1. Recall the time period of a simple pendulum

   For small oscillations, $T = 2\pi\sqrt{\frac{l}{g}}$ where:

   • $T$ = time period
   • $l$ = length of pendulum
   • $g$ = acceleration due to gravity

2. Analyse Assertion (A)

   Assertion says: Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.

   At the top of a mountain, the distance from the center of Earth is greater, so the acceleration due to gravity $g$ is smaller than at the base.

   Since $T = 2\pi\sqrt{\frac{l}{g}},$ if $g$ decreases, then $T$ increases.

   Therefore, the time period at the top of a mountain is indeed larger.

   So, Assertion (A) is true.

3. Analyse Reason (R)

   Reason says: Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

   From $T \propto \frac{1}{\sqrt{g}},$ clearly, if $g$ increases, $T$ decreases; and if $g$ decreases, $T$ increases.

   So, Reason (R) is true.

4. Check whether Reason explains Assertion

   Assertion states that $T$ is longer on the mountain top.

   Reason gives the exact dependence of time period on gravity: increasing $g$ decreases $T$, and decreasing $g$ increases $T$.

   Since gravity is less at the top of a mountain, the reason directly explains why the time period is longer there.

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

5. Evaluate the options

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

Therefore, the correct option is: $\boxed{\text{C}}$

6. Comparison with stored correct answer

   Stored correct answer: C

   My derived answer: C

   They match.