how many right angles are formed by a clock having hour and minute hands ?plese explain

Direct answer: Each hour, the minute hand and the hour hand shall form 90 degree twice.

Approach:
We will formulate the general formula for the angle the hour hand and the minute hand make in terms of the time of the clock(variable) and the speeds of the two arms of the clock. After this, subtract both the angles and equate the formula to 90 degree.

Answer: Speed of hour hand: 0.5 degrees/min
Speed of minute hand: 6 degrees/min

Let the time of the clock be ‘m’ hours ‘n’ minutes.

So, Angular displacement of hour hand= (0.5)(60m+n) degrees= 30m+(n/2)
Angular displacement of minute hand= 6n

Hence, Angle between the two hands(which we want is for 90 degrees)
= 30m+n/2-6n= (+or-)90
We get,
60m-11n=(+or-)180
which is the general equation to to find the times at which the minte hand and the hour hand form 90 degree angle.
m varies from 1 till 12.
Now put the values of m to get the corresponding values of n and hence the times at which the 2 arms make 90 degree angle.

NOTE: Be careful with the negative angles. If you get a negative angle, add 360 degree to the it before finally calculating the value of n.

Approved