _ { A } / I _ { B } = d _ { A } / d _ { B }$ Where $d _ { A }$ and $d _ { B }$ are their densities.

A satellite of mass $m$ revolves around the earth of radius $R$ at a height $x$ from its surface. If $g$ is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is
(1) $g x$
(2) $\frac { g R } { R - x }$
(3) $\frac { g ^ { 2 } } { R + x }$
(4) $\left( \frac { g R ^ { 2 } } { R + x } \right) ^ { 1 / 2 }$

The time period of an earth satellite in circular orbit is independent of
(1) the mass of the satellite
(2) radius of its orbit
(3) both the mass and radius of the orbit
(4) neither the mass of the satellite nor the radius of its orbit.

If $g$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth is
(1) 2 mgR
(2) $\frac { 1 } { 2 } \mathrm { mgR }$
(3) $\frac { 1 } { 4 } \mathrm { mgR }$
(4) mgR

Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
(1) $R ^ { \left( \frac { n + 1 } { 2 } \right) }$
(2) $R ^ { \left( \frac { n - 1 } { 2 } \right) }$
(3) $R ^ { n }$
(4) $\mathrm { R } ^ { \left( \frac { \mathrm { n } - 2 } { 2 } \right) }$

The bob of a simple pendulum executes simple harmonic motion in water with a period $t$, while the period of oscillation of the bob is $t _ { 0 }$ in air. Neglecting frictional force of water and given that the density of the bob is $\left( \frac { 4 } { 3 } \right) \times 1000 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. What relationship between $t$ and $t _ { 0 }$ is true?
(1) $t = t _ { 0 }$
(2) $t = t _ { 0 } / 2$
(3) $t = 2 t _ { 0 }$
(4) $t = 4 t _ { 0 }$

A particle at the end of a spring executes simple harmonic motion with a period $t _ { 1 }$, while the corresponding period for another spring is $t _ { 2 }$. If the period of oscillation with the two springs in series is $T$, then
(1) $T = t _ { 1 } + t _ { 2 }$
(2) $T^2 = t_1^2 + t_2^2$
(3) $\mathrm { T } ^ { - 1 } = \mathrm { t } _ { 1 } ^ { - 1 } + \mathrm { t } _ { 2 } ^ { - 1 }$
(4) $\mathrm { T } ^ { - 2 } = \mathrm { t } _ { 1 } ^ { -2 } + \mathrm { t } _ { 2 } ^ { -2 }$

The total energy of a particle executing simple harmonic motion is
(1) $\propto x$
(2) $\propto x ^ { 2 }$
(3) independent of $x$
(4) $\propto x ^ { 1 / 2 }$

A particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency $\omega _ { 0 }$. An external force $F ( t )$ proportional to $\cos \omega t \left( \omega \neq \omega _ { 0 } \right)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to
(1) $\frac { \mathrm { m } } { \omega _ { 0 } ^ { 2 } - \omega ^ { 2 } }$
(2) $\frac { 1 } { m \left( \omega _ { 0 } ^ { 2 } - \omega ^ { 2 } \right) }$
(3) $\frac { 1 } { m \left( \omega _ { 0 } ^ { 2 } + \omega ^ { 2 } \right) }$
(4) $\frac { m } { \omega _ { 0 } ^ { 2 } + \omega ^ { 2 } }$

In forced oscillation of a particle the amplitude is maximum for a frequency $\omega _ { 1 }$ of the force, while the energy is maximum for a frequency $\omega _ { 2 }$ of the force, then
(1) $\omega _ { 1 } = \omega _ { 2 }$
(2) $\omega _ { 1 } > \omega _ { 2 }$
(3) $\omega _ { 1 } < \omega _ { 2 }$ when damping is small and $\omega _ { 1 } > \omega _ { 2 }$ when damping is large
(4) $\omega _ { 1 } < \omega _ { 2 }$ when damping is large

The displacement y of a particle in a medium can be expressed as $y = 10 ^ { - 6 } \sin ( 110 t + 20 x + \pi / 4 ) m$, where $t$ is in seconds and $x$ in meter. The speed of the wave is
(1) $2000 \mathrm {~m} / \mathrm { s }$
(2) $5 \mathrm {~m} / \mathrm { s }$
(3) $20 \mathrm {~m} / \mathrm { s }$
(4) $5 \pi \mathrm {~m} / \mathrm { s }$

Two spherical conductors $B$ and $C$ having equal radii and carrying equal charges in them repel each other with a force $F$ when kept apart at some distance. A third spherical conductor having same radius as that of $B$ but uncharged is brought in contact with $B$, then brought in contact with $C$ and finally removed away from both. The new force of repulsion between $B$ and $C$ is
(1) $\mathrm { F } / 4$
(2) $3 F / 4$
(3) $F / 8$
(4) $3 \mathrm {~F} / 8$