Let $\vec{F}$ be the force acting on a particle having position vector $\vec{r}$ and $\vec{T}$ be the torque of this force about the origin. Then
(1) $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{T}} = 0$ and $\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{T}} \neq 0$
(2) $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{T}} \neq 0$ and $\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{T}} = 0$
(3) $\vec{r} \cdot \vec{T} \neq 0$ and $\vec{F} \cdot \vec{T} \neq 0$
(4) $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{T}} = 0$ and $\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{T}} = 0$

The time period of a satellite of earth is 5 hours. If the separation between the earth and the satellite is increased to 4 times the previous value, the new time period will become
(1) 10 hours
(2) 80 hours
(3) 40 hours
(4) 20 hours

Two spherical bodies of mass $M$ and $5M$ \& radii $R$ \& $2R$ respectively are released in free space with initial separation between their centres equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is
(1) 2.5 R
(2) 4.5 R
(3) 7.5 R
(4) 1.5 R

The escape velocity for a body projected vertically upwards from the surface of earth is $11 \mathrm{~km/s}$. If the body is projected at an angle of $45^{\circ}$ with the vertical, the escape velocity will be
(1) $11\sqrt{2} \mathrm{~km/s}$
(2) $22 \mathrm{~km/s}$
(3) $11 \mathrm{~km/s}$
(4) $\frac{11}{\sqrt{2}} \mathrm{~km/s}$

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $C_{p}/C_{v}$ for the gas is
(1) $\frac{4}{3}$
(2) 2
(3) $\frac{5}{3}$
(4) $\frac{3}{2}$

A mass M is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period T. If the mass is increased by m, the time period becomes $\frac{5\mathrm{~T}}{3}$. Then the ratio of $\frac{m}{M}$ is
(1) $\frac{3}{5}$
(2) $\frac{25}{9}$
(3) $\frac{16}{9}$
(4) $\frac{5}{3}$

Two particles $A$ and $B$ of equal masses are suspended from two massless springs of spring constant $\mathrm{k}_{1}$ and $\mathrm{k}_{2}$, respectively. If the maximum velocities, during oscillation, are equal, the ratio of amplitude of $A$ and $B$ is
(1) $\sqrt{\frac{\mathrm{k}_{1}}{\mathrm{k}_{2}}}$
(2) $\frac{\mathrm{k}_{2}}{\mathrm{k}_{1}}$
(3) $\sqrt{\frac{\mathrm{k}_{2}}{\mathrm{k}_{1}}}$
(4) $\frac{\mathrm{k}_{1}}{\mathrm{k}_{2}}$

The displacement of a particle varies according to the relation $x = 4(\cos\pi t + \sin\pi t)$. The amplitude of the particle is
(1) $-4$
(2) $4$
(3) $4\sqrt{2}$
(4) $8$

A body executes simple harmonic motion. The potential energy (P.E), the kinetic energy (K.E) and total energy (T.E) are measured as a function of displacement $x$. Which of the following statements is true?
(1) K.E. is maximum when $x = 0$
(2) T.E is zero when $x = 0$
(3) K.E is maximum when x is maximum
(4) P.E. is maximum when $x = 0$

The displacement $y$ of a wave travelling in the $x$-direction is given by $y = 10^{-4}\sin\left(600t - 2x + \frac{\pi}{3}\right)$ metres where x is expressed in metres and t in seconds. The speed of the wave-motion, in $\mathrm{ms}^{-1}$, is
(1) 300
(2) 600
(3) 1200
(4) 200

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) $\mathrm{v}_{1} - \mathrm{v}_{2} = \left[\frac{2h}{\mathrm{~m}}\left(\mathrm{f}_{1} - \mathrm{f}_{2}\right)\right]^{1/2}$

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) $3 \mathrm {~F} / 8$

A charged particle $q$ is shot towards another charged particle $Q$ which is fixed, with a speed $v$. It approaches $Q$ up to a closest distance $r$ and then returns. If $q$ were given a speed $2v$, the closest distance of approach would be
(1) $r$
(2) $2r$
(3) $r / 2$
(4) $r / 4$