The volume of an ideal gas ($\gamma = 1.5$) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is:
(1) $\frac { 16 } { 25 }$
(2) $\frac { 4 } { 5 }$
(3) $\frac { 8 } { 5 \sqrt { 5 } }$
(4) $\frac { 2 } { \sqrt { 5 } }$

Two moles of a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is:
(1) $\frac { 9 } { 4 } R$
(2) $\frac { 7 } { 4 } R$
(3) $\frac { 3 } { 2 } R$
(4) $\frac { 5 } { 2 } R$

Two identical capacitors have same capacitance $C$. One of them is charged to the potential $V$ and other to the potential $2V$. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is:
(1) $\frac { 1 } { 4 } C V ^ { 2 }$
(2) $2 C V ^ { 2 }$
(3) $\frac { 1 } { 2 } C V ^ { 2 }$
(4) $\frac { 3 } { 4 } C V ^ { 2 }$

A galvanometer has a resistance of $50 \Omega$ and it allows maximum current of 5 mA. It can be converted into voltmeter to measure upto 100 V by connecting in series a resistor of resistance.
(1) $5975 \Omega$
(2) $20050 \Omega$
(3) $19950 \Omega$
(4) $19500 \Omega$

In series LCR circuit, the capacitance is changed from $C$ to $4C$. To keep the resonance frequency unchanged, the new inductance should be:
(1) reduced by $\frac { 1 } { 4 } L$
(2) increased by $2L$
(3) reduced by $\frac { 3 } { 4 } L$
(4) increased to $4L$

A monochromatic light of wavelength $6000 \mathrm {~\AA}$ is incident on the single slit of width 0.01 mm. If the diffraction pattern is formed at the focus of the convex lens of focal length 20 cm, the linear width of the central maximum is:
(1) 60 mm
(2) 24 mm
(3) 120 mm
(4) 12 mm

The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2\lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be:
(1) $1 : 8$
(2) $1 : 2$
(3) $4 : 1$
(4) $8 : 1$

The minimum energy required by a hydrogen atom in ground state to emit radiation in Balmer series is nearly:
(1) 1.5 eV
(2) 13.6 eV
(3) 1.9 eV
(4) 12.1 eV

An electron rotates in a circle around a nucleus having positive charge Ze. Correct relation between total energy (E) of electron to its potential energy (U) is:
(1) $\mathrm{E} = \mathrm{U}$
(2) $2\mathrm{E} = \mathrm{U}$
(3) $2\mathrm{E} = 3\mathrm{U}$
(4) $\mathrm{E} = 2\mathrm{U}$

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is:
(1) $\frac { 1 } { 2 }$
(2) $\frac { 10 } { 11 }$
(3) $\frac { 50 } { 11 }$
(4) $\frac { 5 } { 11 }$

One main scale division of a vernier caliper is equal to $m$ units. If $\mathrm { n } ^ { \text {th } }$ division of main scale coincides with $( n + 1 ) ^ { \text {th } }$ division of vernier scale, the least count of the vernier caliper is :
(1) $\frac { n } { ( n + 1 ) }$
(2) $\frac { 1 } { ( n + 1 ) }$
(3) $\frac { m } { ( n + 1 ) }$
(4) $\frac { m } { n ( n + 1 ) }$

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $\dfrac{x}{20}$ kg m$^2$, where the value of $x$ is

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be $\_\_\_\_$ hours 30 minutes.

Each of three blocks $P , Q$ and $R$ shown in figure has a mass of 3 kg . Each of the wire $A$ and $B$ has cross-sectional area $0.005 \mathrm {~cm} ^ { 2 }$ and Young's modulus $2 \times 10 ^ { 11 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$. Neglecting friction, the longitudinal strain on wire $B$ is $\_\_\_\_$ $\times 10 ^ { - 4 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ )

Two blocks of mass 2 kg and 4 kg are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is $4.0 \times 10^{-5}$ m and Young's modulus of the metal is $2.0 \times 10^{11}$ N m$^{-2}$. The longitudinal strain developed in the wire is $\dfrac{1}{\alpha \pi}$. The value of $\alpha$ is $\_\_\_\_$. [Use $g = 10$ m s$^{-2}$]

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of big drop is $\frac { 10 } { x }$. The value of $x$ is $\_\_\_\_$

A plane is in level flight at constant speed and each of its two wings has an area of $40 \mathrm {~m} ^ { 2 }$. If the speed of the air is $180 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ over the lower wing surface and $252 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ over the upper wing surface, the mass of the plane is $\_\_\_\_$ kg. (Take air density to be $1 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ and $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)

An elastic spring under tension of 3 N has a length $a$. Its length is $b$ under tension 2 N. For its length $( 3 a - 2 b )$, the value of tension will be $\_\_\_\_$ N.

The density and breaking stress of a wire are $6 \times 10^4\mathrm{~kg/m}^3$ and $1.2 \times 10^8\mathrm{~N/m}^2$ respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is $\frac{1}{3}^{\text{rd}}$ of the value on the surface of earth. The maximum length of the wire without breaking is $\_\_\_\_$ m (take, $\mathrm{g} = 10\mathrm{~m/s}^2$).

Two persons pull a wire towards themselves. Each person exerts a force of 200 N on the wire. Young's modulus of the material of wire is $1 \times 10 ^ { 11 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$. Original length of the wire is 2 m and the area of cross section is $2 \mathrm {~cm} ^ { 2 }$. The wire will extend in length by $\_\_\_\_$ $\mu \mathrm { m }$.

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s . The maximum velocity of the particle is $\_\_\_\_$ $\mathrm { cm } / \mathrm { s }$.

A tuning fork resonates with a sonometer wire of length 1 m stretched with a tension of 6 N. When the tension in the wire is changed to 54 N, the same tuning fork produces 12 beats per second with it. The frequency of the tuning fork is $\_\_\_\_$ Hz.

A soap bubble is blown to a diameter of 7 cm. 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne $/ \mathrm { cm }$ then the new radius is $\_\_\_\_$ cm. Take $\left( \pi = \frac { 22 } { 7 } \right)$

Three capacitors of capacitances $25\mu\mathrm{F}, 30\mu\mathrm{F}$ and $45\mu\mathrm{F}$ are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored energy is $\frac{9}{x}\mathrm{E}$. The value of $x$ is $\_\_\_\_$.

The distance between charges $+q$ and $-q$ is $2l$ and between $+2q$ and $-2q$ is $4l$. The electrostatic potential at point $P$ at a distance $r$ from centre $O$ is $-\alpha \dfrac{ql}{r^2} \times 10^9$ V, where the value of $\alpha$ is $\_\_\_\_$. (Use $\dfrac{1}{4\pi\varepsilon_0} = 9 \times 10^9$ N$^2$ m$^2$)