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

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 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 $\_\_\_\_$.

Figure below shows a liquid being pushed out of the tube by a piston having area of cross section $2.0 \mathrm {~cm} ^ { 2 }$. The area of cross section at the outlet is $10 \mathrm {~mm} ^ { 2 }$. If the piston is pushed at a speed of $4 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$, the speed of outgoing fluid is $\_\_\_\_$ $\mathrm { cm } \mathrm {~s} ^ { - 1 }$

Two plates $A$ and $B$ have thermal conductivities $84 \mathrm {~W} \mathrm {~m} ^ { - 1 } \mathrm {~K} ^ { - 1 }$ and $126 \mathrm {~W} \mathrm {~m} ^ { - 1 } \mathrm {~K} ^ { - 1 }$ respectively. They have same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of A and B are kept at $100 ^ { \circ } \mathrm { C }$ and $0 ^ { \circ } \mathrm { C }$ respectively, then the temperature of the surface of contact in steady state is $\_\_\_\_$ ${ } ^ { \circ } \mathrm { C }$.

Two identical solid spheres each of mass 2 kg and radii 10 cm are fixed at the ends of a light rod. The separation between the centres of the spheres is 40 cm. The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is $\_\_\_\_$ $\times 10^{-3}\mathrm{~kg~m^2}$.

A solid sphere of mass 500 g radius 5 cm is rotated about one of its diameter with angular speed of $10$ rad s$^{-1}$. If the moment of inertia of the sphere about its tangent is $x \times 10^{-2}$ times its angular momentum about the diameter. Then the value of $x$ will be

A capacitor of capacitance $900~\mu\mathrm{F}$ is charged by a 100 V battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged capacitor connected to negative plate of the charged capacitor. The loss of energy in this process is measured as $x \times 10^{-2}~\mathrm{J}$. The value of $x$ is $\_\_\_\_$.

In an experiment with sonometer when a mass of 180 g is attached to the string, it vibrates with fundamental frequency of 30 Hz. When a mass $m$ is attached, the string vibrates with fundamental frequency of 50 Hz. The value of $m$ is $\_\_\_\_$ g.

Two objects $A$ and $B$ are placed at 15 cm and 25 cm from the pole in front of a concave mirror having radius of curvature 40 cm . The distance between images formed by the mirror is:
(1) 40 cm
(2) 60 cm
(3) 160 cm
(4) 100 cm

A steel rod has a radius of 20 mm and a length of 2.0 m. A force of 62.8 kN stretches it along its length. Young's modulus of steel is $2.0 \times 10^{11}\mathrm{~N~m}^{-2}$. The longitudinal strain produced in the wire is $\_\_\_\_$ $\times 10^{-5}$.

The length of a wire becomes $l_1$ and $l_2$ when 100 N and 120 N tension are applied respectively. If $10l_2 = 11l_1$, then the natural length of wire will be $\frac{1}{x}l_1$. Here the value of $x$ is