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 ?

Correct Option: (d) Between 7:05 hours and 7:10 hours

The hour hand and minute hand are opposite to each other (make an angle of 180 degrees) once every hour, except between 6 and 7 where it happens exactly at 6:00. For the hour of 7:00, the minute hand must gain 7 * 30 + 180 = 390 degrees, or starting from 7:00 (where angle is 210 degrees), the minute hand needs to cover an additional 30 degrees to reach 180 degrees separation. The time taken for minute hand to cover space is given by (210 - 180) * (60/55) = 30 * 12/11 = 360/11 = 32.7 minutes past 7? Wait, let's check when they are at 180 degrees around 7 o'clock. At 7 o'clock, angle is 210 degrees. To be 180 degrees apart, minute hand must be at 210 - 180 = 30 degrees ahead of hour hand, or 210 + 180 = 390 degrees (which is 30 degrees past 12). The minute hand points at 30 degrees (which is 5 minutes mark). So at 7:05, the minute hand is at 5 minutes (30 degrees). Let's check the relative positions at 7:05: minute hand is at 5 min mark (30 deg from 12), hour hand is at 7 hours + 5 minutes, which is 210 + 5 * 0.5 = 212.5 degrees. The difference is 212.5 - 30 = 182.5 degrees, not exactly 180. Wait, let's check the exact formula: Time = (7 * 30 + 180) * (12/11) = 390 * 12 / 11 = 42.5 minutes. But wait, what about option (d) 'Between 7:05 hours and 7:10 hours'? Wait, let's re-calculate: At 7:00, hour hand is at 210 degrees. The minute hand is at 0 degrees. The angle is 210 degrees. To make 180 degrees, the minute hand must travel 210 - 180 = 30 degrees more than the hour hand. Since minute hand gains 5.5 degrees per minute, time = 30 / 5.5 = 60/11 minutes = 5 integer 5/11 minutes past 7. This falls strictly between 7:05 and 7:10 hours! Therefore, option (d) is correct.