JEE Advanced 2012 · Paper 2

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If ρc is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

1. Amore than half-filled if ρc is less than 0.5.
2. Bmore than half-filled if ρc is more than 1.0.
3. Chalf-filled if ρc is more than 0.5.
4. Dless than half-filled if ρc is less than 0.5.

Two moles of ideal helium gas are in a rubber balloon at 30°C. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35°C. The amount of heat required in raising the temperature is nearly (take R = 8.31 J/mol.K)

1. A62 J
2. B104 J
3. C124 J
4. D208 J

A student is performing the experiment of Resonance Column. The diameter of the column tube is 4 cm. The frequency of the tuning fork is 512 Hz. The air temperature is 38°C in which the speed of sound is 336 m/s. The zero of the meter scale coincides with the top end of the Resonance Column tube. When the first resonance occurs, the reading of the water level in the column is

1. A14.0 cm
2. B15.2 cm
3. C16.4 cm
4. D17.6 cm

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)

1. AThe emf induced in the loop is zero if the current is constant.
2. BThe emf induced in the loop is finite if the current is constant.
3. CThe emf induced in the loop is zero if the current decreases at a steady rate.
4. DThe emf induced in the loop is finite if the current decreases at a steady rate.

Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

1. ABoth cylinders P and Q reach the ground at the same time.
2. BCylinder P has larger linear acceleration than cylinder Q.
3. CBoth cylinders reach the ground with same translational kinetic energy.
4. DCylinder Q reaches the ground with larger angular speed.

Two spherical planets P and Q have the same uniform density ρ, masses Mₚ and Mq, and surface areas A and 4A, respectively. A spherical planet R also has uniform density ρ and its mass is (Mₚ + Mq). The escape velocities from the planets P, Q and R, are Vₚ, Vq and Vᵣ, respectively. Then

1. AVq > Vᵣ > Vₚ
2. BVᵣ > Vq > Vₚ
3. CVᵣ/Vₚ = 3
4. DVₚ/Vq = 1/2