1. Given data

• Real depth of point at bottom of each cylinder: $H = 30\text{ cm} = 0.30\text{ m}$
• Refractive index of glass: $n = 1.5$
• Radius of curvature of curved top surfaces: $R = 3\text{ m} = 300\text{ cm}$

We need the apparent depth of the bottom point $X$ when viewed from air through the top surface.

2. Cylinder I: plane top surface

For refraction at a plane surface, apparent depth is

$H_1 = \frac{H}{n} = \frac{30}{1.5} = 20\text{ cm}$

3. Cylinder II: convex top surface

This is refraction at a spherical surface from glass $(n_1=1.5)$ to air $(n_2=1)$.

Use the spherical refraction formula:

$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2-n_1}{R}$

Take Cartesian sign convention with positive direction upward (towards air).

• Object $X$ is inside glass, below the top surface, so $u = -30\text{ cm}$
• For a convex top as seen from inside glass, the center of curvature lies below the surface, so $R = -300\text{ cm}$

Substitute:

$\frac{1}{v} - \frac{1.5}{-30} = \frac{1-1.5}{-300}$

$\frac{1}{v} + 0.05 = \frac{-0.5}{-300} = \frac{1}{600}$

$\frac{1}{v} = \frac{1}{600} - 0.05 = 0.0016667 - 0.05 = -0.0483333$

$v \approx -20.69\text{ cm}$

So the image is inside glass at depth $20.69\text{ cm}$ below the top. Hence

$H_2 \approx 20.69\text{ cm}$

Therefore,

$H_2 > H_1$

and

$H_2 - H_1 \approx 20.69 - 20 = 0.69\text{ cm}$

So option C is false.

4. Cylinder III: concave top surface

Again use

$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2-n_1}{R}$

Now for a concave top as seen from inside glass, the center of curvature lies above the surface, so

$R = +300\text{ cm}$

Also,

$u = -30\text{ cm}$

Substitute:

$\frac{1}{v} - \frac{1.5}{-30} = \frac{1-1.5}{300}$

$\frac{1}{v} + 0.05 = -\frac{1}{600}$

$\frac{1}{v} = -\frac{1}{600} - 0.05 = -0.0016667 - 0.05 = -0.0516667$

$v \approx -19.35\text{ cm}$

Thus,

$H_3 \approx 19.35\text{ cm}$

So,

$H_3 < H_1$

Hence option B is false.

5. Compare $H_2$ and $H_3$

We found:

$H_2 \approx 20.69\text{ cm}, \qquad H_3 \approx 19.35\text{ cm}$

Thus,

$H_2 > H_3$

So option D is true.

6. Evaluate all options

• A: $H_2 > H_1$ ✅ True
• B: $H_3 > H_1$ ❌ False
• C: $0.85\text{ cm} < (H_2-H_1) < 0.9\text{ cm}$ ❌ False, since $H_2-H_1 \approx 0.69\text{ cm}$
• D: $H_2 > H_3$ ✅ True

7. Final answer

The correct options are:

$\boxed{A, D}$

This matches the stored correct answer.