How many degrees are there in the angle between the hour hand and minute hand of a clock. When it is.
(i) `3` o'clock?
(ii) `6` o'clock ?
(iii) `12` o'clock?
(iv) `9` o'clock?

Options

A. (i) `90^@`
(ii) `180^@`
(iii) `0^@`
(iv) `90^@`
B. (i) `80^@`
(ii) `200^@`
(iii) `0^@`
(iv) `90^@`
C. (i) `90^@`
(ii) `180^@`
(iii) `60^@`
(iv) `90^@`
D. (i) `30^@`
(ii) `180^@`
(iii) `60^@`
(iv) `90^@`

Answer

To find the angle between the hour hand and the minute hand of a clock at different times, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Clock Positions**:
- A clock is divided into 12 hours, and each hour represents an angle of 30 degrees (360 degrees / 12 hours).
- The minute hand moves 6 degrees for each minute (360 degrees / 60 minutes).

2. **Calculating the Angle at 3 o'clock**:
- At 3 o'clock, the hour hand is on 3 and the minute hand is on 12.
- The angle between them is calculated as:
\[ \text{Angle} = |(3 \times 30) - (0 \times 6)| = |90 - 0| = 90 \text{ degrees} \]
- **Answer**: 90 degrees.

3. **Calculating the Angle at 6 o'clock**:
- At 6 o'clock, the hour hand is on 6 and the minute hand is on 12.
- The angle between them is calculated as:
\[ \text{Angle} = |(6 \times 30) - (0 \times 6)| = |180 - 0| = 180 \text{ degrees} \]
- **Answer**: 180 degrees.

4. **Calculating the Angle at 12 o'clock**:
- At 12 o'clock, both the hour hand and the minute hand are on 12.
- The angle between them is calculated as:
\[ \text{Angle} = |(12 \times 30) - (0 \times 6)| = |360 - 0| = 0 \text{ degrees} \]
- **Answer**: 0 degrees.

5. **Calculating the Angle at 9 o'clock**:
- At 9 o'clock, the hour hand is on 9 and the minute hand is on 12.
- The angle between them is calculated as:
\[ \text{Angle} = |(9 \times 30) - (0 \times 6)| = |270 - 0| = 270 \text{ degrees} \]
- However, since we usually measure the smaller angle, we can also say:
\[ 360 - 270 = 90 \text{ degrees} \]
- **Answer**: 90 degrees.

### Final Answers:
- (i) 3 o'clock: **90 degrees**
- (ii) 6 o'clock: **180 degrees**
- (iii) 12 o'clock: **0 degrees**
- (iv) 9 o'clock: **90 degrees**