1. Interpret the statement about the plane mirror image

A plane mirror forms an image as far behind the mirror as the object is in front of it.

Given: the image produced by the plane mirror is 5 cm inside the mirror. So, for the mirror, the object must be 5 cm in front of the mirror.

That means the image formed by the lens (which acts as the object for the mirror) lies at a point 5 cm before the mirror.

2. Find the position of the lens image

The mirror is placed at a distance of $20\,\text{cm}$ from the lens. Since the lens image is $5\,\text{cm}$ in front of the mirror, its distance from the lens is

$v = 20 - 5 = 15\,\text{cm}$

So the convex lens alone forms an image at $15\,\text{cm}$ on the other side of the lens.

3. Apply lens formula

For a convex lens,

$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$

Using sign convention:

• $f = +10\,\text{cm}$
• $v = +15\,\text{cm}$
• $u = ?$

Substitute:

$\frac{1}{10} = \frac{1}{15} - \frac{1}{u}$

$\frac{1}{u} = \frac{1}{15} - \frac{1}{10}$

$\frac{1}{u} = \frac{2-3}{30} = -\frac{1}{30}$

Hence,

$u = -30\,\text{cm}$

The negative sign shows the object is on the left side of the lens, which is correct physically. Therefore, the distance of the object from the lens is

$30\,\text{cm}$

4. Compare with stored answer

Derived answer = $30$

Stored correct answer = $30$

They match.