Answer: Option C. -> 38$ 2/11 $min past 4
Answer: (c)
Between 4 and 5 O' clock the hands of the clock will be at right angle twice, first situation will occur when minute hand is 15 min spaces behind the hour hand and the second when minute hand is 15 min spaces ahead of the hour hand.
Fig. (ii) shows the position when minute hand is 15 min spaces behind the hour hand. To come at this position, minute hand has to gain 5 min spaces from the position at 4 O' clock. Now, 55 min are gained by minute hand in 60 min. Therefore, 5 will be gained in$ 60/55 × 5 = 60/11 $min It means that hands of the clock will be at right angle at 5$ 5/11$ min past 5.
Fig, (iii) shows the position when minute hand is 15 min spaces ahead the hour hand. To come at this position, minute hand has to gain 35 min spaces from the position at 4 O' clock Now, 55 min are gained in 60 min.
Therefore, 35 min spaces will be gained in 60 min = $60/55 × 35 min = 420/11$ min It means that second position will come at $38 2/11$ min past 4.
Now, in options 38$ 2/11 $min past 4 is available as option (c).

Answer: Option A. -> 10 $10/11$ min past 8
Answer: (a)
Fig (i) shows the positions of the hands of the clock at 8 O' clock and it is clear that they are 20 min apart. To be in the straight line, they have to be 30 min apart.
So, the minute hand will have to gain 10 min space in order to be 30 min apart from hour hand. Since 55 min are gained in 60 min,
Therefore, 10 min will be gained in $60/55 × 10 = 12/11 × 10 min $
Therefore, the hands will be in straight line but not together at $10 10/11 $min past 8.

Answer: Option D. -> 27$ 3/11$ min past 5
Answer: (d)
From the figure, we find that min hand is 25 min spaces behind the hour hand. In order to coincide, it has to again 25 min spaces.
Now, 55 min are gained by minute hand in 60 min.
Therefore, 25 min will be gained in $60/55 × 25 = 27 3/11$
So, the hands will coincide at 27 $3/11$ min past 5.

Answer: Option D. -> 16$ 4/11$ min past 3
Answer: (d)
At 3 O' clock both the hands of the clock are 15 min apart. Hence, in order to be together, minute hand will have to gain 15 min spaces in order to coincide with the hour hand. Now, 55 min are gained by minute hand in 60 min.
Therefore, 15 min will be gained in $(60/55 × 15) min = (12/11 × 15) min $= $180/11 or 16 4/11 min$
Therefore, the hands will coincide at 16$ 4/11$ min past 3.

Answer: Option C. -> 49 $1/11 $min past 9
Answer: (c)
Both the hands are 15 min spaces apart at 9 O' clock. To be together between 9 and 10 min O' clock hand has to gain 45 min.
Now, minute hand gains 55 min in 60 min. Therefore, it will gain 45 min in $60/55 × 45 = 49 1/11 min$
Therefore, the hands will be together at 49 $1/11$ min past 9.

Answer: Option C. -> 22
Answer: (c)
The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, the coincide only once, i.e. at 12 O' clock).

Answer: Option C. -> 44
Answer: (c)
In 12 hours, they are at right angles 22 times
Therefore, In 24 hours, they are at right angles 44 times.

Answer: Option C. -> 390.5°
Answer: (b)
Clearly, the minute hand traverses 65 minutes in 1 hour.
Therefore, Required angle = (360/60 × 65). = 390.

Answer: Option C. -> 18
Answer: (c)
The duration from 1.00 p.m. to 10.00 p.m. is 9 hours and during each of these 9 hours the hands of a clock are at right angles twice.
So, required number = 9 × 2 = 18.

Answer: Option C. -> 44
Answer: (c)
In 12 hours, the hands coincide or are in opposite direction 22 times.
Therefore, In 24 hours, the hands coincide or are in opposite direction 44 times a day.