GATE CS 2014 Set 2 QGA-10 — General Aptitude

The angle $\theta$ between the hour hand and the minute hand of a clock at $H$ hours and $M$ minutes is given by the formula:

For any time between 6 a.m. and 7 a.m., we have $H = 6$. Substituting this into the formula:

We want to find when this angle is closest to $60^\circ$. Setting $\theta = 60^\circ$ gives two possible cases:

Case 1:

Case 2:

Comparing these target values with the given options:

• Option A (6:22 a.m.): $M = 22$ is extremely close to $21.82$ minutes. The actual angle at 6:22 is $|180 - 5.5(22)| = 59^\circ$, which is only $1^\circ$ away from $60^\circ$.
• Option B (6:27 a.m.): $M = 27$ gives an angle of $|180 - 5.5(27)| = 31.5^\circ$.
• Option C (6:38 a.m.): $M = 38$ gives an angle of $|180 - 5.5(38)| = 29^\circ$.
• Option D (6:45 a.m.): $M = 45$ gives an angle of $|180 - 5.5(45)| = 67.5^\circ$.

Thus, the time closest to making a $60^\circ$ angle is 6:22 a.m.