Q1.

The equation of state of some gases can be expressed as:

(P + a / V²) (V − b) = RT

where P is pressure, V is volume, T is absolute temperature and a, b, R are constants. The dimensions of ‘a’ are:

(a) [ML⁵T^-2]
(b) [M⁰L³T⁰]
(c) [ML^-1T^-2]
(d) [M⁰L⁶T⁰]

Q2.

The time period of oscillation ‘T’ of a gas bubble under water depends upon P, ρ and E, where ‘P’ is the static pressure, ‘ρ’ is the density of water and ‘E’ is the total energy associated with the formation of the bubble. Using the method of dimensions, show that, for constant values of ‘P’ and ‘E’, the time period is directly proportional to the square root of the density ‘ρ’.

Q4.

In the relation, y = r sin(ωt + kx), the dimensional formula for kx or ωt is same as:

(a) r / ω
(b) r / y
(c) ωt / r
(d) yr / ωt

Q5.

Assertion: When we change the unit of measurement of a quantity, its numerical value changes.

Reason: Smaller the unit of measurement smaller is its numerical value.

(a) If both Assertion and Reason are true and Reason is the correct explanation of Assertion.
(b) If both Assertion and Reason are true but Reason is not the correct explanation.
(c) If Assertion is true but Reason is false.
(d) If both Assertion and Reason are false.

Q6.

If P, Q, R are physical quantities having different dimensions, which of the following combinations can never be a meaningful quantity?

(a) (P − Q)/R
(b) PQ − R
(c) PQ/R
(d) (PR − Q²)/R

Q7.

Taking into account of the significant figures, what is the value of 9.99 m − 0.0099 m?

(a) 9.9801 m
(b) 9.98 m
(c) 9.980 m
(d) 9.9 m

Q8.

In the expression P = EL² m^-5 G^-2, E, l, m and G denote energy, angular momentum, mass and universal gravitational constant respectively. Show that P is a dimensionless quantity.

Q9.

Use 𝐀 = î − 2ĵ + k̂, 𝐁 = 2î − k̂ and 𝐂 = î − xĵ as reference vectors.

(i) Find the value of x if (𝐀 + 𝐁) is perpendicular to 𝐂.
(ii) Prove that dot product is distributive over addition using the above vectors.
(iii) Find a vector parallel to 𝐁 and having same magnitude as 𝐀.
(iv) Find a vector which is perpendicular to both 𝐀 & 𝐁.

Q10.

(i) Derive an expression for the magnitude of centripetal acceleration of a body moving with uniform speed v along a circular path of radius r.

(ii) An insect trapped in a circular groove of radius 12 cm moves along the groove steadily and completes seven revolutions in 100 sec. What will be the angular speed and linear speed of motion?

Q11.

State Kepler’s law of areas. Show that it is a consequence of conservation of angular momentum.

Q12.

Derive the law of conservation of linear momentum from Newton’s third law of motion.

Q13.

Derive the following kinematic relations by graphical method:
(a) v = u + at
(b) s = ut + ½at²