Clocks and Calendars

Calendar repeats when the total number of odd days is 0 (for non-leap years) or 7 (accounting for leap years). From 2024 to 2052 is 28 years, containing 7 leap years (2024, 2028, 2032, 2036, 2040, 2044, 2048). Total odd days = 28 + 7 = 35, which is divisible by 7, so the calendar repeats in 2052.

At 2:30, the hour hand is halfway between 2 and 3. Each hour represents 30 degrees, so the hour hand is at 2.5 × 30 = 75 degrees from 12 o'clock. The minute hand points to 6, which is 180 degrees from 12. The angle between them is 180 - 75 = 105 degrees.

In each 12-hour period, the hour and minute hands are exactly opposite (180° apart) 11 times. This happens because they start together at 12:00 and are together 11 more times (every ~65.5 minutes), creating 11 opposition positions. In 24 hours (one full day), this happens 11 × 2 = 22 times.

In 12 hours, the hour and minute hands form a 90° (or 270°) angle 22 times - 11 times when the minute hand is 90° ahead of the hour hand, and 11 times when it's 90° behind. In 24 hours (one full day), this happens 44 times.

At 8:40, the minute hand is at 8 (240 degrees from 12). The hour hand is 2/3 past 8, so at 8 + 40/60 = 8 + 2/3. Hour hand angle: (8 + 2/3) × 30 = 240 + 20 = 260 degrees. Difference: 260 - 240 = 20 degrees. Option A (20°) is correct. Note: Option D has '400' which appears to be a typo for 40°.

The hour and minute hands of a clock form a 180° angle 11 times every 12 hours (not 12 times because they overlap at 6 o'clock which is counted differently). In 24 hours (one full day), this happens 22 times. The hands are at 180° at approximately 12:32, 1:38, 2:43, 3:49, 4:54, 6:00, 7:05, 8:11, 9:16, 10:22, and 11:27 (and the mirrored times in the afternoon/evening). Option B (22) is correct.

Tuesday 1pm: 1 min slow (needs to gain 1 min). Thursday 1pm: 2 min fast (needs to lose 2 min). Total change needed: 3 minutes over 51 hours (from 1pm Tuesday to 1pm Thursday). Since the watch is linear, it gains at 3/51 min per hour. To be correct (go from -1 to 0), it needs 1 hour. But we need to find when it crosses from slow to fast. At Wednesday 5am (40 hours from Tuesday 1pm), it would have gained 40 × 3/51 ≈ 2.35 minutes, crossing zero.

At 9:27, minute hand is at 27×6 = 162° from 12. Hour hand is at 9×30 + 27×0.5 = 270 + 13.5 = 283.5° from 12. Angle between them = 283.5 - 162 = 121.5°.

At 5:25, hour hand position = 5×30 + 25×0.5 = 150 + 12.5 = 162.5 degrees from 12 (each hour = 30°, hour hand moves 0.5° per minute). Minute hand position = 25×6 = 150 degrees from 12 (each minute = 6°). Angle between hands = |162.5 - 150| = 12.5 degrees.

In a 12-hour period, the hour and minute hands coincide 11 times (not 12 at 11:00 because they're together). Since a day has 24 hours, the hands coincide 11 x 2 = 22 times. The first coincidence after 12:00 is at approximately 1:05:27.

In a mirror clock, the reflected time and actual time are symmetric around 12:00 (for hour hand) and 6:00 positions. The sum of reflected and actual positions equals 12. For 4:40, the reflection calculation: hour 4 reflects to 8 (since 4+8=12), minutes 40 reflect to 20 (since 40+20=60). So the actual time is 8:20. Since her appointment was at 7:20, she is exactly on time.

Statement A tells us the time is between 8 and 9, which is insufficient. Statement B gives angle of 72° between hands. Between 8 and 9, the hands make 72° twice - once when minute hand is ahead of hour hand, and once when hour hand is ahead (around 8:12 and 8:24). Without knowing which configuration or additional info, we cannot uniquely determine the time. Both statements together still give two possible times, so the question cannot be solved uniquely.

The hour and minute hands form 90° twice every hour except between 2-4 and 8-10 o'clock when they only form it once. In 12 hours: 10 hours × 2 = 20 times + 2 hours × 1 = 2 times = 22 times. In a full day (24 hours), this happens 22 × 2 = 44 times. Option D is correct.

At 5:50, the minute hand is at 10 (50 minutes), and hour hand is 5 + 50/60 = 5.833. Using formula |30H - 5.5M| = |30(5) - 5.5(50)| = |150 - 275| = 125 degrees. The hour hand moves past 5 toward 6, minute hand at 10, giving 125° between them.

At 10 o'clock, the hour hand is at 10 and the minute hand is at 12. Each hour marking represents 30° (360° ÷ 12 = 30°). The angle between 10 and 12 is 2 hour markings, so the angle is 2 × 30° = 60°. This is the minor angle.