Magnitude of focal length of a spherical mirror is f and magnitude of linear magnification is `1/2`

Options

A. If image is inverted, it is a concave mirror
B. If image is erect, it is a convex mirror
C. Object distance from the mirror may be `3 f`
D. Object distance from the mirror may be f.

Explanation

To solve the problem, we need to analyze the situation for both types of spherical mirrors: concave and convex. We will use the mirror formula and the magnification formula to derive the object distance and determine the type of mirror based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- Magnitude of focal length of a spherical mirror: \( f \)
- Magnitude of linear magnification: \( \frac{1}{2} \)

2. **Determine the Type of Mirror:**
- If the image is inverted, it indicates a concave mirror.
- If the image is erect, it indicates a convex mirror.

3. **Case 1: Convex Mirror**
- For a convex mirror, the object distance \( u \) is negative, so we can denote it as \( u = -x \).
- The magnification \( m \) for a convex mirror is given by:
\[ m = \frac{h'}{h} = \frac{v}{u} \]
where \( v \) is the image distance. Given \( m = \frac{1}{2} \), we have:
\[ \frac{v}{-x} = \frac{1}{2} \implies v = -\frac{x}{2} \]
- The focal length \( f \) for a convex mirror is positive.

4. **Using the Mirror Formula:**
- The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting the values:
\[ \frac{1}{f} = \frac{1}{-\frac{x}{2}} + \frac{1}{-x} \]
- Simplifying:
\[ \frac{1}{f} = -\frac{2}{x} - \frac{1}{x} = -\frac{3}{x} \]
- Therefore, we get:
\[ f = -\frac{x}{3} \]
- Since \( f \) is positive for a convex mirror, this case is valid if \( x = 3f \).

5. **Case 2: Concave Mirror**
- For a concave mirror, the object distance \( u \) is again negative, so we denote it as \( u = -x \).
- The image distance \( v \) for a concave mirror, given the image is inverted (real), is:
\[ v = -\frac{x}{2} \]
- The focal length \( f \) for a concave mirror is negative.

6. **Using the Mirror Formula Again:**
- Substituting into the mirror formula:
\[ \frac{1}{-f} = \frac{1}{-\frac{x}{2}} + \frac{1}{-x} \]
- Simplifying:
\[ -\frac{1}{f} = -\frac{2}{x} - \frac{1}{x} = -\frac{3}{x} \]
- Therefore:
\[ f = \frac{x}{3} \]
- Since \( f \) is negative for a concave mirror, this case is valid if \( x = 3f \).

### Conclusion:
- If the image is erect, it is a convex mirror with object distance \( x = f \).
- If the image is inverted, it is a concave mirror with object distance \( x = 3f \).

### Final Answer:
- The statements are correct for both types of mirrors:
- For a convex mirror: Object distance may be \( f \).
- For a concave mirror: Object distance may be \( 3f \).