At what time between 2 oclock and 3 oclock will the hour hand and minute hand of a clock coincide and be exactly together?

Introduction / Context:
This classic clock question asks when the hands coincide between 2 oclock and 3 oclock. It uses the standard formula involving the relative speed of the hour and minute hands. Such problems are frequently used to test algebraic manipulation and understanding of uniform circular motion on the clock face.

Given Data / Assumptions:

• We are considering a time between 2 oclock and 3 oclock.
• The minute hand moves 6 degrees per minute.
• The hour hand moves 0.5 degree per minute.
• Coincidence means their angles, measured from 12 oclock, are equal.

Concept / Approach:
Between 2 and 3, let t be the minutes after 2 oclock. Then: hour hand angle = 30 * 2 + 0.5 * t = 60 + 0.5 t, minute hand angle = 6 * t. For coincidence, we set these two angles equal and solve for t. We then convert the answer into mixed fraction minutes to match the options.

Step-by-Step Solution:
Step 1: Write the angles. Hour hand angle H = 60 + 0.5 t. Minute hand angle M = 6 t. Step 2: Set H = M for coincidence. 60 + 0.5 t = 6 t. Step 3: Solve for t. 6 t - 0.5 t = 60. 5.5 t = 60. t = 60 / 5.5 = 60 * 2 / 11 = 120 / 11 minutes. 120 / 11 = 10 10/11 minutes. So the hands coincide at 10 10/11 minutes past 2.

Verification / Alternative check:
Substitute t = 120 / 11 into the expressions: Hour hand angle = 60 + 0.5 * 120 / 11 = 60 + 60 / 11 = (660 + 60) / 11 = 720 / 11 degrees. Minute hand angle = 6 * 120 / 11 = 720 / 11 degrees. Both angles are equal, so the hands are indeed together.

Why Other Options Are Wrong:
Options a, c, and d (9 10/11, 11 10/11, 12 10/11) do not satisfy the equality when substituted in place of t. They give slightly different angles for the hour and minute hands, so the hands will not be exactly together at those times.

Common Pitfalls:
Learners sometimes use an approximate rule like 5 minute past the hour for all coincidences, which is incorrect except for specific hours. Another mistake is forgetting that the hour hand moves as time passes and treating its angle as fixed at 60 degrees between 2 and 3. Rounding off t too early can also cause mismatch with the accurate mixed fraction answer.

Final Answer:
The hands of the clock are together at 10 10/11 minutes past 2.