An object is placed at `15` cm in front of a concave mirror whose focal length is `10` cm. The image formed will be

Options

A. magnified and inverted
B. magnified and erect
C. reduced in size and inverted
D. reduced in size and erect.

Explanation

To solve the problem of finding the image characteristics formed by a concave mirror when an object is placed in front of it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Focal length (f) of the concave mirror = -10 cm (negative because it is a concave mirror).
- Object distance (u) = -15 cm (negative as per the sign convention for mirrors).

2. **Use the Mirror Formula**:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) = focal length,
- \( v \) = image distance,
- \( u \) = object distance.

3. **Substitute the Values into the Formula**:
Substitute \( f = -10 \) cm and \( u = -15 \) cm into the mirror formula:
\[ \frac{1}{-10} = \frac{1}{v} + \frac{1}{-15} \]

4. **Rearrange the Equation**:
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{-10} + \frac{1}{15} \]

5. **Find a Common Denominator**:
The common denominator for -10 and 15 is 30. Thus:
\[ \frac{1}{-10} = \frac{-3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
So,
\[ \frac{1}{v} = \frac{-3 + 2}{30} = \frac{-1}{30} \]

6. **Calculate the Image Distance (v)**:
Taking the reciprocal gives:
\[ v = -30 \text{ cm} \]
This means the image is formed at 30 cm in front of the mirror.

7. **Determine the Magnification (m)**:
The magnification formula for mirrors is:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{-30}{-15} = -2 \]

8. **Interpret the Magnification**:
The negative sign indicates that the image is inverted. The magnitude of 2 means the image height is twice that of the object.

### Conclusion:
The image formed is located 30 cm in front of the mirror (real and inverted) and is twice the height of the object.