A capacitor is discharging through a resistor $R$. Consider in time $t_1$, the energy stored in the capacitor reduces to half of its initial value and in time $t_2$, the charge stored reduces to one eighth of its initial value. The ratio $\frac{t_1}{t_2}$ will be
(1) $\frac{1}{2}$
(2) $\frac{1}{3}$
(3) $\frac{1}{4}$
(4) $\frac{1}{6}$

The combination of two identical cells, whether connected in series or parallel combination provides the same current through an external resistance of $2\Omega$. The value of internal resistance of each cell is
(1) $2\Omega$
(2) $4\Omega$
(3) $6\Omega$
(4) $8\Omega$

For $z = a^2 x^3 y^{\frac{1}{2}}$, where '$a$' is a constant. If percentage error in measurement of '$x$' and '$y$' are $4\%$ and $12\%$, respectively, then the percentage error for '$z$' will be $\_\_\_\_$ $\%$.

As per the given figure, two plates $A$ and $B$ of thermal conductivity $K$ and $2K$ are joined together to form a compound plate. The thickness of plates are 4.0 cm and 2.5 cm respectively and the area of cross-section is $120 \mathrm {~cm} ^ { 2 }$ for each plate. The equivalent thermal conductivity of the compound plate is $\left( 1 + \frac { 5 } { \alpha } \right) K$, then the value of $\alpha$ will be $\_\_\_\_$ .

A car covers $AB$ distance with first one-third at velocity $v_{1}$ m s$^{-1}$, second one-third at $v_{2}$ m s$^{-1}$ and last one-third at $v_{3}$ m s$^{-1}$. If $v_{3} = 3v_{1}$, $v_{2} = 2v_{1}$ and $v_{1} = 11$ m s$^{-1}$, then the average velocity of the car is \_\_\_\_ m s$^{-1}$.

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is $I_1$. The same rod is bent into a ring and its moment of inertia about a diameter is $I_2$. If $\frac{I_1}{I_2}$ is $\frac{x\pi^2}{3}$, then the value of $x$ will be $\_\_\_\_$.

A ball of mass 100 g is dropped from a height $h = 10 \mathrm {~cm}$ on a platform fixed at the top of a vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed by a distance $\frac { h } { 2 }$. The spring constant is $\_\_\_\_$ $\mathrm { N } \mathrm { m } ^ { - 1 }$ (Use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)

A curve in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be $30 \mathrm{~m~s}^{-1}$ without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be $\_\_\_\_$ $\mathrm{m~s}^{-1}$.

300 calories of heat is given to a heat engine, and it rejects 225 calories of heat. If source temperature is $227 ^ { \circ } \mathrm { C }$, then the temperature of sink will be $\_\_\_\_$ ${ } ^ { \circ } \mathrm { C }$.

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be $\_\_\_\_$ m. Given, the length of the rod is $10 \sqrt { 3 } \mathrm{~m}$.

If the acceleration due to gravity experienced by a point mass at a height $h$ above the surface of earth is same as that of the acceleration due to gravity at a depth $\alpha h \left( h \ll R _ { e } \right)$ from the earth surface. The value of $\alpha$ will be $\_\_\_\_$. (use $R _ { e } = 6400 \mathrm {~km}$)

A uniform disc with mass $M = 4$ kg and radius $R = 10$ cm is mounted on a fixed horizontal axle as shown in figure. A block with mass $m = 2$ kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is \_\_\_\_ N.
(Take $g = 10$ m s$^{-2}$)

A small spherical ball of radius 0.1 mm and density $10^4 \mathrm{~kg~m}^{-3}$ falls freely under gravity through a distance $h$ before entering a tank of water. If, after entering the water the velocity of ball does not change and it continues to fall with same constant velocity inside water, then the value of $h$ will be $\_\_\_\_$ m. (Given $g = 10 \mathrm{~m~s}^{-2}$, viscosity of water $= 1.0 \times 10^{-5} \mathrm{~N\text{-}s~m}^{-2}$).

