Clocks Quiz

Using $\theta = |30H - 5.5M|$: $|30(10) - 5.5(25)| = |300 - 137.5| = 162.5^\circ$.

The hands coincide 11 times in 12 hours (once between each hour, skipping the 11-1 overlap), meaning 22 times in 24 hours.

Subtract from 11:60. Actual time = $11:60 - 3:40 = 8:20$.

At 3:00, the angle between hands is exactly $90^\circ$. So, at 3 o'clock, the hands are at a right angle.

Using $\theta = |30H - 5.5M|$: $|30(2) - 5.5(30)| = |60 - 165| = 105^\circ$.

$30H - 5.5M = 180 \Rightarrow 120 - 5.5M = 180 \Rightarrow -5.5M = 60 \Rightarrow M = -10.9$ (not possible). So use $5.5M - 30H = 180 \Rightarrow 5.5M - 120 = 180 \Rightarrow 5.5M = 300 \Rightarrow M = 54\frac{6}{11}$ minutes past 4.

In 6 actual hours, the clock loses $6 \times 5 = 30$ minutes. So it shows 6 hours - 30 min = 5:30 PM.

$|30(9) - 5.5M| = 30 \Rightarrow |270 - 5.5M| = 30$. Case 1: $270 - 5.5M = 30 \Rightarrow 5.5M = 240 \Rightarrow M = 43\frac{7}{11}$. Case 2: $270 - 5.5M = -30 \Rightarrow 5.5M = 300 \Rightarrow M = 54\frac{6}{11}$. Between 9 and 10, $M$ must be less than 60, so both are valid. The first occurrence is at $9:43\frac{7}{11}$.

The hands are at right angles 2 times every hour, except between 2-4 and 8-10 where they are right angles 3 times each in 2 hours. Total = $22 \times 2 = 44$ times in 24 hours.

In 24 hours, the clock gains 10 minutes. So at 8 AM Tuesday, it shows 8:10 AM.