In 12 hours, how many times the hours and minutes hands of a clock will coincide with- each other?

Hour hand covers 360° in 12 hour or 720 minutes.

Also, Minute hand covers 360° in 60 minutes.

So, hour hand covers (1/2)° in 1 minute.

and minute hand covers 6° in 1 minute.

Let say, time in the clock when they coincide : hour hand moved H hour and minute hand moved M minutes.

So,

Angle between hands of a clock

When the minute hand is behind the hour hand, the angle between the two hands at M minutes past HH 'o clock

$$=(30H-6M)+(M/2)°$$

$$=(30H-(11M/2))°$$

When the minute hand is ahead of the hour hand, the angle between the two hands at M minutes past HH 'o clock
$$=((6M−30H)−M/2)°$$

$$=((11M/2)-30H)°$$

So, taking mod:

$$|60H-11M|=2\theta$$..........................(1)

So, hour hand and minute hand will coincide 11 times :

between 1 and 2 , 2 and 3 , 3 and 4, 4 and 5 , 5 and 6, 6 and 7 , 7 and 8 , 8 and 9, 9 and 10, 10 and 11, 11 and 12.

1 and 2 :

between 1 and 2 hour hand will pass 1 O'clock,

So,H=1 and angle should be 0.

using equation (1):$$60\times\ 1-11M=0\ or\ M=60\div11=5\frac{5\ }{11}.$$

So,between 1 and 2 , hour hand and minute hand will coincide at $$5\frac{5\ }{11}$$ minutes past 1.

2 and 3:

$$60\times2-11M=0\ or\ M=120\div11=10\frac{10\ }{11}.$$

They will coincide $$10\frac{10\ }{11}$$ minutes past 2.

3 and 4:

$$60\times3-11M=0\ or\ M=\frac{180}{11}=16\frac{4\ }{11}.$$

They will coincide $$16\frac{4\ }{11}$$ minutes past 3.

4 and 5:

$$60\times4-11M=0\ or\ M=\frac{240}{11}=21\frac{9\ }{11}.$$

They will coincide $$21\frac{9\ }{11}$$ minutes past 4.

5 and 6:

$$60\times5-11M=0\ or\ M=\frac{300}{11}=27\frac{3\ }{11}.$$

They will coincide $$27\frac{3\ }{11}$$ minutes past 5.

6 and 7:

$$60\times6-11M=0\ or\ M=\frac{360}{11}=32\ \frac{\ 8}{11}.$$

They will coincide $$32\ \frac{\ 8}{11}$$ minutes past 6.

7 and 8:

$$60\times7-11M=0\ or\ M=\frac{420}{11}=38\frac{2\ }{11}.$$

They will coincide $$38\frac{2\ }{11}$$ minutes past 7.

8 and 9:

$$60\times8-11M=0\ or\ M=\frac{480}{11}=43\frac{7\ }{11}.$$

They will coincide $$43\frac{7\ }{11}$$ minutes past 8.

9 and 10:

$$9\times60-11M=0\ or\ M=\frac{540}{11}=49\frac{1\ }{11}.$$

They will coincide $$49\frac{1\ }{11}$$ minutes past 9.

10 and 11:

$$60\times10-11M=0\ or\ M=\frac{600}{11}=54\frac{6\ }{11}.$$

They will coincide $$54\frac{6\ }{11}$$ minutes past 10.

11 and 12:

$$60\times11-11M=0\ or\ M=\frac{660}{11}=60.$$

So, last time they will coincide 60 minutes past 11O'clock which means at 12 noon.

So, Total 11 times they will coincide.

B is correct choice.