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An ideal gas undergoes a four step cycle as shown in the \( P{-}V \) diagram below. During this cycle, heat is absorbed by the gas in

An extended object is placed at point \( O \), \( 10 \, \text{cm} \) in front of a convex lens \( L_{1} \) and a concave lens \( L_{2} \) is placed \( 10 \, \text{cm} \) behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are \( 20 \, \text{cm} \). The refractive index of both the lenses is \( 1.5 \). The total magnification of this lens system is :

A heavy nucleus \( Q \) of half-life \( 20 \) minutes undergoes alpha-decay with probability of \( 60\% \) and beta-decay with probability of \( 40\% \). Initially, the number of \( Q \) nuclei is \( 1000 \). The number of alpha-decays of \( Q \) in the first one hour is :

A projectile is thrown from a point \( O \) on the ground at an angle \( 45^\circ \) from the vertical and with a speed \( 5\sqrt{2} \, \text{m/s} \). The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, \( 0.5 \, \text{s} \) after the splitting. The other part, \( t \) seconds after the splitting, falls to the ground at a distance \( x \) meters from the point \( O \). The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \).

The value of \( t \) is _____

A projectile is thrown from a point \( O \) on the ground at an angle \( 45^\circ \) from the vertical and with a speed \( 5\sqrt{2} \, \text{m/s} \). The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, \( 0.5 \, \text{s} \) after the splitting. The other part, \( t \) seconds after the splitting, falls to the ground at a distance \( x \) meters from the point \( O \). The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \).

The value of \( x \) is _____

In the circuit shown below, the switch \( S \) is connected to position \( P \) for a long time so that the charge on the capacitor becomes \( q_1 \, \mu\text{C} \). Then \( S \) is switched to position \( Q \). After a long time, the charge on the capacitor is \( q_2 \, \mu\text{C} \).

The magnitude of \( q_1 \) is _____.

In the circuit shown below, the switch \( S \) is connected to position \( P \) for a long time so that the charge on the capacitor becomes \( q_1 \, \mu\text{C} \). Then \( S \) is switched to position \( Q \). After a long time, the charge on the capacitor is \( q_2 \, \mu\text{C} \).

The magnitude of \( q_2 \) is _____.

Two point charges \( -Q \) and \( +Q/\sqrt{3} \) are placed in the \( xy \)-plane at the origin \( (0,0) \) and a point \( (2,0) \), respectively, as shown in the figure. This results in an equipotential circle of radius \( R \) and potential \( V = 0 \) in the \( xy \)-plane with its center at \( (b,0) \). All lengths are measured in meters.

The value of \( R \) is _____ meter.

Two point charges \( -Q \) and \( +Q/\sqrt{3} \) are placed in the \( xy \)-plane at the origin \( (0,0) \) and a point \( (2,0) \), respectively, as shown in the figure. This results in an equipotential circle of radius \( R \) and potential \( V = 0 \) in the \( xy \)-plane with its center at \( (b,0) \). All lengths are measured in meters.

The value of \( b \) is _____ meter.

A horizontal force \( F \) is applied at the center of mass of a cylindrical object of mass \( m \) and radius \( R \), perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is \( \mu \). The center of mass of the object has an acceleration \( a \). The acceleration due to gravity is \( g \). Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

A wide slab consisting of two media of refractive indices \( n_{1} \) and \( n_{2} \) is placed in air as shown in the figure. A ray of light is incident from medium \( n_{1} \) to \( n_{2} \) at an angle \( \theta \), where \( \sin \theta \) is slightly larger than \( \frac{1}{n_{1}} \). Take refractive index of air as \( 1 \). Which of the following statement(s) is(are) correct?

A particle of mass \( M = 0.2 \, \text{kg} \) is initially at rest in the \( xy \)-plane at a point \( (x = -l, \; y = -h) \), where \( l = 10 \, \text{m} \) and \( h = 1 \, \text{m} \). The particle is accelerated at time \( t = 0 \) with a constant acceleration \( a = 10 \, \text{m} \,/ \text{s}^{2} \) along the positive \( x \)-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by \( \vec{L} \) and \( \vec{\tau} \), respectively. \( \hat{i} \), \( \hat{j} \) and \( \hat{k} \) are unit vectors along the positive \( x \), \( y \) and \( z \)-directions, respectively. If \( \hat{k} = \hat{i} \times \hat{j} \) then which of the following statement(s) is(are) correct?

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

A long straight wire carries a current, \( I = 2 \, \text{ampere} \). A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances \( 1 \, \text{cm} \) and \( 4 \, \text{cm} \) from the wire. At time \( t = 0 \), the rod starts moving on the rails with a speed \( v = 3.0 \, \text{m/s} \) (see the figure). A resistor \( R = 1.4 \, \Omega \) and a capacitor \( C_{0} = 5.0 \, \mu \text{F} \) are connected in series between the rails. At time \( t = 0 \), \( C_{0} \) is uncharged. Which of the following statement(s) is(are) correct? \( [ \mu_{0} = 4 \pi \times 10^{-7} \, \text{SI units}. \, \text{Take } \ln 2 = 0.7 ] \)

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration \( a \) along a fixed inclined plane with angle \( \theta = 45^{\circ} \). \( P_{1} \) and \( P_{2} \) are pressures at points \( 1 \) and \( 2 \), respectively located at the base of the tube. Let \( \beta = (P_{1} - P_{2}) / (\rho g d) \), where \( \rho \) is density of water, \( d \) is the inner diameter of the tube and \( g \) is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

An \( \alpha \)-particle (mass \( 4 \, \text{amu} \)) and a singly charged sulfur ion (mass \( 32 \, \text{amu} \)) are initially at rest. They are accelerated through a potential \( V \) and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the \( \alpha \)-particle and the sulfur ion move in circular orbits of radii \( r_{\alpha} \) and \( r_{s} \), respectively. The ratio \( ( \frac{r_{s}}{r_{\alpha}} ) \) is _____

A thin rod of mass \( M \) and length \( a \) is free to rotate in horizontal plane about a fixed vertical axis passing through point \( O \). A thin circular disc of mass \( M \) and of radius \( \frac{a}{4} \) is pivoted on this rod with its center at a distance \( \frac{a}{4} \) from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity \( \Omega \) and the disc rotating about its vertical axis with angular velocity \( 4 \Omega \). The total angular momentum of the system about the point \( O \) is \( \left( \frac{M a^{2} \Omega}{48} \right) n \). The value of \( n \) is _____

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at \( 0 \, \text{K} \). At time \( t = 0 \), the temperature of the object is \( 200 \, \text{K} \). The temperature of the object becomes \( 100 \, \text{K} \) at \( t = t_1 \), and \( 50 \, \text{K} \) at \( t = t_2 \). Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio \( (t_2 / t_1) \) is ____.