A fully charged capacitor $C$ with initial charge $q_{0}$ is connected to a coil of self inductance $L$ at $t=0$. The time at which the energy is stored equally between the electric and the magnetic field is:
(1) $\frac{\pi}{4}\sqrt{LC}$
(2) $2\pi\sqrt{LC}$
(3) $\sqrt{LC}$
(4) $\pi\sqrt{LC}$

Consider the following statements $P$: Suman is brilliant $Q$: Suman is rich $R$: Suman is honest The negation of the statement ``Suman is brilliant and dishonest if and only if Suman is rich'' can be expressed as
(1) $\sim(\mathrm{Q}\leftrightarrow(\mathrm{P}\wedge\sim\mathrm{R}))$
(2) $\sim Q\leftrightarrow\sim P\wedge R$
(3) $\sim(P\wedge\sim R)\leftrightarrow Q$
(4) $\sim P\wedge(Q\leftrightarrow\sim R)$

Let $R$ be the set of real numbers. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: $A=\{(x,y)\in R\times R: y-x \text{ is an integer}\}$ is an equivalence relation on $R$. Statement-2: $B=\{(x,y)\in R\times R: x=\alpha y \text{ for some rational number }\alpha\}$ is an equivalence relation on $R$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

The amount of heat produced in an electric circuit depends upon the current ( $I$ ), resistance ( $R$ ) and time ( $t$ ). If the error made in the measurements of the above quantities are $2 \% , 1 \%$ and $1 \%$ respectively then the maximum possible error in the total heat produced will be
(1) $1 \%$
(2) $2 \%$
(3) $6 \%$
(4) $3 \%$

Sand is being dropped on a conveyer belt at the rate of 2 kg per second. The force necessary to keep the belt moving with a constant speed of $3 \mathrm {~ms} ^ { - 1 }$ will be
(1) 12 N
(2) 6 N
(3) zero
(4) 18 N

A particle of mass m is at rest at the origin at time $\mathrm{t} = 0$. It is subjected to a force $\mathrm{F}(\mathrm{t}) = \mathrm{F}_{0}\mathrm{e}^{-\mathrm{bt}}$ in the $x$ direction. Its speed $v(t)$ is depicted by which of the following curves?

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements. If two springs $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ of force constants $k_{1}$ and $k_{2}$, respectively, are stretched by the same force, it is found that more work is done on spring $\mathrm{S}_{1}$ than on spring $\mathrm{S}_{2}$. Statement 1: If stretched by the same amount, work done on $\mathrm{S}_{1}$ will be more than that on $\mathrm{S}_{2}$. Statement 2: $k_{1} < k_{2}$
(1) Statement 1 is false, Statement 2 is true
(2) Statement 1 is true, Statement 2 is false
(3) Statement 1 is true, Statement 2 is the correct explanation for statement 1
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for statement 1.

Two point masses of mass $m _ { 1 } = f M$ and $m _ { 2 } = ( 1 - f ) M ( f < 1 )$ are in outer space (far from gravitational influence of other objects) at a distance $R$ from each other. They move in circular orbits about their centre of mass with angular velocities $\omega _ { 1 }$ for $m _ { 1 }$ and $\omega _ { 2 }$ for $m _ { 2 }$. In that case
(1) $( 1 - f ) \omega _ { 1 } = f \omega$
(2) $\omega _ { 1 } = \omega _ { 2 }$ and independent of $f$
(3) $f \omega _ { 1 } = ( 1 - f ) \omega _ { 2 }$
(4) $\omega _ { 1 } = \omega _ { 2 }$ and depend on $f$

Two cars of masses $m_{1}$ and $m_{2}$ are moving in circles of radii $r_{1}$ and $r_{2}$, respectively. Their speeds are such that they make complete circles in the same time $t$. The ratio of their centripetal acceleration is
(1) $m_{1}r_{1} : m_{2}r_{2}$
(2) $m_{1} : m_{2}$
(3) $r_{1} : r_{2}$
(4) $1 : 1$

The mass of a spaceship is 1000 kg. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10\,\mathrm{m/s}^{2}$ and 6400 km respectively. The required energy for this work will be:
(1) $6.4 \times 10^{11}$ Joules
(2) $6.4 \times 10^{8}$ Joules
(3) $6.4 \times 10^{9}$ Joules
(4) $6.4 \times 10^{10}$ Joules

This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: When moment of inertia $I$ of a body rotating about an axis with angular speed $\omega$ increases, its angular momentum $L$ is unchanged but the kinetic energy $K$ increases if there is no torque applied on it. Statement 2: $L = I\omega$, kinetic energy of rotation $= \frac{1}{2}I\omega^{2}$
(1) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of the Statement 1.
(4) Statement 1 is true, Statement 2 is false.

