Assertion (A) : The angle between the two arms of a watch when the time is 3.10 pm, will be 35°
Reason (R) : When the hour arm of the watch is ahead of the minute arm, then angle between the two arms at the time t hour and x minute is $\left[30\left(t-\frac{x}{5}\right)+\frac{x}{2}\right]^\circ$
In the light of the above statements, choose the most appropriate answer from the options given below :
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).
Important Questions from Clock and Calendar (Notes)
- From the given options, at what angle are the hands of a clock inclined at 10 minutes to 2 (Smaller angle) ?
- If $1^{st}$ January, 2001 was a Monday, what was the day on $26^{th}$ January, 2003 ?
- Today is Friday. Before $43$ days, it was?
- How many times starting at 1:00 pm would the minute and hour hands of a clock make an angle of $40^\circ$ with each other in the next 6 hours?
- A man starts his journey at 0100 Hrs local time to reach another country at 0900 Hrs local time on the same date. He starts a return journey on the same night at 2100 Hrs local time to his original place, taking the same time to travel back. If the time zone of his country of visit lags by 10 hours, the duration for which the man was away from his place is
