1. Key idea: a thin film on a spherical surface behaves like two refracting spherical surfaces in contact
   The film has refractive index $n_1=1.4$, the cylinder has refractive index $n_2=1.5$, and outside medium is air $(n=1)$.

Because the film is thin and of uniform thickness, the two spherical interfaces are essentially concentric and have nearly the same radius $R$. Hence we can add the powers of the two refracting surfaces.

For a spherical refracting surface,

with Cartesian sign convention. Here the surface is convex as seen from air, and the center lies inside the cylinder, so $R>0.$

2. Case 1: Rays go from air $\to$ film $\to$ glass
   Parallel rays are incident from air.

• First surface: air $(1)$ to film $(1.4)$

• Second surface: film $(1.4)$ to glass $(1.5)$

So total power is

This is equivalent to a single refracting surface from air to glass:

For parallel rays from air, image is formed in glass at distance $f_1$ from the surface. For a refracting surface,

Here $u=\infty$, $n_0=1$, $n_2=1.5$, so

Hence,

So A is correct and B is incorrect.

3. Case 2: Rays go from glass $\to$ film $\to$ air
   Now parallel rays are incident from inside the glass toward air.

Again add powers:

• First surface: glass $(1.5)$ to film $(1.4)$

• Second surface: film $(1.4)$ to air $(1)$

Thus total power:

Equivalent to single refracting surface from glass to air:

For parallel rays inside glass, using refraction formula:

would be incorrect because object space is glass. Correctly,

with $u=\infty$:

Therefore,

So C is correct and D is incorrect.

4. Option check

• A: $|f_1|=3R$ ✅
• B: $|f_1|=2.8R$ ❌
• C: $|f_2|=2R$ ✅
• D: $|f_2|=1.4R$ ❌

5. Final answer The correct options are: