A body of mass \( m \) is dropped from a height \( h \). What is its kinetic energy just before it hits the ground?

• (A) \( mgh \)
• (B) \( \frac{1}{2} mgh \)
• (C) \( 2mgh \)
• (D) \( mgh^2 \)

Two charges \( q_1 \) and \( q_2 \) are placed at a distance \( r \). If the distance between them is doubled, the electrostatic force between them becomes:

• (A) One-fourth of the original force
• (B) Half of the original force
• (C) Four times the original force
• (D) Twice the original force

A light ray enters a glass slab of refractive index \( \mu = 1.5 \) from air. What is the speed of light inside the glass slab? (Speed of light in air \( c = 3 \times 10^8 m/s \))

• (A) \( 2 \times 10^8 m/s \)
• (B) \( 4.5 \times 10^8 m/s \)
• (C) \( 3 \times 10^8 m/s \)
• (D) \( 1.5 \times 10^8 m/s \)

A gas undergoes an adiabatic process. During this process:

• (A) No heat is exchanged with the surroundings
• (B) The temperature of the gas remains constant
• (C) The pressure of the gas remains constant
• (D) The volume of the gas remains constant

Using the standard electrode potential, find out the pair between which redox reaction is not feasible. \( E^{\ominus} \) values: \( Fe^{3+}/Fe^{2+} = +0.77V \); \( I_2/I^- = +0.54V \); \( Cu^{2+}/Cu = +0.34V \); \( Ag^+/Ag = +0.80V \).

• (A) \( Fe^{3+} \) and \( I^- \)
• (B) \( Ag^+ \) and \( Cu \)
• (C) \( Fe^{3+} \) and \( Cu \)
• (D) \( Ag \) and \( Fe^{3+} \)

The reaction \( A(g) \rightarrow P(g) + Q(g) + R(g) \) follows first-order kinetics with a half-life of \( 69.3 \) s at \( 500^{\circ}C \). Starting with pure \( A \) in a container at \( 500^{\circ}C \) and a pressure of \( 0.4 \) atm, what will be the total pressure of the system after \( 230 \) s?

• (A) \( 1.15 atm \)
• (B) \( 1.32 atm \)
• (C) \( 1.22 atm \)
• (D) \( 1.12 atm \)

Electron affinity is positive, when:

• (A) \( O^- \rightarrow O^- \)
• (B) \( O^- \rightarrow O^{2-} \)
• (C) \( O \rightarrow O^+ \)
• (D) \( O \rightarrow O^{2+} \)

The ionic radii in Å of \( N^{3-} \), \( O^{2-} \), and \( F^- \) are respectively:

• (A) \( 1.71, 1.40 \) and \( 1.36 \)
• (B) \( 1.71, 1.36 \) and \( 1.40 \)
• (C) \( 1.36, 1.40 \) and \( 1.71 \)
• (D) \( 1.36, 1.71 \) and \( 1.40 \)

Find the sum of the series \( \left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \cdots \) up to \( n \) terms.

• (A) \( \frac{x^{2n}-1}{x^2-1} \times \frac{x^{2n+2}+1}{x^{2n}} + 2n \)
• (B) \( \frac{x^{2n}+1}{x^2+1} \times \frac{x^{2n+2}-1}{x^{2n}} - 2n \)
• (C) \( \frac{x^{2n}-1}{x^2-1} \times \frac{x^{2n}-1}{x^{2n}} - 2n \)
• (D) \( None of these \)

A person invites 10 friends to dinner and places them such that 4 are at one round table and 6 are at another round table. The total number of ways in which he can arrange the guests is:

• (A) \( 10!/6! \)
• (C) \( 9!/24 \)
• (D) \( None of these \)

If \( z_1, z_2, \ldots, z_n \) are complex numbers such that \( |z_1| = |z_2| = \ldots = |z_n| = 1 \), then \( |z_1 + z_2 + \ldots + z_n| \) is equal to:

• (A) \( |z_1 z_2 z_3 \ldots z_n| \)
• (B) \( |z_1| + |z_2| + \ldots + |z_n| \)
• (C) \( \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \)
• (D) \( n \)

Evaluate the definite integral: \( \int_{0}^{\pi/2} \sin^2(x) \, dx \).

• (A) \( \frac{\pi}{4} \)
• (B) \( \frac{\pi}{2} \)
• (C) \( 1 \)
• (D) \( 0 \)