Explanation:
To find the number of times the hands of a clock are at a right angle in a day, we need to consider the possible positions of the hour and minute hands where they form a right angle.

Position of the hour hand:
The hour hand moves 360 degrees in 12 hours, so in 1 hour it moves 30 degrees (360 degrees divided by 12). In 1 minute it moves 0.5 degrees (30 degrees divided by 60).

Position of the minute hand:
The minute hand moves 360 degrees in 60 minutes, so in 1 minute it moves 6 degrees (360 degrees divided by 60).

Calculating the angles:
To find the angle between the hour and minute hands at a specific time, we need to calculate the difference between their positions.

Possible angles:
A right angle measures 90 degrees. So, we need to find the times when the difference between the positions of the hour and minute hands is either 90 degrees or 270 degrees (as the hands can be in the opposite direction).

Calculating the number of times:
In a day, there are 24 hours and 60 minutes in each hour. So, there are a total of 24 x 60 = 1440 minutes in a day.

For a right angle (90 degrees):
The hour hand moves 0.5 degrees in 1 minute, and the minute hand moves 6 degrees in 1 minute. So, the difference between their positions must be either 90 degrees or 270 degrees.

To calculate the number of times this occurs in a day, we divide 90 degrees by the difference in their movements:
90 degrees / (6 degrees - 0.5 degrees) = 90 degrees / 5.5 degrees = 16.36

So, the hands of the clock are at a right angle 16 times for a 90 degree angle in a day.

For an obtuse angle (270 degrees):
Using the same calculation, we can find the number of times the hands are at a right angle for a 270 degree angle:
270 degrees / (6 degrees - 0.5 degrees) = 270 degrees / 5.5 degrees = 49.09

So, the hands of the clock are at a right angle 49 times for a 270 degree angle in a day.

Total number of times:
To find the total number of times the hands of the clock are at a right angle in a day, we add the number of times for a 90 degree angle and a 270 degree angle:
16 + 49 = 65

Therefore, the correct answer is option C) 44.