A body having specific charge $8 \mu \mathrm { C } \mathrm { g } ^ { - 1 }$ is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of $100 \mathrm {~V} \mathrm {~m} ^ { - 1 }$ is applied horizontally towards the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be $\_\_\_\_$ s.

A person standing on a spring balance inside a stationary lift measures 60 kg. The weight of that person if the lift descends with uniform downward acceleration of $1.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ will be $\_\_\_\_$ $\mathrm { N } . \left[ g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right]$

A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $10 ^ { - 6 } \mathrm {~m} ^ { 2 }$ stretched by an amount 0.04 m. The velocity of the projected stone is $\mathrm { m } \mathrm { s} ^ { - 1 }$. (Young's modulus of rubber $= 0.5 \times 10 ^ { 9 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$)

The coefficient of static friction between two blocks is 0.5 and the table is smooth. The maximum horizontal force that can be applied to move the blocks together is $\_\_\_\_$ N (take $g = 10\text{ m s}^{-2}$)

The volume $V$ of a given mass of monoatomic gas changes with temperature $T$ according to the relation $V = K T ^ { \frac { 2 } { 3 } }$. The workdone when temperature changes by 90 K will be $x R$. The value of $x$ is [ $R$ universal gas constant]

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere, of same mass $m$ and radius $R$ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is [Mark the body as per their respective numbering given in the question]

The water is filled up to a height of 12 m in a tank having vertical sidewalls. A hole is made in one of the walls at a depth $h$ below the water level. The value of $h$ for which the emerging stream of water strikes the ground at the maximum range is $\_\_\_\_$ m.

As shown in the figure, a block of mass $\sqrt { 3 } \mathrm {~kg}$ is kept on a horizontal rough surface of coefficient of friction $\frac { 1 } { 3 \sqrt { 3 } }$. The critical force to be applied on the vertical surface as shown at an angle $60 ^ { \circ }$ with horizontal such that it does not move, will be $3x$. The value of $x$ will $\left[ g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } ; \sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 } ; \cos 60 ^ { \circ } = \frac { 1 } { 2 } \right]$ $\mu = \frac { 1 } { 3 \sqrt { 3 } }$

Two simple harmonic motions are represented by the equations $x_1 = 5\sin\left(2\pi t + \frac{\pi}{4}\right)$ and $x_2 = 5\sqrt{2}(\sin 2\pi t + \cos 2\pi t)$. The amplitude of the second motion is $\_\_\_\_$ times the amplitude in the first motion.

A body of mass $(2M)$ splits into four masses $\{m, M-m, m, M-m\}$, which are rearranged to form a square as shown in the figure. The ratio of $\frac{M}{m}$ for which, the gravitational potential energy of the system becomes maximum is $x:1$. The value of $x$ is $\_\_\_\_$.

A particle executes S.H.M. with amplitude $A$ and time period $T$. The displacement of the particle when its speed is half of maximum speed is $\frac { \sqrt { x } A } { 2 }$. The value of $x$ is

A solid disc of radius $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac { 2 } { b } g \sin \theta$, where $b$ is $\_\_\_\_$. (Round off to the Nearest Integer) ( $g =$ acceleration due to gravity) ( $\theta =$ angle as shown in figure)

The radius in kilometer to which the present radius of earth ($R = 6400 \mathrm {~km}$) to be compressed so that the escape velocity is increased 10 times is $\_\_\_\_$.

A circular disc reaches from top to bottom of an inclined plane of length $L$. When it slips down the plane, it takes time $t _ { 1 }$. When it rolls down the plane, it takes time $t _ { 2 }$. The value of $\frac { t _ { 2 } } { t _ { 1 } }$ is $\sqrt { \frac { 3 } { x } }$. The value of $x$ will be

In the given figure, two wheels $P$ and $Q$ are connected by a belt $B$. The radius of $P$ is three times that of $Q$. In the case of the same rotational kinetic energy, the ratio of rotational inertias $\left( \frac { I _ { 1 } } { I _ { 2 } } \right)$ will be $x : 1$. The value of $x$ will be $\_\_\_\_$.

A container is divided into two chambers by a partition. The volume of first chamber is 4.5 litre and second chamber is 5.5 litre. The first chamber contain 3.0 moles of gas at pressure 2.0 atm and second chamber contain 4.0 moles of gas at pressure 3.0 atm. After the partition is removed and the mixture attains equilibrium, then, the common equilibrium pressure existing in the mixture is $x \times 10 ^ { - 1 } \mathrm {~atm}$. Value of $x$ (nearest integer) is $\_\_\_\_$

Two waves are simultaneously passing through a string and their equations are: $y_1 = A_1 \sin k(x - vt)$, $y_2 = A_2 \sin k(x - vt + x_0)$. Given amplitudes $A_1 = 12\text{ mm}$ and $A_2 = 5\text{ mm}$, $x_0 = 3.5\text{ cm}$ and wave number $k = 6.28\text{ cm}^{-1}$. The amplitude of resulting wave will be $\_\_\_\_$ mm.

A rod $CD$ of thermal resistance $10.0\mathrm{~KW}^{-1}$ is joined at the middle of an identical rod $AB$ as shown in figure. The ends $A$, $B$ and $D$ are maintained at $200^{\circ}\mathrm{C}$, $100^{\circ}\mathrm{C}$ and $125^{\circ}\mathrm{C}$ respectively. The heat current in $CD$ is $P\mathrm{~W}$. The value of $P$ is $\_\_\_\_$.

Time period of a simple pendulum is $T$. The time taken to complete $\frac { 5 } { 8 }$ oscillations starting from mean position is $\frac { \alpha } { 12 } T$. The value of $\alpha$ is $\_\_\_\_$ .

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be $\frac { x } { 5 } \frac { G M ^ { 2 } } { R }$ where $x$ is $\_\_\_\_$. (Round off to the Nearest Integer) ( $M$ is the mass of earth, $R$ is the radius of earth, $G$ is the gravitational constant)

Two separate wires $A$ and $B$ are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire $B$ is 4 times that of the radius of wire $A$. The length of the wires $A$ and $B$ are in the ratio of $a : b$. Then $\frac { a } { b }$ can be expressed as $\frac { 1 } { x }$, where $x$ is $\_\_\_\_$.

In the reported figure, heat energy absorbed by a system in going through a cyclic process is $\_\_\_\_$ $\pi \mathrm { J }$.

A particle executes simple harmonic motion represented by displacement function as $x ( t ) = A \sin ( \omega t + \phi )$. If the position and velocity of the particle at $t = 0 \mathrm {~s}$ are 2 cm and $2 \omega \mathrm {~cm} \mathrm {~s} ^ { - 1 }$ respectively, then its amplitude is $x \sqrt { 2 } \mathrm {~cm}$ where the value of $x$ is $\_\_\_\_$.

The width of one of the two slits in a Young's double slit experiment is three times the other slit. If the amplitude of the light coming from a slit is proportional to the slit-width, the ratio of minimum to maximum intensity in the interference pattern is $x$ : 4 where $x$ is $\_\_\_\_$.

The mass per unit length of a uniform wire is $0.135 \mathrm {~g} \mathrm {~cm} ^ { - 1 }$. A transverse wave of the form $y = - 0.21 \sin ( x + 30 t )$ is produced in it, where $x$ is in meter and $t$ is in second. Then, the expected value of tension in the wire is $x \times 10 ^ { - 2 } \mathrm {~N}$. Value of $x$ is $\_\_\_\_$ (Round-off to the nearest integer)