The surface tension of soap solution is $3.5 \times 10 ^ { - 2 } \mathrm {~N} \mathrm {~m} ^ { - 1 }$. The amount of work done required to increase the radius of soap bubble from 10 cm to 20 cm is $\_\_\_\_$ $\times 10 ^ { - 4 } \mathrm {~J}$. $\left($ take $\left. \pi = \frac { 22 } { 7 } \right)$

The elastic potential energy stored in a steel wire of length 20 m stretched through 2 cm is 80 J. The cross sectional area of the wire is $\_\_\_\_$ mm$^2$. (Given, $Y = 2.0 \times 10^{11}$ N m$^{-2}$)

A Carnot engine operating between two reservoirs has efficiency $\frac { 1 } { 3 }$. When the temperature of cold reservoir raised by $x$, its efficiency decreases to $\frac { 1 } { 6 }$. The value of $x$, if the temperature of hot reservoir is $99 ^ { \circ } C$, will be
(1) 16.5 K
(2) 33 K
(3) 66 K
(4) 62 K

A car $P$ travelling at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ sounds its horn at a frequency of 400 Hz. Another car $Q$ is travelling behind the first car in the same direction with a velocity $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The frequency heard by the passenger of the car $Q$ is approximately [Take, velocity of sound $= 360 \mathrm {~m} \mathrm {~s} ^ { - 1 }$]
(1) 421 Hz
(2) 471 Hz
(3) 485 Hz
(4) 514 Hz

A wire of density $8 \times 10 ^ { 3 } \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ is stretched between two clamps 0.5 m apart. The extension developed in the wire is $3.2 \times 10 ^ { - 4 } \mathrm {~m}$. If $Y = 8 \times 10 ^ { 10 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$, the fundamental frequency of vibration in the wire will be $\_\_\_\_$ Hz

At a given point of time the value of displacement of a simple harmonic oscillator is given as $y = A\cos\left(30^\circ\right)$. If amplitude is 40 cm and kinetic energy at that time is 200 J, the value of force constant $1.0 \times 10^x$ N m$^{-1}$. The value of $x$ is $\_\_\_\_$.

The ratio of intensities at two points $P$ and $Q$ on the screen in a Young's double slit experiment where phase difference between two waves of same amplitude are $\frac { \pi } { 3 }$ and $\frac { \pi } { 2 }$, respectively are
(1) $2 : 3$
(2) $1 : 3$
(3) $3 : 1$
(4) $3 : 2$

If $V$ is the gravitational potential due to sphere of uniform density on its surface, then its value at the centre of sphere will be:
(1) $\frac { 4 } { 3 } V$
(2) $\frac { V } { 2 }$
(3) $V$
(4) $\frac { 3 V } { 2 }$

The half life of a radioactive substance is $T$. The time taken, for disintegrating $\frac { 7 ^ { \text {th } } } { 8 }$ part of its original mass will be:
(1) $2 T$
(2) $3 T$
(3) $T$
(4) $8 T$

A metallic surface is illuminated with radiation of wavelength $\lambda$, the stopping potential is $V_0$. If the same surface is illuminated with radiation of wavelength $2\lambda$, the stopping potential becomes $\frac{V_0}{4}$. The threshold wavelength for this metallic surface will be
(1) $3\lambda$
(2) $4\lambda$
(3) $\frac{3}{2}\lambda$
(4) $\frac{\lambda}{4}$

Equivalent resistance between the adjacent corners of a regular $n$-sided polygon of uniform wire of resistance $R$ would be :
(1) $\frac { ( n - 1 ) R } { n ^ { 2 } }$
(2) $\frac { ( n - 1 ) R } { ( 2 n - 1 ) }$
(3) $\frac { n ^ { 2 } R } { n - 1 }$
(4) $\frac { ( n - 1 ) R } { n }$

