Question

When will both the hands of the clock be in the same line between 10 and 11 O'clock?

• A. \(10 : 54 \frac{6}{11}\) O'clock
• B. \(10 : 55 \frac{6}{11}\) O'clock
• C. \(10 : 21 \frac{9}{11}\) O'clock
• D. A & C both

Hint:

To find when clock hands align or oppose, set the angle difference to 0 or 180 degrees and solve for time.

Answer 1

The Correct Option is D

Step 1: Calculate the position of the hour hand at 10 O'clock. At 10 O'clock, the hour hand is at: \[ 10 \times 30 = 300^\circ \]
Step 2: Calculate the positions after \(x\) minutes past 10: Hour hand moves at \(0.5^\circ\) per minute, so position after \(x\) minutes: \[ 300 + 0.5x \] Minute hand moves at \(6^\circ\) per minute, so position after \(x\) minutes: \[ 6x \]
Step 3: Condition for the hands to be in the same line: They can be overlapping (angle difference = 0) or opposite (angle difference = 180). \textit{Case 1: Overlapping} \[ 6x = 300 + 0.5x \implies 5.5x = 300 \implies x = \frac{300}{5.5} = \frac{600}{11} = 54 \frac{6}{11} \] \textit{Case 2: Opposite line} \[ |6x - (300 + 0.5x)| = 180 \] Two possibilities: \[ 6x - 0.5x - 300 = 180 \implies 5.5x = 480 \implies x = \frac{960}{11} = 87 \frac{3}{11} \text{ (not between 10 and 11)} \] or \[ 300 + 0.5x - 6x = 180 \implies 300 - 5.5x = 180 \implies 5.5x = 120 \implies x = \frac{240}{11} = 21 \frac{9}{11} \]
Step 4: Conclusion The hands are in the same line at: \[ 10 : 54 \frac{6}{11} \quad \text{and} \quad 10 : 21 \frac{9}{11} \] Hence, options (1) and (3) are correct, so option (4) is the correct answer. \[ \boxed{10 : 54 \frac{6}{11} \text{ and } 10 : 21 \frac{9}{11}} \]