Clock Puzzle: Diametrically Opposite Hands

Question: At what time between 2 pm and 3 pm will the hour and minute hands of a clock be diametrically opposite?

Step 1: Understand the relative speed and initial position

The minute hand moves 6 degrees per minute, and the hour hand moves 0.5 degrees per minute.
The relative speed of the minute hand with respect to the hour hand is:

6 - 0.5 = 5.5 degrees per minute.

At 2 pm, the hour hand is at 60 degrees (2 × 30 degrees) from the 12 o'clock position, and the minute hand is at 0 degrees.
For the hands to be diametrically opposite, the angle between them should be 180 degrees.

Step 2: Calculate the angle to be covered

At 2 pm, the minute hand is 60 degrees behind the hour hand.
To be opposite, it must gain an extra 180 degrees.
Total angle to be covered = 60 + 180 = 240 degrees.

Step 3: Calculate the time taken

Time taken (t) = Angle ÷ Relative Speed
t = 240 ÷ 5.5 = 240 ÷ (11/2) = 480 ÷ 11 minutes.

Step 4: Convert into minutes and fraction

480 ÷ 11 = 43 ⁷⁄₁₁ minutes.

Step 5: Final Time

The time will be 2 hours and 43 ⁷⁄₁₁ minutes past 2 pm.

Answer: The correct answer is (D) 2:43 ⁷⁄₁₁ pm.