1. Examine the Assertion (A)

   Assertion: In aqueous solution, $\mathrm{Cr}^{2+}$ is reducing while $\mathrm{Mn}^{3+}$ is oxidising.

   • Chromium: $\mathrm{Cr}: [Ar]3d^5 4s^1$ $\mathrm{Cr}^{2+}: [Ar]3d^4$ $\mathrm{Cr}^{3+}: [Ar]3d^3$

     In aqueous medium, $\mathrm{Cr}^{2+}$ tends to get oxidised to $\mathrm{Cr}^{3+}$. Since it causes reduction of other species while itself gets oxidised, $\mathrm{Cr}^{2+}$ acts as a reducing agent.

   • Manganese: $\mathrm{Mn}: [Ar]3d^5 4s^2$ $\mathrm{Mn}^{2+}: [Ar]3d^5$ $\mathrm{Mn}^{3+}: [Ar]3d^4$

     In aqueous medium, $\mathrm{Mn}^{3+}$ tends to get reduced to $\mathrm{Mn}^{2+}$. Since it oxidises other species while itself gets reduced, $\mathrm{Mn}^{3+}$ acts as an oxidising agent.

   Therefore, Assertion (A) is true.

2. Examine the Reason (R)

   Reason: Extra stability to half-filled electronic configuration is observed than incompletely filled electronic configuration.

   This is a standard fact in transition elements:

   • Half-filled configurations like $d^5$ are especially stable.
   • Thus species tend to convert to configurations with extra stability whenever possible.

   Hence, Reason (R) is true.

3. Check whether Reason explains Assertion

   • $\mathrm{Cr}^{2+}$ has configuration $3d^4$ and tends to change to the more stable $\mathrm{Cr}^{3+}$ state.
   • $\mathrm{Mn}^{3+}$ has configuration $3d^4$ and tends to change to the more stable $\mathrm{Mn}^{2+}$ state, which is $3d^5$ (half-filled, extra stable).

   The key explanation of the behavior comes from the enhanced stability of the half-filled $d^5$ configuration relative to $d^4$.

   Thus, (R) is the correct explanation of (A).

4. Evaluate the options

   • A: (A) true, (R) false → Incorrect
   • B: Both true and (R) is the correct explanation → Correct
   • C: Both true but (R) not correct explanation → Incorrect
   • D: (A) false, (R) true → Incorrect

Final Answer

The correct option is: $\boxed{\text{B}}$