Question

Between 1 o'clock and 2 o'clock find the time when the minute hand and an hour hand are
(a) coinciding
(b) straight i.e. in opposite direction
(c) at angle of 90${}^{\circ }$ for the first time

Answer 1

(a) At 1 o'clock the minute hand is 5 minutes behind the hour hand The minute hand gains 5 minutes in $5\times \frac{12}{11}$ minutes i.e. in $5\frac{5}{11}$ minutes Thus the hands will coincide at $5\frac{5}{11}$ minutes past 1 o'clock
(b) when thehands are in opposite direction the difference is 30 minutes so the minutes hand has to gain (5+30) min i.e. 35 minutes which it can do in $35\times \frac{12}{11}$ minutes
(c) when the hands are at right angle they are 15 minutes apart So the hands are at right angle for the first time when the minute hand will gain (5+15) minutes