Question

How many times between 4 a.m. and 5 a.m the minute and hour hands of a clock will be at right angle?

• A. $2$
• B. $3$
• C. $4$
• D. $1$

Answer 1

Answer: (A)

$2$

At 4 o'clock, the minute hand will be 20 min. spaces behind the hour hand.

Now, when the two hands are at right angles, they are 15 min. spaces apart

So, they are at right angles in following two cases.

Case I :

When minute hand in 15 min. spaces behind the hour hand:

In this case min. hand will have to gain (20-15) =5 min. spaces. 55 min. spaces are gained by it in 60 min.

5 min. spaces will be gained by it in $(\frac{60}{55}\times 5)min.=5\frac{5}{11}min$

$\therefore$ They are at right angles at $5\frac{5}{11}$ min. past 4.

Case II:

When the minute hand is 15 min. spaces ahead of the hour hand:

To be in this position, the minute hand will have to gain (20+15)=35 minute spaces.

55 min. spaces are gained in 60 min.

35 min spaces are gained in $(\frac{60}{55}\times 35)min=38\frac{2}{11}min.$

$\therefore$ They are at right angles at $38\frac{2}{11}min.past4.$

Hence 2 times.