Step 1: Calculate Watch Angle for Assertion (A)

We need to find the angle between the arms of a watch at 3:10 pm. Let the hour ($t$) be 3 and the minute ($x$) be 10.

• Hour Hand Position: The hour hand moves 360° in 12 hours (30° per hour) and 0.5° per minute. At 3:10, its position is $(3 \times 30^\circ) + (10 \times 0.5^\circ) = 90^\circ + 5^\circ = 95^\circ$ from the 12.
• Minute Hand Position: The minute hand moves 360° in 60 minutes (6° per minute). At 10 minutes past the hour, its position is $10 \times 6^\circ = 60^\circ$ from the 12.
• Angle: The angle between the arms is $|95^\circ - 60^\circ| = 35^\circ$.

Assertion (A) states the angle is 35°, which is correct.

Step 2: Analyze the Formula in Reason (R)

Reason (R) provides a formula for the angle when the hour arm is ahead of the minute arm: $\left[30\left(t-\frac{x}{5}\right)+\frac{x}{2}\right]^\circ$.

• Simplify the Formula: The formula simplifies to $30t - \frac{30x}{5} + \frac{x}{2} = 30t - 6x + \frac{x}{2} = 30t - \frac{11x}{2}$.
• Condition Check: The standard angle formula is $|30t - \frac{11x}{2}|$. When the hour hand is ahead (i.e., $30t + \frac{x}{2} > 6x$), the angle is exactly $30t - \frac{11x}{2}$.
• The simplified formula from Reason (R) matches the correct angle calculation for the specified condition. Thus, Reason (R) is correct.

Step 3: Link Reason (R) to Assertion (A)

• At 3:10 pm, the hour hand (95°) is ahead of the minute hand (60°), satisfying the condition in Reason (R).
• Applying the formula from Reason (R) to 3:10 ($t=3, x=10$): Angle = $\left[30\left(3-\frac{10}{5}\right)+\frac{10}{2}\right]^\circ = [30(3-2) + 5]^\circ = [30(1) + 5]^\circ = 35^\circ$.
• This result matches Assertion (A), confirming that Reason (R) correctly explains Assertion (A).

Final Conclusion

Both Assertion (A) and Reason (R) are correct, and Reason (R) serves as the correct explanation for Assertion (A).