Eight identical spherical liquid drops, each falling with a terminal velocity \(v\), coalesce to form a single large drop. The terminal velocity of the new large drop is:

• (A) \(2v\)
• (B) \(4v\)
• (C) \(8v\)
• (D) \(16v\)

A uniform solid sphere of mass \(M\) and radius \(R\) is placed on a smooth horizontal surface. It is struck by a horizontal cue at a height \(h\) above the center. For the sphere to roll without slipping immediately after the impact, the value of \(h\) must be:

• (A) \(\frac{R}{2}\)
• (B) \(\frac{2R}{5}\)
• (C) \(\frac{2R}{3}\)
• (D) \(\frac{3R}{5}\)

A disc of mass \(M\) and radius \(R\) has a concentric hole of radius \(\frac{R}{2}\). Its moment of inertia about an axis passing through its center and perpendicular to its plane is:

• (A) \(\frac{15}{32} MR^2\)
• (B) \(\frac{13}{32} MR^2\)
• (C) \(\frac{5}{16} MR^2\)
• (D) \(\frac{3}{8} MR^2\)

The work done in increasing the radius of a soap bubble from \(R\) to \(2R\) is \(W\). The work done in further increasing its radius from \(2R\) to \(3R\) will be:

• (A) \(\frac{5}{3} W\)
• (B) \(\frac{4}{3} W\)
• (C) \(\frac{7}{3} W\)
• (D) \(W\)

Water flows through a horizontal pipe. At one point, the velocity is \(2 m/s\) and the pressure is \(2 m\) of water column. What is the pressure at another point where the velocity is \(3 m/s\)? (Density of water = \(10^3 kg/m^3\))

• (A) \(1.75 m\) of water
• (B) \(1.2 m\) of water
• (C) \(1 m\) of water
• (D) \(1.5 m\) of water

Identify the type of intermolecular force present between benzene and ammonia.

• (A) Hydrogen bonding
• (B) Dipole-induced dipole interaction
• (C) Dipole-dipole interaction
• (D) Ion-dipole interaction

What is the SI unit for rate of diffusion?

• (A) \(cm^3 minute^{-1}\)
• (B) \(dm^3 hour^{-1}\)
• (C) \(dm^3 s^{-1}\)
• (D) \(mL hour^{-1}\)

Which of the following pair of liquids has dipole-induced dipole interaction?

• (A) Water and ammonia
• (B) Water and chloroform
• (C) Water and benzene
• (D) Water and ethyl alcohol

A certain mass of a gas occupies a volume of \(2.5 dm^3\) at NTP. Calculate the change in volume of gas at the same temperature, if pressure of gas is changed to \(1.25 atm\).

• (A) \(3.0 dm^3\)
• (B) \(0.5 dm^3\)
• (C) \(4.5 dm^3\)
• (D) \(1.5 dm^3\)

A gas occupies a volume of \(4.2 dm^3\) at \(101 kPa\) pressure. What volume will the gas occupy if the pressure is increased to \(235 kPa\) keeping the temperature constant?

• (A) \(1.8 dm^3\)
• (B) \(3.6 dm^3\)
• (C) \(0.9 dm^3\)
• (D) \(2.1 dm^3\)

Let \(f(x) = \int \frac{x\sqrt{x}}{(1 + x)^2} dx \quad (x \geq 0)\). Then \(f(3) - f(1)\) is equal to:

• (A) \(-\frac{1}{12} + \frac{1}{2} + \frac{\sqrt{3}}{4}\)
• (B) \(\frac{1}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}\)
• (C) \(-\frac{1}{6} + \frac{1}{2} + \frac{\sqrt{3}}{4}\)
• (D) \(\frac{1}{6} + \frac{1}{2} - \frac{\sqrt{3}}{4}\)

Gas is being pumped into a spherical balloon at the rate of \(30 ft^3/min\). Find the rate at which the radius increases when the radius becomes \(15 ft\).

• (A) \(\frac{1}{30} ft/min\)
• (B) \(\frac{1}{15} ft/min\)
• (C) \(\frac{1}{30\pi} ft/min\)
• (D) \(\frac{1}{20\pi} ft/min\)

Let the function \(f(x) = \frac{x - |x|}{x}\). Then:

• (A) the function is continuous everywhere
• (B) the function is not continuous
• (C) the function is continuous when \(x < 0\)
• (D) the function is continuous for all \(x\) except zero

If the direction ratios of two lines are given by \(l + m + n = 0\) and \(mn - 2ln + lm = 0\), then the angle between the lines is:

• (A) \(\frac{\pi}{4}\)
• (B) \(\frac{\pi}{3}\)
• (C) \(\frac{\pi}{2}\)
• (D) \(0\)

The equation of normal to the curve \(y = \log_e x\) at the point \(P(1, 0)\) is:

• (A) \(2x + y = 2\)
• (B) \(x - 2y = 1\)
• (C) \(x - y = 1\)
• (D) \(x + y = 1\)