When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.

At 7 o'clock, they are 25 min. spaces apart.

So, Minute hand will have to gain only 5 min. spaces.

55 min. spaces are gained in 60 min.

5 min. spaces are gained in \(\left(\frac{60}{55}\times5\right)min.=5\frac{5}{11}min\)

Required time =   \(5\frac{5}{11}min. past 7\)

At 5 o'clock, the hands are 25 min. spaces apart.

To be at right angles and that too between 5.30 and 6, the minute hand has to gain (25 + 15) = 40 min. spaces.

55 min. spaces are gained in 60 min.

40 min. spaces are gained in \(\left(\frac{60}{55}\times40\right)min.=43\frac{7}{11}min\)

So, =    \(43\frac{7}{11}min. past 5\)

Angle traced by hour hand in  \(\frac{13}{3}hrs = \left(\frac{360}{12}\times\frac{13}{3}\right)^{0}=130^{0}\)

Angle traced by min. hand in 20 min \(\left(\frac{360}{60}\times20\right)^{0}=120^{0}\)

So, Required angle = (130 - 120)º = 10º.

Angle traced by hour hand in \(\frac{21}{4}hrs = \left(\frac{360}{12}\times\frac{21}{4}\right)^{0}=157\frac{1}{2}^{0} \)

Angle traced by min. hand in 15 min. = \(\left(\frac{360}{60}\times15\right)^{0}=90^{0} \)

So, Required angle = \(\left(157\frac{1}{2}\right)^{0}-90^{0} =67\frac{1}{2}^{0}\)

Angle traced by hour hand in 12 hrs. = 360°.

Angle traced by it in\(\frac{11}{3}hrs = \left(\frac{360}{12}\times\frac{11}{3}\right)^{0}=110^{0}.\)  

Angle traced by minute hand in 60 min. = 360°.

Angle traced by it in 40 min. = \(\left(\frac{360}{60}\times40\right)^{0}=240^{0}. \)

So, Required angle (240 - 110)° = 130°.

In 12 hours, they are at right angles 22 times.

So, In 24 hours, they are at right angles 44 times.

Angle traced by hour hand in \(\frac{17}{7}hrs = \left(\frac{360}{12}\times\frac{17}{7}\right)^{0}=255^{0}\)

Angle traced by min. hand in 30 min. = \(\left(\frac{360}{60}\times30\right)^{0}=180^{0}.\)

So, Required angle = (255 - 180)º = 75º

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.

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.

So, Minute hand will have to gain 50 min. spaces.

55 min. spaces are gained in 60 min.

50 min. spaces are gained in \(\left(\frac{60}{55}\times50\right)min. or 54\frac{6}{11}min. \)

So, Required time = \(54\frac{6}{11} min. past 4\)

To be together between 9 and 10 o'clock, the minute hand has to gain 45 min. spaces.

55 min. spaces gained in 60 min.

45 min. spaces are gained in \(\left(\frac{60}{55}\times45\right)min. or 49\frac{1}{11}min.\)

So, The hands are together at \(49\frac{1}{11} min. past 9\)