Ravi is observing a wall clock while waiting for his job interview at exactly 6:00 a.m. At some point, he notices that the hour and minute hands form a straight angle (180°). At what time (in full minutes) does this happen, after 6:00 a.m.?
(a) 06:50 a.m.
(b) 07:00 a.m.
(c) 07:05 a.m.
(d) 07:20 a.m.

Question

Ravi is observing a wall clock while waiting for his job interview at exactly 6:00 a.m. At some point, he notices that the hour and minute hands form a straight angle (180&deg;). At what time (in full minutes) does this happen, after 6:00 a.m.?
(a) 06:50 a.m.
(b) 07:00 a.m.
(c) 07:05 a.m.
(d) 07:20 a.m.

ONLiNE UPSC · Accepted Answer

Let the time be 6:M, i.e., M minutes after 6:00.
We use the formula for angle between the hour and minute hands:
Angle = |30H &minus; 5.5M|
Where:
H = 6 (since it's after 6:00)
M = number of minutes after 6:00
Set the angle equal to 180&deg;:
|30 * 6 &minus; 5.5M| = 180
|180 &minus; 5.5M| = 180
Now solve:
Case 1:
180 &minus; 5.5M = 180
&rArr; 5.5M = 0
&rArr; M = 0
&rarr; At 6:00, angle = 0&deg;, not 180&deg;
Case 2:
5.5M &ndash; 180 = 180
5.5M = 360
M = 360/5.5 = 720/11 =65 minutes and 5/11​
Therefore,
The hour and minute hands form a 180&deg; angle exactly at
65 minutes and 5/11 seconds after 6:00 a.m.,
i.e., at 7:05 a.m. approximately.

Hence, option c is correct.

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M Question 1	⏱ 0
Ravi is observing a wall clock while waiting for his job interview at exactly 6:00 a.m. At some point, he notices that the hour and minute hands form a straight angle (180°). At what time (in full minutes) does this happen, after 6:00 a.m.? (a) 06:50 a.m. (b) 07:00 a.m. (c) 07:05 a.m. (d) 07:20 a.m.