This is a matching question where we need to associate processes from Column II with the physical changes described in Column I.

Step-by-step analysis of each process in Column II:

1. Process (P): A capacitor, initially uncharged, is connected to a battery.

   • When an uncharged capacitor is connected to a battery, the battery does work to move charges onto the capacitor plates. This process stores electrical potential energy in the capacitor.
   • The energy stored in the capacitor increases from zero to $U = \frac{1}{2}CV^2$, where C is the capacitance and V is the voltage of the battery.
   • Thus, the energy of the system (the capacitor) is increased.
   • This matches statement (A).

2. Process (Q): A gas in an adiabatic container is compressed by pushing the piston.

   • The container is adiabatic, which means there is no heat exchange with the surroundings ($Q = 0$).
   • The gas is compressed by a piston, which means external mechanical work is done on the gas ($W_{\text{on}} > 0$).
   • According to the First Law of Thermodynamics, the change in internal energy is $\Delta U = Q + W_{\text{on}}$.
   • Since $Q=0$, we have $\Delta U = W_{\text{on}}$. Because work is done on the gas, $\Delta U > 0$. The internal energy of the system increases. The internal energy of a gas is the kinetic energy of its randomly moving molecules.
   • This matches statement (A) (The energy of the system is increased).
   • It also matches statement (B) (Mechanical energy is provided to the system, which is converted into energy of random motion of its parts).

3. Process (R): A gas in a rigid container gets cooled.

   • The container is rigid, so its volume is constant (isochoric process). This means no work is done by or on the gas ($W = \int P dV = 0$).
   • The gas is cooled, which means heat flows out of the system ($Q < 0$).
   • From the First Law of Thermodynamics, $\Delta U = Q - W$. With $W=0$, we get $\Delta U = Q < 0$. The internal energy of the system decreases.
   • The mass of the gas remains constant (ignoring negligible relativistic effects).
   • This process does not match any of the statements in Column I.

4. Process (S): A heavy nucleus at rest fissions into two fragments and some neutrons.

   • In nuclear fission, the total mass of the products (fragments and neutrons) is less than the mass of the original heavy nucleus. This difference in mass is called the mass defect.
   • Therefore, the mass of the system is decreased. This matches statement (D).
   • According to Einstein's mass-energy equivalence ($E=mc^2$), the lost mass is converted into a large amount of energy. This energy is released primarily as the kinetic energy of the fission fragments and neutrons.
   • The rest mass energy of the nucleus is a form of internal energy. This internal energy is converted into the kinetic energy of the fragments, which is a form of mechanical energy.
   • This matches statement (C) (Internal energy of the system is converted into its mechanical energy).

5. Process (T): A resistive wire loop is placed in a time-varying magnetic field.

   • According to Faraday's Law of Induction, a time-varying magnetic flux through the loop induces an electromotive force (EMF).
   • This EMF drives an electric current in the resistive wire.
   • The flow of current through the resistor causes Joule heating ($P = I^2R$), dissipating energy as heat.
   • This heat increases the temperature and thus the internal energy (thermal energy) of the wire loop.
   • The energy is supplied by the external time-varying magnetic field.
   • Therefore, the energy of the system (the wire loop) is increased. This matches statement (A).

Summary of Matches:

• (A) The energy of the system is increased. -> Matches with (P), (Q), and (T).
• (B) Mechanical energy is provided to the system, which is converted into energy of random motion of its parts. -> Matches with (Q).
• (C) Internal energy of the system is converted into its mechanical energy. -> Matches with (S).
• (D) Mass of the system is decreased. -> Matches with (S).

Matching with Options:

Let's check the given options based on our derived matches:

• A: (A) $\to$(P), (S), (T); (B) $\to$(Q); (C) $\to$(S); (D) $\to$(S), (R) - Incorrect because (A) should include (Q) and (D) should not include (R).
• B: (A) $\to$(P), (Q), (T); (B) $\to$(Q); (C) $\to$(S); (D) $\to$(S) - This perfectly matches our analysis.
• C: (A) $\to$(P), (Q), (T); (B) $\to$(T); (C) $\to$(S); (D) $\to$(S) - Incorrect because (B) matches (Q), not (T).
• D: (A) $\to$(S), (Q), (T); (B) $\to$(Q); (C) $\to$(S); (D) $\to$(S) - Incorrect because (A) should include (P).

The correct option is B.