Assertion : Time taken by a particle in SHM to move from `x = A` to `x = sqrt(3A)/(2)` is same as the time taken by the particle to move from `x = sqrt(3A)/(2)` to `x = (A)/(2)`.
Reason : Corresponding angles rotated is the reference circle are same in the given time intervals.

Options

A. (a)If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. (b)If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. (c)If Assertion is true, but the Reason is false.
D. (d)If Assertion is false but the Reason is true.

Explanation

To solve the problem, we need to analyze the assertion and reason given in the question step by step.

### Step-by-Step Solution:

1. **Understanding the Positions**:
- We have a particle in Simple Harmonic Motion (SHM) with maximum displacement (amplitude) \( A \).
- The particle moves from \( x = A \) to \( x = \frac{\sqrt{3}A}{2} \) and then from \( x = \frac{\sqrt{3}A}{2} \) to \( x = \frac{A}{2} \).

2. **Finding Angles**:
- For the position \( x = A \), the angle \( \theta_1 \) corresponding to this position is \( 0^\circ \) (since it is at maximum displacement).
- For the position \( x = \frac{\sqrt{3}A}{2} \):
\[ \cos(\theta_1) = \frac{x}{A} = \frac{\sqrt{3}A/2}{A} = \frac{\sqrt{3}}{2} \implies \theta_1 = 30^\circ \]
- For the position \( x = \frac{A}{2} \):
\[ \cos(\theta_2) = \frac{x}{A} = \frac{A/2}{A} = \frac{1}{2} \implies \theta_2 = 60^\circ \]

3. **Calculating the Angle Covered**:
- The angle covered when moving from \( x = A \) to \( x = \frac{\sqrt{3}A}{2} \) is:
\[ \Delta \theta_1 = \theta_1 - 0 = 30^\circ \]
- The angle covered when moving from \( x = \frac{\sqrt{3}A}{2} \) to \( x = \frac{A}{2} \) is:
\[ \Delta \theta_2 = \theta_2 - \theta_1 = 60^\circ - 30^\circ = 30^\circ \]

4. **Time Calculation**:
- The total angle in one complete cycle (one period \( T \)) is \( 360^\circ \).
- The time taken to cover \( 30^\circ \) is:
\[ t = \frac{30^\circ}{360^\circ} \times T = \frac{1}{12} T \]
- Thus, the time taken for both intervals is the same, \( \frac{1}{12} T \).

5. **Conclusion**:
- The assertion states that the time taken to move from \( x = A \) to \( x = \frac{\sqrt{3}A}{2} \) is the same as the time taken to move from \( x = \frac{\sqrt{3}A}{2} \) to \( x = \frac{A}{2} \), which we found to be true.
- The reason states that the corresponding angles rotated in the reference circle are the same, which is also true.

### Final Answer:
Both the assertion and reason are correct.

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