A `10 mm` long awl pin is placed vertically in front of a concave mirror. A `5 mm` long image of the awl pin is formed at `30 cm` in front of the mirror. The focal length of this mirror is

Options

A. `- 30 cm`
B. `- 20 cm`
C. ` - 40 cm`
D. `- 60 cm`

Explanation

To find the focal length of the concave mirror given the information about the awl pin and its image, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of the awl pin (object height, \( h_o \)) = 10 mm
- Length of the image (image height, \( h_i \)) = 5 mm
- Image distance from the mirror (v) = -30 cm (negative because it is in front of the concave mirror)

2. **Calculate the Magnification (m):**
The magnification formula is given by:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the values:
\[ m = \frac{5 \text{ mm}}{10 \text{ mm}} = \frac{1}{2} \]

3. **Relate Magnification to Image and Object Distances:**
From the magnification formula, we can write:
\[ \frac{1}{2} = -\frac{-30}{u} \]
Rearranging gives:
\[ u = -30 \times 2 = -60 \text{ cm} \]
(The negative sign indicates that the object is in front of the mirror.)

4. **Use the Mirror Formula:**
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{-30} + \frac{1}{-60} \]

5. **Calculate the Right-Hand Side:**
Finding a common denominator (which is 60):
\[ \frac{1}{f} = \frac{-2}{60} + \frac{-1}{60} = \frac{-3}{60} = \frac{-1}{20} \]

6. **Find the Focal Length (f):**
Taking the reciprocal gives:
\[ f = -20 \text{ cm} \]

### Final Answer:
The focal length of the concave mirror is \( -20 \text{ cm} \).