1. Initial entry into the rectangular block

The ray enters the rectangular block at normal incidence, so it goes undeviated up to the prism.

The prism is a right-angled prism, so the ray inside the prism strikes the slant face at an angle of incidence $i=45^\circ.$

Thus, the behavior is decided by refraction / total internal reflection at the prism-block interface and then possibly at the block-medium interface.

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2. Critical condition at prism-block interface

For light going from prism $(\mu_1)$ to block $(\mu_2)$:

• Total internal reflection occurs if $\mu_1>\mu_2$ and $i > c, \qquad \sin c = \frac{\mu_2}{\mu_1}.$

Since $i=45^\circ$, TIR occurs when $\sin 45^\circ > \frac{\mu_2}{\mu_1}$ $\frac{1}{\sqrt2} > \frac{\mu_2}{\mu_1}$ $\mu_1 > \sqrt2\,\mu_2.$

So condition 1 corresponds to reflection at the slant face, giving path $e \to i$. Hence, $S \to 1.$

This already leaves only Option D among the choices, but let us verify all others.

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3. Case $\mu_1=\mu_2$

If the prism and block have equal refractive indices, there is no refraction at the prism boundary. The prism is optically invisible to the ray.

So the ray continues straight through along path $e \to g$. Hence, $Q \to 3.$

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4. Case $\mu_2>\mu_1$ and $\mu_2>\mu_3$

Here, on entering the prism from the block, the ray goes from denser to rarer? Actually at the prism interaction the decisive issue is that no TIR can occur inside prism because $\mu_1<\mu_2$.

So the ray refracts through the prism and reaches the top surface of the rectangular block. Since $\mu_2>\mu_3$, it can emerge or reflect depending on angle, but from the geometry of the standard prism path, this corresponds to the upward emergent path $e \to f$.

Thus, $P \to 2.$

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5. Case $\mu_2<\mu_1<\sqrt2\,\mu_2$ and $\mu_2>\mu_3$

At the prism-block slant face, $\mu_1<\sqrt2\,\mu_2,$ so TIR does not occur.

The ray refracts out of the prism into the block. Since $\mu_2>\mu_3$, at the outer surface of the block the ray finally emerges in the path corresponding to $e \to h$.

Therefore, $R \to 4.$

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6. Matching summary

So we get:

• $P : e\to f \to 2$
• $Q : e\to g \to 3$
• $R : e\to h \to 4$
• $S : e\to i \to 1$

Thus the correct matching is $P-2,\; Q-3,\; R-4,\; S-1.$

So the correct option is D.

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7. Comparison with stored answer

Stored correct answer = D.

My derived answer = D.

They agree.