Between 2:00 and 3:00, at what exact time do the hour hand and the minute hand of a clock coincide?

Introduction / Context:
This problem asks for the exact time between 2:00 and 3:00 when the hour and minute hands overlap. Questions about the coincidence of clock hands are a standard topic in quantitative aptitude, because they require using a general formula for the relative motion of the hands. Understanding how often and when the hands coincide within each hour builds a deeper intuition about continuous movement on the clock dial rather than only discrete positions at exact hours.

Given Data / Assumptions:

- We are looking between 2:00 and 3:00. - The clock has 12 equal hour divisions, corresponding to 360 degrees total. - The minute hand moves 6 degrees per minute. - The hour hand moves 0.5 degrees per minute. - We want the moment when both hands are exactly at the same position.

Concept / Approach:
Let m be the number of minutes after 2:00. We compute the angular position of the hour hand and the minute hand with respect to the 12 o clock position, then set them equal because coincidence means the hands are at the same angle. This gives a linear equation in m. Solving for m yields the exact fractional number of minutes after 2:00 at which the coincidence occurs. This value is usually written as a mixed fraction like 10 10/11 minutes to match standard exam conventions.

Step-by-Step Solution:
Step 1: Let m be the minutes past 2:00. Step 2: At 2:00, the hour hand is at 2 * 30 = 60 degrees from 12. Step 3: In m minutes, the hour hand moves an extra 0.5 * m degrees, so its angle is 60 + 0.5 * m degrees. Step 4: The minute hand at m minutes is at 6 * m degrees from 12. Step 5: For coincidence, set the two angles equal: 60 + 0.5 * m = 6 * m. Step 6: Rearrange to get 60 = 6 * m - 0.5 * m = 5.5 * m. Step 7: Therefore m = 60 / 5.5 = 120 / 11 minutes. Step 8: 120 / 11 minutes is equal to 10 10/11 minutes, so the hands coincide at 10 10/11 minutes past 2:00.

Verification / Alternative check:
We can check reasonableness by recalling a general result: the hands coincide every 65 5/11 minutes and there are 11 coincidences in 12 hours. The approximate time between coincidences is a little more than one hour. After the 2:00 coincidence is reached, the next overlap will occur slightly after 3:00 plus an additional fraction of an hour. The value 10 10/11 minutes past 2:00 fits the pattern because it is the first coincidence after 1:00 and before 3:00, aligning with the known distribution of overlaps.

Why Other Options Are Wrong:
Values like 9 10/11 or 11 10/11 minutes past 2:00 do not satisfy the equation that equates the hour and minute hand angles. Times past 3:00 do not lie between 2:00 and 3:00, so they are automatically not valid for this question. Only 10 10/11 minutes past 2:00 arises from the correct solution of the equation 60 + 0.5 * m = 6 * m, so the other listed times are incorrect.

Common Pitfalls:
A frequent mistake is to assume that the overlap happens at exactly half past the hour or at some simple round minute mark. Another pitfall is forgetting that the hour hand continues to move during the minutes after 2:00, leading to equations that treat the hour hand as fixed. Always use the correct speeds of both hands, set the positions equal, and explicitly solve the resulting equation to find the exact fractional minute value.

Final Answer:
Between 2:00 and 3:00, the clock hands coincide at 10 10/11 minutes past 2:00.