At which one of the following times, do the hour hand and the minute hand of the clock make an angle of 180° with each other?

The minute hand moves 360 degrees in 60 minutes, so its speed is 6 degrees per minute. The hour hand moves 360 degrees in 12 hours (720 minutes), so its speed is 0.5 degrees per minute.

Let's analyze the options:

A) At 7:00 hours: At 7:00, the minute hand is at 12 (0 degrees). The hour hand is at 7. Since each hour mark is 30 degrees (360/12), the hour hand is at 7 * 30 = 210 degrees from 12. The angle between them is 210 degrees (or 360 - 210 = 150 degrees, taking the smaller angle). This is not 180 degrees. So, A is incorrect.

For the hands to make an angle of 180 degrees, they must be opposite each other. This happens approximately at 6:00, 7:05, 8:10, etc.

Let's find the exact time 't' minutes past 7:00 when the angle is 180 degrees. Position of minute hand from 12 = 6t degrees. Position of hour hand from 12 = (7 * 30) + (0.5t) = 210 + 0.5t degrees.

For the hands to be 180 degrees apart, the difference in their positions must be 180 degrees. (210 + 0.5t) - 6t = 180 (assuming hour hand is ahead of minute hand, which it is initially after 7:00) 210 - 5.5t = 180 210 - 180 = 5.5t 30 = 5.5t t = 30 / 5.5 = 300 / 55 = 60 / 11 minutes.

So, the exact time is 7 hours and 60/11 minutes. 60/11 minutes is approximately 5.45 minutes (5 minutes and about 27 seconds).

Now let's check where this time falls: B) Between 7:00 hours and 7:05 hours: The calculated time (7:05 and 60/11 minutes) is after 7:05, so this range is incorrect. C) At 7:05 hours: At 7:05, the minute hand is at 5 (30 degrees). The hour hand is at 7 + (5/60)*30 = 210 + 2.5 = 212.5 degrees. The angle is 212.5 - 30 = 182.5 degrees. This is not exactly 180 degrees. So, C is incorrect. D) Between 7:05 hours and 7:10 hours: The calculated exact time of 7:05 and 60/11 minutes (approximately 7:05:27) falls precisely within this range. So, D is correct.

The final answer is D.