JEE Advanced 2015 · Paper 1

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

1. AIf the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm.
2. BIf the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.005 mm.
3. CIf the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm.
4. DIf the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.005 mm.

Planck’s constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are)

1. AM ∝ √c
2. BM ∝ √G
3. CL ∝ √h
4. DL ∝ √G

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature T. Assuming the gases are ideal, the correct statement(s) is(are)

1. AThe average energy per mole of the gas mixture is 2RT.
2. BThe ratio of speed of sound in the gas mixture to that in helium gas is √(6/5).
3. CThe ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/2.
4. DThe ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/√2.

Match the nuclear processes given in column I with the appropriate option(s) in column II. Column I:

1. ANuclear fusion
2. BFission in a nuclear reactor
3. Cβ-decay
4. Dγ-ray emission. Column II: (P) Absorption of thermal neutrons by ²³⁵₉₂U; (Q) ⁶⁰₂₇Co nucleus; (R) Energy production in stars via hydrogen conversion to helium; (S) Heavy water; (T) Neutrino emission

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U₀ are constants). Match the potential energies in column I to the corresponding statement(s) in column II. Column I:

1. AU₁(x) = U₀/2 [1 − (x/a)²]²
2. BU₂(x) = U₀/2 (x/a)²
3. CU₃(x) = U₀/2 (x/a)² exp[−(x/a)²]
4. DU₄(x) = U₀/2 [x/a − 1/3 (x/a)³]. Column II: (P) The force acting on the particle is zero at x = a.; (Q) The force acting on the particle is zero at x = 0.; (R) The force acting on the particle is zero at x = −a.; (S) The particle experiences an attractive force towards x = 0 in the region |x| < a.; (T) The particle with total energy U₀/4 can oscillate about the point x = −a.