Clock - 01

Explanation:-

In 12 hours, the hands coincide or are in opposite direction 22 times.
∴ In 24 hours, the hands coincide or are in opposite direction 44 times a day.

Explanation:-

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock).
AM
12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

PM
12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55
The hands overlap about every 65 minutes, not every 60 minutes.
∴ The hands coincide 22 times in a day.

Explanation:-

At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand.
To be coincident, it must gain 15 min. spaces.
55 min. are gained in 60 min. $$\eqalign{ & 15\,{\text{min}}{\text{.}}\,{\text{are}}\,{\text{gained}}\,{\text{in}}\,\left( {\frac{{60}}{{55}} \times 15} \right)\,{\text{min}} \cr & = 16\frac{4}{{11}}\,{\text{min}} \cr & \therefore {\text{The}}\,{\text{hands}}\,{\text{are}}\,{\text{coincident}}\,{\text{at}}\,16\frac{4}{{11}}\,{\text{min}}{\text{.}}\,{\text{past}}\,3 \cr}$$

Explanation:-

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 $$\eqalign{ & \left( {\frac{{60}}{{55}} \times 45} \right){\text{min}}{\text{.}}\,{\text{or}}\,49\frac{1}{{11}}\,{\text{min}}. \cr & \therefore {\text{The}}\,{\text{hands}}\,{\text{are}}\,{\text{together}}\,{\text{at}}\,49\frac{1}{{11}}{\text{min}}{\text{.}}\,{\text{past}}\,9 \cr}$$

Explanation:-

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.

Explanation:-

$$\eqalign{ & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{hour}}\,{\text{hand}}\,{\text{in}}\,\frac{{17}}{2}\,{\text{hrs}} \cr & \,{\text{ = }}\,{\left( {\frac{{360}}{{12}} \times \frac{{17}}{2}} \right)^ \circ } = 255 \cr & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{min}}{\text{.}}\,{\text{hand}}\,{\text{in}}\,{\text{30}}\,{\text{min}} \cr & = {\left( {\frac{{360}}{{60}} \times 30} \right)^ \circ } = 180 \cr & \therefore {\text{Required}}\,{\text{angle}} \cr & = {\left( {255 - 180} \right)^ \circ } = {75^ \circ } \cr}$$

Explanation:-

In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

Explanation:-

$$\eqalign{ & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{hour}}\,{\text{hand}}\,{\text{in}}\,12\,{\text{hrs}}\,{\text{ = }}\,{\text{36}}{{\text{0}}^ \circ } \cr & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{it}}\,{\text{in}}\,\frac{{11}}{3}\,{\text{hrs}} \cr & = {\left( {\frac{{360}}{{12}} \times \frac{{11}}{3}} \right)^ \circ } = {110^ \circ } \cr & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{min}}{\text{.}}\,{\text{hand}}\,{\text{in}}\,60\,{\text{min}}\,{\text{ = }}\,{\text{36}}{{\text{0}}^ \circ } \cr & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{it}}\,{\text{in}}\,{\text{40}}\,{\text{min}}. \cr & = {\left( {\frac{{360}}{{60}} \times 40} \right)^ \circ } = {240^ \circ } \cr & \therefore {\text{Required}}\,{\text{angle}} \cr & = {\left( {240 - 110} \right)^ \circ } = {130^ \circ } \cr}$$

Explanation:-

$$\eqalign{ & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{hour}}\,{\text{hand}}\,{\text{in}}\,\frac{{13}}{3}\,{\text{hrs}} \cr & = {\left( {\frac{{360}}{{12}} \times \frac{{13}}{3}} \right)^ \circ } = {130^ \circ } \cr & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{min}}{\text{.}}\,{\text{hand}}\,{\text{in}}\,{\text{20}}\,{\text{min}} \cr & = {\left( {\frac{{360}}{{60}} \times 20} \right)^ \circ } = {120^ \circ } \cr & \therefore {\text{Required}}\,{\text{angle}} \cr & = {\left( {130 - 120} \right)^ \circ } \cr & = {10^ \circ } \cr}$$

Explanation:-

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 $$\eqalign{ & \left( {\frac{{60}}{{55}} \times 40} \right)\,\min = 43\frac{7}{{11}}\,\min \cr & \therefore {\text{Required}}\,{\text{time}} = 43\frac{7}{{11}}\,\min .\,past\,5 \cr}$$