At what time between 7 oclock and 8 oclock will the hands of a clock lie in the same straight line but not coincide, that is, be 180 degrees apart?

Introduction / Context:
The question asks for the time between 7 and 8 oclock when the clock hands are in one straight line but not together. This means the angle between them must be 180 degrees. It is similar to the earlier opposite direction problem but in a different hour interval. It checks the use of angular positions and solving a linear equation involving the time in minutes after an hour.

Given Data / Assumptions:

• We consider a time between 7:00 and 8:00.
• The hands must form a straight line but not overlap, so the angle between them is 180 degrees.
• Minute hand moves 6 degrees per minute.
• Hour hand moves 0.5 degree per minute and starts from 210 degrees at 7:00.

Concept / Approach:
Let t be the number of minutes after 7:00. Then: hour hand angle = 30 * 7 + 0.5 * t = 210 + 0.5 t, minute hand angle = 6 * t. We want the angle between the hands to be 180 degrees, so the absolute difference of these two angles must be 180 degrees. We choose the correct sign based on which hand is ahead in that interval.

Step-by-Step Solution:
Step 1: Set up the separation condition. |6 t - (210 + 0.5 t)| = 180. Step 2: Choose sign. Shortly after 7:00, the hour hand is initially far ahead. The minute hand starts behind and moves faster, so for the first time when they are opposite, the hour hand is still ahead. Thus we use: (210 + 0.5 t) - 6 t = 180. Step 3: Solve the equation. 210 + 0.5 t - 6 t = 180. 210 - 5.5 t = 180. 5.5 t = 210 - 180 = 30. t = 30 / 5.5 = 30 * 2 / 11 = 60 / 11 minutes. 60 / 11 = 5 5/11 minutes. Therefore, the time is 7 hours 5 5/11 minutes.

Verification / Alternative check:
At t = 60 / 11: Hour hand angle = 210 + 0.5 * 60 / 11 = 210 + 30 / 11 = (2310 + 30) / 11 = 2340 / 11 degrees. Minute hand angle = 6 * 60 / 11 = 360 / 11 degrees. Difference = (2340 / 11) - (360 / 11) = 1980 / 11 = 180 degrees, confirming the hands are opposite.

Why Other Options Are Wrong:
Option b (6 5/11 minutes) gives a different separation when substituted into the formula. Options c and d (exactly 7:00 or 8:00) correspond to the hands being together or at right angles depending on the time, but not 180 degrees apart in this context. Only 5 5/11 minutes past 7 produces an exact 180 degree separation.

Common Pitfalls:
A frequent mistake is to forget the 0.5 degree per minute movement of the hour hand and treat it as fixed at the 7 oclock position. Some learners also use the wrong sign in the absolute difference equation, leading to the second solution where t exceeds 60 minutes, which is not within the hour under consideration.

Final Answer:
The hands of the clock are in a straight line but not together at 5 5/11 minutes past 7.