Focal length of a convex mirror is 10 cm

Options

A. image of an object placed at 20 cm is also at 20 cm
B. image of an object placed at 10 cm is at infinity
C. both (a) and (b) are wrong
D. both (a) and (b) are correct

Answer

To solve the problem regarding the focal length of a convex mirror, we will analyze the given options step by step using the mirror formula.

### Given:
- Focal length (F) of the convex mirror = +10 cm (positive because it is a convex mirror)

### Mirror Formula:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length of the mirror
- \( v \) = image distance (positive for virtual images in convex mirrors)
- \( u \) = object distance (negative for real objects in front of the mirror)

### Analyzing the Options:

#### Option 1: Image of an object placed at 20 cm is also at 20 cm.
1. **Object Distance (u)**: Since the object is placed at 20 cm, we take \( u = -20 \) cm.
2. **Using the mirror formula**:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{10} = \frac{1}{v} + \frac{1}{-20} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{10} + \frac{1}{20} \]
Finding a common denominator (20):
\[ \frac{1}{v} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \]
Therefore:
\[ v = \frac{20}{3} \text{ cm} \]
This means the image distance is \( \frac{20}{3} \) cm, not 20 cm. Thus, **Option 1 is incorrect**.

#### Option 2: Image of an object placed at 10 cm is at infinity.
1. **Object Distance (u)**: For an object placed at 10 cm, we take \( u = -10 \) cm.
2. **Using the mirror formula**:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{10} = \frac{1}{v} + \frac{1}{-10} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \]
Therefore:
\[ v = 5 \text{ cm} \]
This means the image distance is 5 cm, not at infinity. Thus, **Option 2 is also incorrect**.

### Conclusion:
Since both options A and B are incorrect, the correct answer is:
**Option 3: Both A and B are wrong.**