Two mirrors at an angle `theta^(@)` produce 5 images of a point. The number of images produced when `theta` is decreased to `theta^(@) - 30^(@)` is

Options

A. 9
B. 10
C. 11
D. 12

Explanation

To solve the problem, we need to determine how many images are produced by two mirrors when the angle between them is decreased from \( \theta \) to \( \theta - 30^\circ \).

### Step-by-Step Solution:

1. **Understanding the Formula for Number of Images**:
The formula to calculate the number of images \( N \) formed by two mirrors inclined at an angle \( \theta \) is given by:
\[ N = \frac{360^\circ}{\theta} - 1 \]

2. **Given Information**:
We know that when the angle \( \theta \) produces 5 images:
\[ N = 5 \]
Therefore, we can set up the equation:
\[ \frac{360^\circ}{\theta} - 1 = 5 \]

3. **Solving for \( \theta \)**:
Rearranging the equation gives:
\[ \frac{360^\circ}{\theta} = 6 \]
Multiplying both sides by \( \theta \) and then by 6 gives:
\[ 360^\circ = 6\theta \]
Dividing both sides by 6:
\[ \theta = \frac{360^\circ}{6} = 60^\circ \]

4. **Finding the New Angle**:
The problem states that \( \theta \) is decreased by \( 30^\circ \):
\[ \theta' = \theta - 30^\circ = 60^\circ - 30^\circ = 30^\circ \]

5. **Calculating the Number of Images for the New Angle**:
Now, we will use the same formula to find the number of images when \( \theta' = 30^\circ \):
\[ N' = \frac{360^\circ}{\theta'} - 1 = \frac{360^\circ}{30^\circ} - 1 \]
Calculating this gives:
\[ N' = 12 - 1 = 11 \]

6. **Conclusion**:
Therefore, the number of images produced when the angle is decreased to \( \theta - 30^\circ \) is \( 11 \).

### Final Answer:
The number of images produced when \( \theta \) is decreased to \( \theta - 30^\circ \) is **11**.

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