1. What time between $14:10$ and $15:10$ do the clocks show when the angle between the minute and hour hands is $90^{\circ}$?
Answer: 15:10, 14 hours $27 \frac{3}{11}$ minutes $=14 \frac{5}{11}$ hours

1) If the clock shows $15: 00$, then obviously the angle between the minute and hour hands is $90^{\circ}$.
2) In $n$ minutes, the minute hand passes through an angle of $(6 n)^{\circ}$ degrees, and the hour hand through $\left(\frac{n}{2}\right)^{\circ}$ degrees. At 14:00, the angle between the minute and hour hands is $60^{\circ}$ degrees, and let's say that after $n$ minutes from $14: 00$ the angle becomes $90^{\circ}$. Then

$$
6 n-\left(\frac{n}{2}+60^{\circ}\right)=90^{\circ}
$$

from which $n=\frac{300}{11}=27 \frac{3}{11}$ minutes.