A thin convex lens \(L\) (refractive index \(= 1.5\)) is placed on a plane mirror \(M\). When a pin is placed at \(A\), such that \(OA = 18\, cm\), its real inverted image is formed at \(A\) itself, as shown in figure. When a liquid of refractive index \({\mu _1}\) is put between the lens and the mirror, the pin has to be moved to \(A’\). such that \(OA’ = 27\,cm\), to get its inverted real image at \(A'\) itself. The value of \({\mu _1}\) will be

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.