A convex lens of focal length 20cm is placed in front of convex...

For lens $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$

$⇒ \dfrac{1}{v} - \dfrac{1}{\left(- 60\right)} = \dfrac{1}{20} ⇒ \dfrac{1}{v} + \dfrac{1}{60} = \dfrac{1}{20}$

$v = 30cm$

For the final image to be formed on the object itself, after refraction from lens the ray should meet the mirror perpendicularly and the image by lens should be on the centre of curvature of mirror.

$R = 30 - 10 = 20cm$.

Now the focal length of mirror ${f}_{m} = \dfrac{R}{2} = 10cm$