Angle Between Hands — Mains Questions

A clock gains 5 minutes every hour. If it shows the correct time at 12:00 noon, at what actual time will it show 6:00 PM? Also, find the angle between the hands when the clock shows 6:00 PM.

Derive the general formula for finding all times in a 12-hour period when clock hands form an angle of θ degrees. Apply this formula to find all times when the angle is 45°.

Two identical clocks start at 12:00. Clock A runs normally while Clock B runs 10% fast. After how many hours will both clocks show the same angle between their hands? Explain the mathematical reasoning.