(a) (i) Definition of Refractive Index [1 Mark]

The refractive index ($\mu$) of a medium is defined as the ratio of the speed of light in vacuum ($c$) to the speed of light in that medium ($v$).

$$\mu = \frac{c}{v}$$

(a) (ii) Derivation of Prism Formula [2 Marks]

Step 1 (Governing Laws): [0.5 Mark]
For a prism, the angle of deviation $\delta$ is given by $\delta = (i + e) - A$. At the angle of minimum deviation ($\delta_m$), we have $i = e$ and $r_1 = r_2 = r$.

Step 2 (Mathematical Working): [1 Mark]
From the geometry of the prism, $A = r_1 + r_2 = 2r \implies r = \frac{A}{2}$.
Substituting $i = e$ in the deviation formula: $\delta_m = 2i - A \implies i = \frac{A + \delta_m}{2}$.

Step 3 (Final Concluded Answer): [0.5 Mark]
Applying Snell's Law at the first surface: $\mu = \frac{\sin i}{\sin r}$. Substituting the values of $i$ and $r$:

(a) (iii) Ray Tracing and Explanation [2 Marks]

Step 1 (Given Data & Analysis): [0.5 Mark]
The ray is incident normally on face BC. Thus, the angle of incidence at the first surface is $0^\circ$. The ray passes undeviated into the prism and strikes face AC at an angle of incidence equal to the prism angle $A$.

Step 2 (Mathematical Working): [1 Mark]
Assuming an equilateral prism ($A = 60^\circ$), the angle of incidence at face AC is $i' = 60^\circ$. The critical angle ($i_c$) for the prism ($\mu = 1.5$) is:

Since $i' (60^\circ) > i_c (41.8^\circ)$, the ray undergoes Total Internal Reflection (TIR) at face AC.

Step 3 (Final Concluded Answer): [0.5 Mark]
The ray reflects at an angle of $60^\circ$ (Law of Reflection) and strikes face AB at an angle of $30^\circ$. It then emerges into the air, bending away from the normal.