A virtual, erect and magnified image of an object is to be produced with a concave mirror of focal length 12 cm. Which of the following object distance should be chosen for this purpose?

Options

A. 10 cm
B. 14 cm
C. 18 cm
D. 24 cm

Answer

To solve the problem of finding the object distance (u) for a concave mirror that produces a virtual, erect, and magnified image, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given data**:
- Focal length (f) of the concave mirror = -12 cm (negative because it is a concave mirror).
- The image is virtual, erect, and magnified.

2. **Understand the characteristics of the image**:
- A virtual image formed by a concave mirror occurs when the object is placed between the focal point and the mirror (i.e., u < f).
- Since the image is virtual, the image distance (v) will be positive.

3. **Use the mirror formula**:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging this gives:
\[ \frac{1}{u} = \frac{1}{f} - \frac{1}{v} \]

4. **Substituting known values**:
Substitute \( f = -12 \) cm into the equation:
\[ \frac{1}{u} = \frac{1}{-12} - \frac{1}{v} \]

5. **Determine the sign of v**:
Since the image is virtual, \( v \) is positive. Therefore, we can express \( \frac{1}{v} \) as a positive value.

6. **Establish conditions for u**:
For a virtual image, we know that:
\[ \frac{1}{u} = \frac{1}{-12} - \frac{1}{v} \]
Since \( \frac{1}{v} \) is positive, we can conclude that:
\[ \frac{1}{u} > \frac{1}{-12} \]
This implies that \( u < -12 \) cm (since u is always negative).

7. **Finding the range for u**:
The condition \( u < -12 \) cm indicates that the object distance must be less than the absolute value of the focal length. Therefore, we need to find an object distance that is less than 12 cm in magnitude.

8. **Choose the correct option**:
Among the options provided, we need to select the one that is less than 12 cm (in the negative direction). The only option that fulfills this condition is -10 cm (or 10 cm in absolute terms).

### Conclusion:
The object distance that should be chosen for producing a virtual, erect, and magnified image with a concave mirror of focal length 12 cm is **10 cm**.