Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?
(1) $d$
(2) $\frac{m_2}{m_1}$ d
(3) $\frac{m_1}{m_1 + m_2} d$
(4) $\frac{m_1}{m_2} d$

A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1, -1)$ is
(1) $-F(\hat{i} - \hat{j})$
(2) $F(\hat{i} - \hat{j})$
(3) $-F(\hat{i} + \hat{j})$
(4) $F(\hat{i} + \hat{j})$

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $\omega' =$
(1) $\frac{\omega m}{(m + 2M)}$
(2) $\frac{\omega(m + 2M)}{m}$
(3) $\frac{\omega(m - 2M)}{(m + 2M)}$
(4) $\frac{\omega m}{(m + M)}$

Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is
(1) $m\ell^2$
(2) $2m\ell^2$
(3) $3m\ell^2$
(4) $3m\ell^2$

A wire elongates by $\ell$ mm when a load $W$ is hanged from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends, the elongation of the wire will be (in mm)
(1) $\ell/2$
(2) $\ell$
(3) $2\ell$
(4) zero

Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be $75\%$ of the total energy?
(1) $\frac{1}{12}$ s
(2) $\frac{1}{6}$ s
(3) $\frac{1}{4}$ s
(4) $\frac{1}{3}$ s

The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm, is $4.4$ m/s. The period of oscillation is
(1) 100 s
(2) 0.01 s
(3) 10 s
(4) 0.1 s

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $\omega$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time
(1) at the highest position of the platform
(2) at the mean position of the platform
(3) for an amplitude of $\frac{g}{\omega^2}$
(4) for an amplitude of $\frac{g^2}{\omega^2}$

A whistle producing sound waves of frequencies 9500 Hz and above is approaching a stationary person with speed $v$ ms$^{-1}$. The velocity of sound in air is $300$ ms$^{-1}$. If the person can hear frequencies upto a maximum of $10{,}000$ Hz, the maximum value of $v$ upto which he can hear the whistle is
(1) $30$ ms$^{-1}$
(2) $15\sqrt{2}$ ms$^{-1}$
(3) $15/\sqrt{2}$ ms$^{-1}$
(4) $15$ ms$^{-1}$

A string is stretched between fixed points separated by 75 cm. It is observed to have resonant frequencies of 420 Hz and 315 Hz. There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is
(1) 10.5 Hz
(2) 105 Hz
(3) 1.05 Hz
(4) 1050 Hz

The flux linked with a coil at any instant '$t$' is given by $\phi = 10t^2 - 50t + 250$. The induced emf at $t = 3$ s is
(1) 190 V
(2) $-190$ V
(3) $-10$ V
(4) 10 V

In a series resonant LCR circuit, the voltage across $R$ is 100 volts and $R = 1$ k$\Omega$ with $C = 2\,\mu$F. The resonant frequency $\omega$ is $200$ rad/s. At resonance the voltage across $L$ is
(1) $4 \times 10^{-3}$ V
(2) $2.5 \times 10^{-2}$ V
(3) 40 V
(4) 250 V

An inductor ($L = 100$ mH), a resistor ($R = 100\,\Omega$) and a battery ($E = 100$ V) are initially connected in series. After a long time the battery is disconnected after short circuiting the points $A$ and $B$. The current in the circuit 1 ms after the short circuit is
(1) 1 A
(2) $1/e$ A
(3) $e$ A
(4) 0.1 A

An alpha nucleus of energy $\frac{1}{2}mv^2$ bombards a heavy nuclear target of charge $Ze$. Then the distance of closest approach for the alpha nucleus will be proportional to
(1) $\frac{1}{Ze}$
(2) $v^2$
(3) $\frac{1}{m}$
(4) $\frac{1}{v^4}$

If the binding energy per nucleon in ${}_{3}^{7}\mathrm{Li}$ and ${}_{2}^{4}\mathrm{He}$ nuclei are 5.60 MeV and 7.06 MeV respectively, then in the reaction $\mathrm{p} + {}_{3}^{7}\mathrm{Li} \rightarrow 2\,{}_{2}^{4}\mathrm{He}$, the energy of the proton must be
(1) 39.2 MeV
(2) 28.24 MeV
(3) 17.28 MeV
(4) 1.46 MeV

If the ratio of the concentration of electrons to that of holes in a semiconductor is $\frac{7}{5}$ and the ratio of currents is $\frac{7}{4}$, then what is the ratio of their drift velocities?
(1) $\frac{4}{7}$
(2) $\frac{5}{8}$
(3) $\frac{4}{5}$
(4) $\frac{5}{4}$

