Reasoning — Clock

At 3:30, the hour hand is between 3 and 4, and the minute hand is at 6. Angle of minute hand from 12 = 30 minutes * 6°/minute = 180°. Angle of hour hand from 12 = 3 hours * 30°/hour + 30 minutes * 0.5°/minute = 90° + 15° = 105°. The angle between them = |180° - 105°| = 75°. Alternatively, using the formula: Angle = |(11/2)M - 30H| = |(11/2)*30 - 30*3| = |11*15 - 90| = |165 - 90| = 75°.

To find the mirror image time, subtract the given time from 11:60. 11:60 - 7:20 = 4:40.

The hands of a clock coincide 11 times in 12 hours. They coincide once every hour, except between 11 and 1 o'clock, where they coincide only once (at 12 o'clock).

The total duration from Monday 8 AM to Wednesday 8 AM is 48 hours. In 1 hour, the clock gains 10 seconds. In 48 hours, the clock will gain = 48 * 10 seconds = 480 seconds. Convert seconds to minutes: 480 seconds / 60 seconds/minute = 8 minutes. So, when the actual time is 8 AM on Wednesday, the faulty clock will show 8 minutes past 8 AM, i.e., 8:08 AM.

The formula for the angle between hands is θ = |(11/2)M - 30H|. Here, H = 4, θ = 90°. 90 = |(11/2)M - 30*4| 90 = |(11/2)M - 120| Case 1: (11/2)M - 120 = 90 (11/2)M = 210 M = 420/11 = 38 2/11 minutes. Case 2: (11/2)M - 120 = -90 (11/2)M = 30 M = 60/11 = 5 5/11 minutes. Both 4:05 5/11 minutes and 4:38 2/11 minutes are valid times. From the options, 4:05 5/11 minutes is present.

At 10:10, H=10, M=10. Angle = |(11/2)M - 30H| = |(11/2)*10 - 30*10| = |55 - 300| = |-245| = 245°. Since the question asks for the smaller angle, we subtract this from 360°. Smaller angle = 360° - 245° = 115°.

The hands of a clock point in opposite directions (180 degrees) 11 times in a 12-hour period. This happens once every hour, except between 5 and 7 o'clock, where it happens only once (at 6 o'clock). Therefore, in a 24-hour period, they will point in opposite directions 11 * 2 = 22 times.

For the clock to show the correct time again, it must gain a total of 12 hours (720 minutes). The clock gains 15 minutes in 24 hours. Time taken to gain 1 minute = 24/15 hours. Time taken to gain 720 minutes = (24/15) * 720 hours = (8/5) * 720 hours = 8 * 144 hours = 1152 hours. Convert hours to days: 1152 hours / 24 hours/day = 48 days. So, the clock will show the correct time again 48 days after Monday 10 AM. Monday 10 AM + 48 days = Sunday 10 AM (since 48 days is 6 weeks and 6 days, Monday + 6 days = Sunday).

If the minute hand is 4 minutes ahead of the hour hand, the angle between them is 4 * 6° = 24°. Since the minute hand is ahead, we use the formula: (11/2)M - 30H = θ. Here, H = 5, θ = 24°. (11/2)M - 30*5 = 24 (11/2)M - 150 = 24 (11/2)M = 174 M = 348/11 = 31 7/11 minutes. So, the time is 5:31 7/11 minutes.

The hands of a clock form a right angle (90 degrees) twice every hour, except between 2-3 o'clock and 8-9 o'clock, where they form a right angle only once each. So, in 12 hours, they form a right angle 22 times (24 - 2 = 22).