At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?

The difference between the hour and minute hand of a clock is given by $$\left|30H-5.5m\right|$$. Here H is the current hour and m represents the number of completed minutes in the current hour.

In the given time frame of 2: 30 to 3: 00 pm.

At 2 : 30 pm the angle = $$\left|30\cdot2-5.5\cdot30\right|\ =\ 105\ \deg rees$$

At 3: 00 pm the angle = $$\left|30\cdot3-5.5\cdot0\right|\ =\ 90\ \deg rees$$

The function of $$\left|30\cdot H-5.5\cdot m\right|\ =\ $$ constantly increases as the value of m increases from 31, 32................ 59.

Because of the modulus function, the net value of the function remains positive

Between 2: 30 to 2: 59 the angle is constantly increasing. The minimum value is 2: 30 which is equal to 105 degrees which is greater than the 90 degrees when the time is 3: 00.

Hence 90 degrees is the minimum angle.