At what time between 2 o'clock and 3 o'clock will the hour and minutes hands of a watch be together?

The minute hand moves 360° in 60 minutes, giving a rate of 6° per minute.

The hour hand moves 360° in 12 hours (720 minutes), giving a rate of 0.5° per minute.

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At 2 o'clock, the hour hand is at 60° from 12 o'clock position (since $2 \times 30° = 60°$), while the minute hand is at 0°.

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Let $t$ be the number of minutes past 2 o'clock when the hands coincide.

Position of minute hand = $6t$ degrees

Position of hour hand = $60 + 0.5t$ degrees

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For the hands to be together:

$6t = 60 + 0.5t$

$6t - 0.5t = 60$

$5.5t = 60$

$t = \dfrac{60}{5.5}$

$t = \dfrac{60}{\frac{11}{2}}$

$t = \dfrac{120}{11}$

$t = 10\dfrac{10}{11}$ minutes

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Therefore, the hour and minute hands coincide at $10\dfrac{10}{11}$ minutes past 2 o'clock.

The answer is Option 3: $10\dfrac{10}{11}$ min past 2.