Question

Given below are two statements: Assertion (A): The angle between the two arms of a watch when the time is \(3.10\) pm will be \(35^\circ\). 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 \(\left[30\left(t-\frac{x}{5}\right)+\frac{x{2}\right]^\circ\).}

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For clock angle problems, substitute hour and minute carefully in the angle formula.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This problem asks us to determine the truth values of the Assertion (concerning the angle of a clock's hands at 3:10) and the Reason (an algebraic representation of the angle), and establish if the Reason logically explains the Assertion.

Step 2: Detailed Explanation:
Let us analyze both statements:
• Verification of Assertion (A):
At 3:10 PM, the hour hand has moved past 3, and the minute hand is exactly on 2. We can calculate the angle using the standard clock formula: \[ \theta = \left| 30H - \frac{11}{2}M \right| \] where $H = 3$ and $M = 10$: \[ \theta = \left| 30(3) - 5.5(10) \right| = |90 - 55| = 35^\circ \] Thus, Assertion (A) is correct.
• Verification of Reason (R):
The angular position of the hour hand from the 12 o'clock mark at $t$ hours and $x$ minutes is: \[ \text{Hour Position} = (30t + 0.5x)^\circ \] The angular position of the minute hand is: \[ \text{Minute Position} = (6x)^\circ \] If the hour hand is ahead of the minute hand, the angle is the difference between these positions: \[ \theta = 30t + 0.5x - 6x = 30t - 5.5x \] Now, let us simplify the formula given in Reason (R): \[ 30\left(t - \frac{x}{5}\right) + \frac{x}{2} = 30t - \frac{30x}{5} + 0.5x \] \[ = 30t - 6x + 0.5x = 30t - 5.5x \] Both expressions are mathematically identical. Thus, the formula in Reason (R) is correct.
• Testing (A) with (R):
Substitute $t = 3$ and $x = 10$ into Reason (R)'s formula: \[ \text{Angle} = 30\left(3 - \frac{10}{5}\right) + \frac{10}{2} = 30(3 - 2) + 5 = 35^\circ \] Since this general formula yields the specific result mentioned in Assertion (A), the Reason is the correct explanation of the Assertion.

Step 3: Final Answer:
Both (A) and (R) are correct, and (R) is the correct explanation of (A), which corresponds to option (A).