Consider the following statements:
Which of the above statements are correct?
Statement 1: To determine when the hour hand and minute hand coincide, the angle between them must be 0 degrees. The formula for the angle `θ` between the hour hand and minute hand is given by ``, where `x` represents the minutes past the hour and `H` represents the hour. For the hands to coincide after 3 o'clock (`H=3`), we set `θ=0`:
`0 = \frac{11}{2}(x) - (3)(30)`
`0 = \frac{11}{2}(x) - 90`
`\frac{11}{2}(x) = 90`
`x = \frac{180}{11} \approx 16.36` minutes.
This calculation indicates that the hour hand and minute hand coincide at approximately 3:16.36 pm, which falls within the interval between 3:16 pm and 3:17 pm. Therefore, statement 1 is correct.
Statement 2: The second hand completes one full rotation (360 degrees) every minute. During this same minute, the minute hand advances by 6 degrees (360 degrees / 60 minutes). Due to the significant difference in their speeds, the second hand will always overtake and coincide with the minute hand exactly once within every one-minute interval. Consequently, between 4:58 pm and 4:59 pm, the minute hand and second hand will coincide. Therefore, statement 2 is correct.
Options B, C, and D are incorrect because both statement 1 and statement 2 are factually correct. Option A accurately identifies that both statements are correct, whereas the other options either incorrectly exclude a correct statement or refer to non-existent statements.