A uniform heavy rod of mass 20 kg, cross sectional area $0.4 \mathrm{~m}^{2}$ and length 20 m is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is $x \times 10^{-9}$ m. The value of $x$ is $\_\_\_\_$. (Given: Young's modulus $Y = 2 \times 10^{11} \mathrm{~Nm}^{-2}$ and $g = 10 \mathrm{~ms}^{-2}$)

Four identical discs each of mass '$M$' and diameter '$a$' are arranged in a small plane as shown in figure. If the moment of inertia of the system about $OO'$ is $\frac{x}{4}Ma^2$. Then, the value of $x$ will be \_\_\_\_. [Figure]

A liquid of density $750$ kg m$^{-3}$ flows smoothly through a horizontal pipe that tapers in cross-sectional area from $A_{1} = 1.2 \times 10^{-2}$ m$^{2}$ to $A_{2} = \frac{A_{1}}{2}$. The pressure difference between the wide and narrow sections of the pipe is 4500 Pa. The rate of flow of liquid is \_\_\_\_ $\times 10^{-3}$ m$^{3}$ s$^{-1}$.

In an experiment to determine the velocity of sound in air at room temperature using a resonance tube, the first resonance is observed when the air column has a length of 20.0 cm for a tuning fork of frequency 400 Hz is used. The velocity of the sound at room temperature is $336 \mathrm{~m~s}^{-1}$. The third resonance is observed when the air column has a length of $\_\_\_\_$ cm.

Moment of Inertia (M.I.) of four bodies having same mass $M$ and radius $2R$ are as follows $I_1 =$ M.I. of solid sphere about its diameter $I_2 =$ M.I. of solid cylinder about its axis $I_3 =$ M.I. of solid circular disc about its diameter $I_4 =$ M.I. of thin circular ring about its diameter If $2I_2 + I_3 + I_4 = xI_1$ then the value of $x$ will be $\_\_\_\_$.

The position vector of 1 kg object is $\vec{r} = (3\hat{\mathrm{i}} - \hat{\mathrm{j}})\mathrm{~m}$ and its velocity $\vec{v} = (3\hat{\mathrm{j}} + \hat{\mathrm{k}})\mathrm{~m\,s^{-1}}$. The magnitude of its angular momentum is $\sqrt{x}\mathrm{~N\,m\,s}$, where $x$ is $\_\_\_\_$.

Two coils require 20 minutes and 60 minutes respectively to produce same amount of heat energy when connected separately to the same source. If they are connected in parallel arrangement to the same source; the time required to produce same amount of heat by the combination of coils, will be $\_\_\_\_$ min.

Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the $\sqrt{3}$ times of amplitude of individual motions. The phase difference between the two motions is $\_\_\_\_$ (degree)

A tunning fork of frequency 340 Hz resonates in the fundamental mode with an air column of length 125 cm in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is \_\_\_\_ cm.
(Velocity of sound in air is $340$ m s$^{-1}$)

The displacement current of $4.425 \mu\mathrm{A}$ is developed in the space between the plates of parallel plate capacitor when voltage is changing at a rate of $10^6 \mathrm{~V~s}^{-1}$. The area of each plate of the capacitor is $40 \mathrm{~cm}^2$. The distance between each plate of the capacitor is $x \times 10^{-3}$ m. The value of $x$ is $\_\_\_\_$. (Permittivity of free space, $\varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$)

A heat engine operates with the cold reservoir at temperature 324 K. The minimum temperature of the hot reservoir, if the heat engine takes 300 J heat from the hot reservoir and delivers 180 J heat to the cold reservoir per cycle, is $\_\_\_\_$ K.

Two satellites $S_1$ and $S_2$ are revolving in circular orbits around a planet with radius $R_1 = 3200 \mathrm{~km}$ and $R_2 = 800 \mathrm{~km}$ respectively. The ratio of speed of satellite $S_1$ to the speed of satellite $S_2$ in their respective orbits would be $\frac{1}{x}$ where $x =$ $\_\_\_\_$.