Q.1.
(a) What is the gradient of a scalar function? Derive an expression for grad φ. (2,4)
(b) Prove that the scalar quadruple product can be expressed as: (10)

(Â × B̂) · (Ĉ × D̂) =

 · Ĉ    · D̂
B̂ · Ĉ   B̂ · D̂

(c) Show that the vectors  = î + ĵ + 2k̂ and B̂ = 2î + 2ĵ + 4k̂ are parallel to each other. (4)

Q.2.
(a) Write a note on various law of conservations. (8)
(b) Which law of conservation guides the balanced motion of a bicycle? (2)
(c) If the potential energy of two particles U = ^a⁄_{(x1 – x2)ⁿ} – b(x₁ – x₂)². Show that they exerts equal and opposite force on each other. (10)

Q.3.
(a) A stick of length l is at rest in one system and oriented at an angle θ with respect to x-axis; What are the length and orientation angle of the stick as viewed by the observer moving with the speed v with respect to the first system. (12)
(b) State Postulate of Special Theory of relativity. (4)
(c) “There was a girl named Ms. Bright. She could travel faster than light. She went out one day in an Einsteinian way and came back the previous night.” Justify it. (4)

Q.4.
(a) How is the negative results of Michelson-Morley experiment interpreted according to Einstein theory of relativity. (12)
(b) A wave has a speed of 243 m/s and a wavelength of 3.27cm. calculate the frequency and the period of the wave. (8)

Q.5.
(a) Viscosity is an example of a transport phenomenon. What property is being transported? Can you think of other transport phenomena and their corresponding properties? (6)
(b) How much work is done by pressure in forcing 1.4 m³ of water through a 13-mm internal diameter pipe if the difference in pressure at the two ends of the pipe is 1.2 atm? (8)
(c) Give a molecular explanation of why surface tension decreases with increasing temperature. (6)

Q.6.
(a) When two waves interfere, does one alter the progress of the other. (5)
(b) Two waves of the same amplitude and frequency are travelling on the same string. At a certain instant the string looks like a straight line. Are the two waves necessarily travelling in the same direction? What is the phase relationship between the two waves? (10)
(c) if two waves differ only in amplitude and are propagated in opposite directions through a medium, will they produce standing waves? Is energy transported? Are there any nodes? (5)

Q.7.
(a) List examples of Brownian motion in physical phenomena. (8)
(b) Show that a volume per mole of a gas increases, the van der Waals equation tends to the equation of state of an ideal gas. (6)
(c) State laws of thermodynamics. (6)

Q.8.
(a) Explain Maxwell distribution of molecular velocities. (7)
(b) Discuss Fermi-Dirac statistics in detail. (7)
(c) What is the significance of the Fermi energy in a fermion system at 0K? at T > 0K? (6)