Answer: Option D. -> 649°
Time from 12 noon to 9 : 38 A.M.
= 12 hours + 9 hours 38 minutes
= 21 hours 38 minutes
$$\eqalign{
& {\text{ = 21}}\frac{{38}}{{60}}{\text{ hours}} \cr
& {\text{ = 21}}\frac{{19}}{{30}}{\text{ hours}} \cr
& {\text{ = }}\frac{{649}}{{30}}{\text{hours}} \cr} $$
Angle traced by the hour hand in 12 hours = 360°
Angle traced by the minute hand in
$$\eqalign{
& \Leftrightarrow \frac{{649}}{{30}}{\text{hours}} \cr
& = {\left( {\frac{{360}}{{12}} \times \frac{{649}}{{30}}} \right)^ \circ } \cr
& = {649^ \circ } \cr} $$

Answer: Option C. -> 4557.67 CM
Number of rounds completed by the minute hand in 3 days 5 hours
$$\eqalign{
& = \left( {3 \times 24 + 5} \right) \cr
& = 77 \cr} $$
Number of rounds completed by the hour hand in 3 days 5 hours
$$\eqalign{
& = \left( {3 \times 2 + \frac{5}{{12}}} \right) \cr
& = 6\frac{5}{{12}} \cr} $$
∴ Difference between the distance traversed
$${\text{ = }}\left[ {77 \times \left( {2 \times \frac{{22}}{7} \times 10} \right) - 6\frac{5}{{12}} \times \left( {2 \times \frac{{22}}{7} \times 7} \right)} \right]{\text{cm}}$$
$$\eqalign{
& = \left( {4840 - 282.33} \right){\text{ cm}} \cr
& = 4557.67{\text{ cm}} \cr} $$

Answer: Option A. -> 126 minutes
One clock show 10 pm, on 21st January 2010
One clock gains = 2 minutes
Other clock loses = 5 minutes
Time period between 10 pm and 4 pm = 18 hours
∴ Required difference
= (2 × 18 + 5 × 18 ) minutes
= 126 minutes

Answer: Option B. -> 24 minutes past 5
Since the time read by the lady was 57 minutes earlier than the correct time, so the minute hand is (60 - 57) = 3 minutes spaces behind the hour hand.
Now, at 5 o'clock, the minute hand is 25 minutes spaces behind the hour hand.
To be 3 minutes spaces behind, it must gain (25 - 3) = 22 minutes spaces.
55 minutes spaces are gained in 60 minutes.
22 minutes spaces are gained in $$\left( {\frac{{60}}{{55}} \times 22} \right)$$   = 24 minutes
Hence, the correct time was 24 minutes past 5.

Answer: Option A. -> 6 hours
To be together between 8 am and 9 am, the minute hand has to gain 40 minutes spaces.
55 minutes spaces are gained in 60 minutes.
40 minutes space are gained in $$\left( {\frac{{60}}{{55}} \times 40} \right)$$  minutes = $${\text{43}}\frac{7}{{11}}$$  minutes
So, Henry started his trip at $${\text{43}}\frac{7}{{11}}$$  minutes past 8 am.
Now, to be 180° apart, the hands must be 30 minutes spaces apart.
At 2 pm, they are 10 minutes spaces apart.
∴ The minute hand will have to gain (10 + 30) = 40 minutes spaces.
As calculate above, 40 minutes spaces are gained in $${\text{43}}\frac{7}{{11}}$$  minutes.
So, Henry's trip ended at $${\text{43}}\frac{7}{{11}}$$  minutes past 2 pm
∴ Duration of travel = Duration from $${\text{43}}\frac{7}{{11}}$$  minutes past 8 am to $${\text{43}}\frac{7}{{11}}$$  minutes past 2 pm = 6 hours

Answer: Option C. -> 11 : 36 am
Time gained in 1 hour = 6 minutes
Time gained in 6 hours = (6 × 6) minutes = 36 minutes
After 6 hours, the correct time is 11 : 00 am and the watch will show 11 : 36 am.

Answer: Option B. -> 5 : 00 am on Wednesday
Time from 1 pm on Wednesday to 1 pm on Thursday = 48 hours
So, the watch gains (1 + 2) minute or 3 minutes in 48 hours.
Now, 3 minutes are gained in 48 hours
So, 1 minute is gained in $$\left( {\frac{{48}}{3}} \right)$$   = 16 hours.
Thus, the watch showed the correct time 16 hours after 1 pm on Tuesday, i.e., 5 am on Wednesday

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

Answer: Option D. -> $$54\frac{6}{{11}}$$ min. past 4
At 4 o'clock, the hands of the watch are 20 min. spaces apart.
To be in opposite directions, they must be 30 min. spaces apart.
∴ Minute hand will have to gain 50 min. spaces.
55 min. spaces are gained in 60 min.
50 min. spaces are gained in
$$\eqalign{
& \left( {\frac{{60}}{{55}} \times 50} \right){\text{min}}{\text{.}}\,{\text{or}}\,54\frac{6}{{11}}\,{\text{min}}. \cr
& \therefore {\text{Required}}\,{\text{time}} \cr
& = 54\frac{6}{{11}}{\text{min}}{\text{.}}\,{\text{past}}\,4 \cr} $$

Answer: Option B. -> 22
The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. (Because between 5 and 7 they point in opposite directions at 6 o'clcok only).
So, in a day, the hands point in the opposite directions 22 times.