Step 1: Understanding the Question:
This question involves understanding the standard directions on a clock face and applying a rotation based on the given condition.
We are told that at 12:00 noon, the minute hand (which normally points North) points North-East. We need to find the position of the hour hand at 1:30 p.m. under this same rotation.
Step 2: Key Formula or Approach:
1. Identify the normal direction of hands at the given times.
2. Determine the angle of rotation from the standard North direction.
3. Apply this rotation to the normal direction of the hour hand at 1:30 p.m.
Step 3: Detailed Explanation:
Normal State at 12:00 Noon: Usually, at 12:00, the minute hand points exactly at the number 12, which represents North (\(0^{\circ}\) or \(360^{\circ}\)).
Given Condition: The hand points North-East. On a compass, North-East is halfway between North and East, representing a \(45^{\circ}\) clockwise rotation from North.
Normal State at 1:30 P.M.: At 1:30 p.m., the minute hand is at 6 (South), and the hour hand is exactly halfway between 1 and 2.
Angle of the Hour Hand at 1:30: Each hour mark on a clock represents \(30^{\circ}\) (\(360/12\)). At 1:00, the hour hand is at \(30^{\circ}\) from North. At 1:30, it has moved half of the next hour's distance (\(15^{\circ}\)). So, the hour hand's normal position is \(30^{\circ} + 15^{\circ} = 45^{\circ}\) clockwise from North.
Applying the Rotation: We found that the clock's North (12) is actually pointing North-East (\(45^{\circ}\) from true North). Since the hour hand at 1:30 is naturally at a \(45^{\circ}\) angle relative to the 12 o'clock mark, we add this to the orientation of the 12 mark.
Final Calculation: True Direction = (Direction of 12) + (Angle of hand from 12).
True Direction = \(45^{\circ} \text{ (North-East)} + 45^{\circ} \text{ (position at 1:30)} = 90^{\circ}\).
Result: \(90^{\circ}\) clockwise from North is exactly East.
Step 4: Final Answer:
The hour hand at 1:30 p.m. will point towards the East direction. Option (C) is correct.