Assuming the earth to be a sphere of uniform density, the acceleration due to gravity inside the earth at a distance of $r$ from the centre is proportional to
(1) $r$
(2) $r^{-1}$
(3) $r^{2}$
(4) $r^{-2}$

Water is flowing through a horizontal tube having cross-sectional areas of its two ends being $A$ and $A^{\prime}$ such that the ratio $A/A^{\prime}$ is 5. If the pressure difference of water between the two ends is $3\times10^{5} \mathrm{~N\,m}^{-2}$, the velocity of water with which it enters the tube will be (neglect gravity effects)
(1) $5 \mathrm{~m\,s}^{-1}$
(2) $10 \mathrm{~m\,s}^{-1}$
(3) $25 \mathrm{~m\,s}^{-1}$
(4) $50\sqrt{10} \mathrm{~m\,s}^{-1}$

The pressure of an ideal gas varies with volume as $P = \alpha V$, where $\alpha$ is a constant. One mole of the gas is allowed to undergo expansion such that its volume becomes ' $m$ ' times its initial volume. The work done by the gas in the process is
(1) $\frac { \alpha V } { 2 } \left( m ^ { 2 } - 1 \right)$
(2) $\frac { \alpha ^ { 2 } V ^ { 2 } } { 2 } \left( m ^ { 2 } - 1 \right)$
(3) $\frac { \alpha } { 2 } \left( m ^ { 2 } - 1 \right)$
(4) $\frac { \alpha V ^ { 2 } } { 2 } \left( m ^ { 2 } - 1 \right)$

A given ideal gas with $\gamma = \frac{C_{p}}{C_{v}} = 1.5$ at a temperature $T$. If the gas is compressed adiabatically to one-fourth of its initial volume, the final temperature will be
(1) $2\sqrt{2}\,T$
(2) $4T$
(3) $2T$
(4) $8T$

$n$ moles of an ideal gas undergo a process $A \rightarrow B$ as shown in the figure. Maximum temperature of the gas during the process is
(1) $\frac{9P_{0}V_{0}}{nR}$
(2) $\frac{3P_{0}V_{0}}{2nR}$
(3) $\frac{9P_{0}V_{0}}{2nR}$
(4) $\frac{9P_{0}V_{0}}{4nR}$

If a simple pendulum has significant amplitude (up to a factor of $1/e$ of original) only in the period between $t = 0\,\mathrm{s}$ to $t = \tau\,\mathrm{s}$, then $\tau$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with $b$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds:
(1) $\dfrac{0.693}{b}$
(2) $b$
(3) $\dfrac{1}{b}$
(4) $\dfrac{2}{b}$

A ring is suspended from a point $S$ on its rim as shown in the figure. When displaced from equilibrium, it oscillates with time period of 1 second. The radius of the ring is (take $g = \pi ^ { 2 }$ )
(1) 0.15 m
(2) 1.5 m
(3) 1.0 m
(4) 0.5 m

A uniform tube of length 60.5 cm is held vertically with its lower end dipped in water. A sound source of frequency 500 Hz sends sound waves into the tube. When the length of tube above water is 16 cm and again when it is 50 cm , the tube resonates with the source of sound. Two lowest frequencies (in Hz ), to which tube will resonate when it is taken out of water, are (approximately).
(1) 281,562
(2) 281,843
(3) 276,552
(4) 272,544

A wave represented by the equation $y_{1} = a\cos(kx - \omega t)$ is superimposed with another wave to form a stationary wave such that the point $x = 0$ is node. The equation for the other wave is
(1) $a\cos(kx - \omega t + \pi)$
(2) $a\cos(kx + \omega t + \pi)$
(3) $a\cos\left(kx + \omega t + \frac{\pi}{2}\right)$
(4) $a\cos\left(kx - \omega t + \frac{\pi}{2}\right)$

A charge of total amount $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R ( R > r )$ such that the surface charge densities on the two spheres are equal. The electric potential at the common centre is
(1) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R - r ) Q } { \left( R ^ { 2 } + r ^ { 2 } \right) }$
(2) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R + r ) Q } { 2 \left( R ^ { 2 } + r ^ { 2 } \right) }$
(3) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R + r ) Q } { \left( R ^ { 2 } + r ^ { 2 } \right) }$
(4) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R - r ) Q } { 2 \left( R ^ { 2 } + r ^ { 2 } \right) }$

A series combination of $n_{1}$ capacitors, each of capacity $C_{1}$ is charged by source of potential difference 4V. When another parallel combination of $n_{2}$ capacitors each of capacity $C_{2}$ is charged by a source of potential difference $V$, it has the same total energy stored in it as the first combination has. The value of $C_{2}$ in terms of $C_{1}$ is then
(1) $16\frac{n_{2}}{n_{1}}C_{1}$
(2) $\frac{2C_{1}}{n_{1}n_{2}}$
(3) $2\frac{n_{2}}{n_{1}}C_{1}$
(4) $\frac{16C_{1}}{n_{1}n_{2}}$

A resistance $R$ and a capacitance $C$ are connected in series to a battery of negligible internal resistance through a key. The key is closed at $t = 0$. If after $t$ sec the voltage across the capacitance was seven times the voltage across $R$, the value of $t$ is
(1) $3RC\ln 2$
(2) $2RC\ln 2$
(3) $2RC\ln 7$
(4) $3RC\ln 7$

We wish to make a microscope with the help of two positive lenses both with a focal length of 20 mm each and the object is positioned 25 mm from the objective lens. How far apart the lenses should be so that the final image is formed at infinity?
(1) 20 mm
(2) 100 mm
(3) 120 mm
(4) 80 mm

The first diffraction minimum due to the single slit diffraction is seen at $\theta = 30^{\circ}$ for a light of wavelength $5000\,\AA$ falling perpendicularly on the slit. The width of the slit is
(1) $2.5\times10^{-5}\mathrm{~cm}$
(2) $1.25\times10^{-5}\mathrm{~cm}$
(3) $10\times10^{-5}\mathrm{~cm}$
(4) $5\times10^{-5}\mathrm{~cm}$