An object is placed in front of a thin convex lens of focal length `30cm`and a plane mirror is placed`15cm`behind the lens.If the final image of the object coincides with the object the distance of the object from the lens is

Options

A. `60cm
B. `30cm`
C. `15cm`
D. `25cm`

Explanation

To solve the problem, we need to determine the distance of the object from the lens such that the final image coincides with the object itself. We have a thin convex lens with a focal length of \( f = 30 \, \text{cm} \) and a plane mirror placed \( 15 \, \text{cm} \) behind the lens.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a convex lens with a focal length \( f = 30 \, \text{cm} \).
- A plane mirror is placed \( 15 \, \text{cm} \) behind the lens.

2. **Using the Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length, \( v \) is the image distance from the lens, and \( u \) is the object distance from the lens.

3. **Finding the Image Distance**:
- The image formed by the lens will act as a virtual object for the plane mirror.
- Since the mirror is \( 15 \, \text{cm} \) behind the lens, if the image distance from the lens is \( v \), the distance from the image to the mirror is \( (v + 15) \, \text{cm} \).

4. **Condition for Coinciding Images**:
- For the final image to coincide with the object, the distance of the object from the lens must equal the focal length of the lens, i.e., \( u = f \).
- Thus, we can set \( u = 30 \, \text{cm} \).

5. **Substituting into the Lens Formula**:
- Now substituting \( u = -30 \, \text{cm} \) (the object distance is taken as negative in lens formula convention):
\[ \frac{1}{30} = \frac{1}{v} - \frac{1}{(-30)} \]
- Simplifying this gives:
\[ \frac{1}{30} = \frac{1}{v} + \frac{1}{30} \]
- Rearranging gives:
\[ \frac{1}{v} = 0 \]
- This implies \( v \) approaches infinity, meaning the rays are parallel.

6. **Conclusion**:
- Since the object must be at the focus for the rays to exit parallel, the distance of the object from the lens is \( 30 \, \text{cm} \).

### Final Answer:
The distance of the object from the lens is \( 30 \, \text{cm} \).