In a common base mode of a transistor, the collector current is 5.488 mA for an emitter current of 5.60 mA. The value of the base current amplification factor $(\beta)$ will be
(1) 48
(2) 49
(3) 50
(4) 51

A body is at rest at $x = 0$. At $t = 0$, it starts moving in the positive $x$-direction with a constant acceleration. At the same instant another body passes through $x = 0$ moving in the positive $x$ direction with a constant speed. The position of the first body is given by $\mathrm { x } _ { 1 } ( \mathrm { t } )$ after time ' t ' and that of the second body by $x _ { 2 } ( t )$ after the same time interval. Which of the following graphs correctly describes $\left( x _ { 1 } - x _ { 2 } \right)$ as a function of time ' $t$ '?
(1), (2), (3), (4) [see graphs in original]

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range
(1) $200 \mathrm{~J} - 500 \mathrm{~J}$
(2) $2 \times 10 ^ { 5 } \mathrm{~J} - 3 \times 10 ^ { 5 } \mathrm{~J}$
(3) $20,000 \mathrm{~J} - 50,000 \mathrm{~J}$
(4) $2,000 \mathrm{~J} - 5,000 \mathrm{~J}$

A thin rod of length ' $L$ ' is lying along the $x$-axis with its ends at $x = 0$ and $x = L$. Its linear density (mass/length) varies with $x$ as $k\left( \frac { x } { L } \right) ^ { n }$, where $n$ can be zero or any positive number. If the position $x _ { \mathrm { CM } }$ of the centre of mass of the rod is plotted against ' $n$ ', which of the following graphs best approximates the dependence of $x _ { \mathrm { CM } }$ on $n$?
(1), (2), (3), (4) [see graphs in original]

A body of mass $m = 3.513 \mathrm{~kg}$ is moving along the $x$-axis with a speed of $5.00 \mathrm{~ms} ^ { - 1 }$. The magnitude of its momentum is recorded as
(1) $17.6 \mathrm{~kg~ms} ^ { - 1 }$
(2) $17.565 \mathrm{~kg~ms} ^ { - 1 }$
(3) $17.56 \mathrm{~kg~ms} ^ { - 1 }$
(4) $17.57 \mathrm{~kg~ms} ^ { - 1 }$

Consider a uniform square plate of side ' $a$ ' and mass ' $m$ '. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is
(1) $\frac { 5 } { 6 } m a ^ { 2 }$
(2) $\frac { 1 } { 12 } m a ^ { 2 }$
(3) $\frac { 7 } { 12 } m a ^ { 2 }$
(4) $\frac { 2 } { 3 } m a ^ { 2 }$

A planet in a distant solar system is 10 times more massive than the earth and its radius is 10 times smaller. Given that the escape velocity from the earth is $11 \mathrm{~kms} ^ { - 1 }$, the escape velocity from the surface of the planet would be
(1) $1.1 \mathrm{~kms} ^ { - 1 }$
(2) $11 \mathrm{~kms} ^ { - 1 }$
(3) $110 \mathrm{~kms} ^ { - 1 }$
(4) $0.11 \mathrm{~kms} ^ { - 1 }$

This question contains Statement - 1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement - I: For a mass $M$ kept at the centre of a cube of side ' $a$ ', the flux of gravitational field passing through its sides is $4\pi GM$.
Statement - II: If the direction of a field due to a point source is radial and its dependence on the distance ' $r$ ' from the source is given as $1/r^{2}$, its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.
(1) Statement - 1 is false, Statement - 2 is true.
(2) Statement - 1 is true, Statement - 2 is true; Statement - 2 is correct explanation for Statement-1.
(3) Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement-1.
(4) Statement - 1 is true, Statement - 2 is False.

A spherical solid ball of volume $V$ is made of a material of density $\rho _ { 1 }$. It is falling through a liquid of density $\rho _ { 2 }$ $\left( \rho _ { 2 } < \rho _ { 1 } \right)$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F _ { \text{viscous} } = - k v ^ { 2 }$ $(k > 0)$. The terminal speed of the ball is
(1) $\sqrt { \frac { Vg \left( \rho _ { 1 } - \rho _ { 2 } \right) } { k } }$
(2) $\frac { Vg \rho _ { 1 } } { k }$
(3) $\sqrt { \frac { V g \rho _ { 1 } } { k } }$
(4) $\frac { Vg \left( \rho _ { 1 } - \rho _ { 2 } \right) } { k }$