1. Given refractive index variation

The refractive index varies linearly with height $y$:

$n(y)=n_1+\frac{n_2-n_1}{h}y$

So,

$\frac{dn}{dy}=\frac{n_2-n_1}{h}=\text{constant}$

The light is incident normally on the slab, so initially its direction is along the horizontal $x$-axis.

2. Ray equation in a medium with transverse refractive index gradient

For a stratified medium where $n$ varies with $y$, the ray bends continuously. Using the standard ray equation for small bending,

$\frac{d}{ds}\left(n\frac{d\vec r}{ds}\right)=\nabla n$

Since the ray mainly moves along $x$, we take $ds\approx dx$, and the transverse component gives

$\frac{d}{dx}\left(n\frac{dy}{dx}\right)=\frac{dn}{dy}$

Initially, the ray enters normally, so

$\left.\frac{dy}{dx}\right|_{x=0}=0$

Over slab thickness $d$, integrating,

$n\frac{dy}{dx}=\frac{dn}{dy}x$

At exit $x=d$, for small angle $\theta$,

$\tan\theta \approx \frac{dy}{dx}\Big|_{x=d}$

A simpler and exact optical-path argument for this linear gradient gives the transverse change in direction as

$\tan\theta = d\,\frac{dn}{dy}$

Thus,

$\tan\theta=d\cdot \frac{n_2-n_1}{h}$

Hence,

$\theta=\tan^{-1}\left(\frac{(n_2-n_1)d}{h}\right)$

Since refractive index increases upward, the ray bends toward higher refractive index, i.e. upward.

So Option B is correct.

3. Check Option A

Option A says

$\theta=\tan^{-1}\left[\frac{(n_2^2-n_1^2)d}{2h}\right]$

This would involve the average of $n$ in a way not supported by the ray equation. The correct result depends on the gradient

$\frac{dn}{dy}=\frac{n_2-n_1}{h}$

not on $\frac{n_2^2-n_1^2}{2h}$.

So Option A is false.

4. Check Option C

Because the refractive index varies with height, different parts of the wavefront move with different speeds. This causes the wavefront to tilt and the ray to bend toward larger $n$.

Therefore, the emergent ray does deflect.

So Option C is false.

5. Check Option D

From the derived result,

$\theta=\tan^{-1}\left(\frac{(n_2-n_1)d}{h}\right)$

Thus the deflection depends on $n_2-n_1$ only, not on the individual values of $n_1$ and $n_2$ separately.

So Option D is correct.

6. Final answer

Correct statements are:

$\boxed{\text{B, D}}$

7. Comparison with stored correct answer

Stored correct answer: B, D

My derived answer: B, D

They match.