Discussion

When will the clock’s hands be in a straight line but pointing in opposite directions between 2 am and 3 am?

Answer: A

The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is

\[ \theta = \left| 30H - 5.5M \right| \]

Opposite directions means a straight line with \( \theta = 180 \), and \( H = 2 \):

\[ \left| 60 - 5.5M \right| = 180 \]

\[ 5.5M = 240 \quad\Longrightarrow\quad M = \frac{480}{11} = 43\tfrac{7}{11} \]

The other root, \( 5.5M = -120 \), is negative and so discarded.

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