Step 1: Find the image made by the lens.
Use the lens formula with sign rules. \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] Here $f = 10$ cm and $u = -15$ cm.
Step 2: Solve for the image distance.
\[ \frac{1}{v} = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30} \] \[ v = 30\,\text{cm} \] So the lens alone would form the image $30$ cm to its right.
Step 3: Locate where the mirror sits.
The convex mirror is placed at the focus, that is $10$ cm to the right of the lens.
Step 4: Find where the lens image falls relative to the mirror.
The lens image is at $30$ cm, the mirror is at $10$ cm, so the image would form $30 - 10 = 20$ cm behind the mirror.
Step 5: Use the retrace condition.
For the final image to come back exactly on the object, light must hit the mirror straight on and return along itself. This happens when the rays head toward the centre of curvature. So the centre of curvature is $20$ cm away, meaning \[ R = 20\,\text{cm} \]
Step 6: Convert to focal length.
\[ f = \frac{R}{2} = 10\,\text{cm} \] \[ \boxed{10\,\text{cm}} \]