Two radioactive elements $A$ and $B$ initially have same number of atoms. The half life of $A$ is same as the average life of $B$. If $\lambda_A$ and $\lambda_B$ are decay constants of $A$ and $B$ respectively, then choose the correct relation from the given options.
(1) $\lambda_A \ln 2 = \lambda_B$
(2) $\lambda_A = \lambda_B$
(3) $\lambda_A = \lambda_B \ln 2$
(4) $\lambda_A = 2\lambda_B$

A potential $V_0$ is applied across a uniform wire of resistance $R$. The power dissipation is $P_1$. The wire is then cut into two equal halves and a potential of $V_0$ is applied across the length of each half. The total power dissipation across two wires is $P_2$. The ratio of $P_2 : P_1$ is $\sqrt{x} : 1$. The value of $x$ is $\_\_\_\_$.

When a resistance of $5\,\Omega$ is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of 250 mA, however when $1050\,\Omega$ resistance is connected with it in series, it gives full scale deflection for 25 volt. The resistance of galvanometer is $\_\_\_\_$ $\Omega$.

By what percentage will the transmission range of a TV tower be affected when the height of the tower is increased by $21\%$?
(1) $15\%$
(2) $12\%$
(3) $10\%$
(4) $14\%$

A transmitting antenna is kept on the surface of the earth. The minimum height of receiving antenna required to receive the signal in line of sight at 4 km distance from it is $x \times 10^{-2}$ m. The value of $x$ is $\_\_\_\_$. (Let, radius of earth $R = 6400$ km)
(1) 125
(2) 1250
(3) 12.5
(4) 1.25

A horse rider covers half the distance with $5 \mathrm{~m~s}^{-1}$ speed. The remaining part of the distance was travelled with speed $10 \mathrm{~m~s}^{-1}$ for half the time and with speed $15 \mathrm{~m~s}^{-1}$ for other half of the time. The mean speed of the rider averaged over the whole time of motion is $\frac{x}{7} \mathrm{~m~s}^{-1}$. The value of $x$ is $\_\_\_\_$.

If the maximum load carried by an elevator is 1400 kg ( 600 kg -Passengers + 800 kg -elevator) , which is moving up with a uniform speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the frictional force acting on it is 2000 N , then the maximum power used by the motor is $\_\_\_\_$ $\mathrm { kW } . \quad g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

The length of a metallic wire is increased by $20\%$ and its area of cross-section is reduced by $4\%$. The percentage change in resistance of the metallic wire is $\_\_\_\_$.

In the given figure, an inductor and resistor are connected in series with a battery of emf $E$ volt. $\frac{E^a}{2b}$ J s$^{-1}$ represents the maximum rate at which the energy is stored in the magnetic field (inductor). The numerical value of $\frac{b}{a}$ will be $\_\_\_\_$.

A thin uniform rod of length 2 m, cross sectional area $A$ and density $d$ is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $\omega$. If value of $\omega$ in terms of its rotational kinetic energy $E$ is $\sqrt{\frac{\alpha E}{Ad}}$, then the value of $\alpha$ is $\_\_\_\_$.

A force of $- P \hat { k }$ acts on the origin of the coordinate system. The torque about the point $( 2 , - 3 )$ is $P ( a \hat { i } + b \hat { j } )$ , The ratio of $\frac { a } { b }$ is $\frac { x } { 2 }$. The value of $x$ is

A light rope is wound around a hollow cylinder of mass 5 kg and radius 70 cm. The rope is pulled with a force of 52.5 N. The angular acceleration of the cylinder will be $\_\_\_\_$ rad $s ^ { - 2 }$.

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively $\left( k _ { s p h } : k _ { c y l } \right)$ is $2 : \sqrt { x }$. The value of $x$ is $\_\_\_\_$.

The general displacement of a simple harmonic oscillator is $x = A \sin \omega t$. Let $T$ be its time period. The slope of its potential energy $(U)$ - time $(t)$ curve will be maximum when $t = \frac{T}{\beta}$. The value of $\beta$ is $\_\_\_\_$.