A converging lens of focal length 12 cm and a diverging mirror of...

For the final image to coincide with the object, the light rays must retrace their path after reflection from the diverging (convex) mirror.

This is possible only if the rays refracted by the convex lens fall normally on the convex mirror. For a convex mirror, rays fall normally if they are directed towards its center of curvature.

The radius of curvature of the convex mirror is $R = 2f = 2 \times 7.5 = 15 \text{ cm}$.

Thus, the image formed by the lens must be at the center of curvature of the mirror, which is $15 \text{ cm}$ behind the mirror.

The distance of this image from the lens is $v = 5.0 + 15 = 20 \text{ cm}$.

Using the lens formula, $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$:

$\dfrac{1}{20} - \dfrac{1}{u} = \dfrac{1}{12}$

$\dfrac{1}{u} = \dfrac{1}{20} - \dfrac{1}{12} = \dfrac{3 - 5}{60} = -\dfrac{1}{30}$

$u = -30 \text{ cm}$

The object should be placed $30 \text{ cm}$ from the lens, further away from the mirror.