While deriving the lens maker’s formula for a thin double convex lens…

The correct answer is Option (3) → acts as a virtual object for the second surface, since the rays are refracted again at the second surface before they can meet at \( I_1 \)

The lens maker’s formula is found in two steps. The first surface alone would bend the rays from O to meet at \( I_1 \), which lies beyond the second surface. The rays reach the second surface first and are refracted again, so they never actually meet at \( I_1 \). For the second surface they are rays converging towards a point, and such a point is a virtual object.

Option (1) acts as a real object for the second surface, since it is formed by rays that have actually been refracted — Incorrect. A real object is a point from which rays actually diverge towards the surface. Here the rays are still converging when they strike the second surface.

Option (2) is the final image formed by the lens, since the lens is thin — Incorrect. A thin lens still has two refracting surfaces. Thinness only allows all distances to be measured from the optical centre. The final image I is formed by the second surface.

Option (3) acts as a virtual object for the second surface, since the rays are refracted again at the second surface before they can meet at \( I_1 \) — Correct. The rays meet the second surface before they reach \( I_1 \). So \( I_1 \) is only the point towards which they converge.

Option (4) acts as a virtual object for the second surface, since it lies on the same side of the lens as the object — Incorrect. The role is right but the reason is not. \( I_1 \) is formed on the far side of the lens, beyond the second surface, and that is why the rays cannot reach it.

NCERT (2026-27):
Refraction by a Lens – “… (i) The first refracting surface forms the image I₁ of the object O … The image I₁ acts as a virtual object for the second surface that forms the image at I …”