Q. How many times from 4 p.m. to 10 p.m. the hands are at right angles ?

Explanation:

Minute hand speed = 6°/min, hour hand speed = 0.5°/min, so relative speed = 5.5°/min. At t minutes past h o’clock, the angle between the hands is |6t − (30h + 0.5t)| = |5.5t − 30h|. For a right angle, set |5.5t − 30h| = 90, giving t = (30h ± 90)/5.5 within [0,60).
– Between 4–5, 5–6, 6–7, 7–8: both solutions lie in the hour → 2 each.
– Between 8–9: t = 27.27 min and t = 60 min (i.e., 9:00). Count 9:00 once.
– Between 9–10: only t = 32.73 min lies in the hour (the other is >60).
Total unique occurrences from 4 p.m. to 10 p.m.: 2 + 2 + 2 + 2 + 2 (including 9:00 once) + 1 = 11.
Equivalently, in 12 hours the hands are at right angles 22 times; over 6 hours it’s 11 (and 4:00 and 10:00 are not right angles).