Consider the following statements: 1. Between 3:16 p.m. and 3:17 p.m., both hour hand and minute hand coincide. 2. Between 4:58 p.m. and 4:59 p.m., both minute hand and second hand coincide. Which of the above statements is/are correct?

To determine the correctness of the statements, we need to calculate the exact times of coincidence for the clock hands.

Statement 1: Between 3:16 p.m. and 3:17 p.m., both hour hand and minute hand coincide.

• The minute hand moves at 6 degrees per minute.
• The hour hand moves at 0.5 degrees per minute.
• Their relative speed is 6 - 0.5 = 5.5 degrees per minute.
• At 3:00 p.m., the hour hand is at 90 degrees (from the 12 position) and the minute hand is at 0 degrees.
• For them to coincide, the minute hand must cover the 90-degree gap.
• Time taken = Initial angle difference / Relative speed = 90 degrees / 5.5 degrees/minute = 900/55 minutes = 180/11 minutes.
• 180/11 minutes is approximately 16.36 minutes.
• So, the hands coincide at approximately 3:16.36 p.m. (i.e., 3:16 and 4/11 minutes).
• This time falls between 3:16 p.m. and 3:17 p.m.
• Therefore, statement 1 is correct.

Statement 2: Between 4:58 p.m. and 4:59 p.m., both minute hand and second hand coincide.

• The second hand moves at 6 degrees per second.
• The minute hand moves at 0.1 degrees per second (or 6 degrees per minute).
• Their relative speed is 6 - 0.1 = 5.9 degrees per second.
• At 4:58:00 p.m., the minute hand is at the 58-minute mark. Its position is (58/60) * 360 = 348 degrees from the 12 position. The second hand is at 0 degrees.
• For them to coincide, the second hand must cover the 348-degree gap.
• Let 's' be the number of seconds past 4:58:00 p.m.
• Angle of second hand = 6s.
• Angle of minute hand = 348 + 0.1s (since it moves slightly during these seconds).
• For coincidence, 6s = 348 + 0.1s.
• 5.9s = 348.
• s = 348 / 5.9 = 3480 / 59 seconds.
• 3480 / 59 seconds is approximately 58.98 seconds.
• So, the hands coincide at approximately 4:58:58.98 p.m.
• This time falls between 4:58 p.m. and 4:59 p.m.
• Therefore, statement 2 is correct.

Since both statements are correct, the answer is C.

The final answer is C) Both 1 and 2