Step 1: Understanding the Question:
The subject of this problem is Clock Geometry, a popular topic in competitive reasoning exams. We are required to find a specific time from the given choices where the physical separation between the hour hand and the minute hand corresponds to an angle of exactly 60 degrees. To solve this, one must understand that a clock face is a circular plane divided into twelve equal parts. As the hands move, they sweep across these divisions at different speeds, creating various angles. In this problem, we focus on stationary "on-the-hour" positions where the minute hand is always at the 12 o'clock mark.
Step 2: Key Formulas and approach:
A standard clock is a circle representing 360 degrees. Since there are 12 hour markings, the angular distance between any two consecutive hour marks is calculated as $360 / 12 = 30$ degrees. When a clock shows a time that is exactly on the hour (e.g., 1:00, 2:00), the minute hand is at the zero-minute position (the 12). The angle $\theta$ between the hands can be calculated by the simple formula $\theta = 30 \times H$, where $H$ is the number of hours past 12. For more complex times involving minutes, the formula $\theta = |30H - (11/2)M|$ is used. Our approach here will be to apply the hourly multiplier to each option to see which one yields 60.
Step 3: Detailed Explanation:
We start by analyzing Option A, which is 2:00. At this moment, the hour hand is exactly on the 2 and the minute hand is on the 12. The number of hour spaces between them is 2. Multiplying 2 by 30 degrees per space gives $2 \times 30 = 60$ degrees. This matches the requirement.
Next, we look at Option B, which is 3:00. The hour hand is at the 3 and the minute hand is at the 12. This represents 3 hour spaces. The angle is $3 \times 30 = 90$ degrees, forming a right angle.
Moving to Option C, the time is 4:00. The hour hand has traveled to the 4 o'clock position. The distance from the 12 is 4 hour spaces. The angle is $4 \times 30 = 120$ degrees.
Finally, we check Option D, which is 6:00. The hour hand is at 6 and the minute hand is at 12. They are exactly opposite each other, covering 6 hour spaces. The angle is $6 \times 30 = 180$ degrees, which is a straight line.
Through this systematic comparison, we can see that only the first option results in the desired 60-degree measurement.
Step 4: Final Answer:
The time at which the hands of the clock form a 60-degree angle is 2:00.