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?

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

• At 3:00 p.m., the hour hand is at 3 (90 degrees from 12) and the minute hand is at 12 (0 degrees). The angle between them is 90 degrees.
• The minute hand moves 6 degrees per minute. 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.
• To coincide, the minute hand must cover the 90-degree gap.
• Time taken = 90 degrees / 5.5 degrees/minute = 90 / (11/2) = 180/11 minutes.
• 180/11 minutes = 16 and 4/11 minutes.
• So, the hands coincide at approximately 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.

• Let's consider the interval from 4:58:00 to 4:59:00.
• At 4:58:00, the minute hand is at the 58-minute mark (58 * 6 = 348 degrees from 12). The second hand is at the 0-minute mark (0 degrees).
• The second hand moves 360 degrees per minute (or 6 degrees per second). The minute hand moves 6 degrees per minute (or 0.1 degrees per second).
• Let 't' be the number of seconds past 4:58:00 when they coincide.
• Position of second hand = 6t degrees.
• Position of minute hand = (58 * 6) + (t/60 * 6) = 348 + t/10 degrees.
• For them to coincide: 6t = 348 + t/10
• 6t - t/10 = 348
• (60t - t)/10 = 348
• 59t/10 = 348
• t = 3480 / 59 seconds.
• 3480 / 59 = 58 and 58/59 seconds.
• So, the hands coincide at approximately 4:58:58 and 58/59 seconds. 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