Problem:
A prism of prism angle (A=60^(@)) and refractive index n_(2) is placed in liquid of refractive index n_(1). A light ray is incident on face AB at constant angle of incidence I and it emerges at surface AC at angle of emergent e. Somehow n_(1) is increasing at rate N_(1) and n_(2) is increasing at rate n _(2). Choose the correct option(s)?

Solution:
Let's consider the given problem step by step.

Step 1: Draw the diagram

Step 2: Find the relation between the angle of incidence I and the angle of emergence e.
We know that the angle of incidence I and the angle of emergence e are related by the formula:

n_(1)sin(I) = n_(2)sin(e)

Step 3: Find the rate of change of angle of incidence I and the angle of emergence e with respect to time.
We can find the rate of change of angle of incidence I and the angle of emergence e with respect to time by differentiating the above equation with respect to time:

n_(1)cos(I)dI/dt = n_(2)cos(e)de/dt

Step 4: Find the rate of change of refractive indices n_(1) and n_(2) with respect to time.
We are given that n_(1) is increasing at a rate of N_(1) and n_(2) is increasing at a rate of n _(2). Therefore, we have:

dn_(1)/dt = N_(1)
dn_(2)/dt = n _(2)

Step 5: Find the relation between the rates of change of angle of incidence I and the angle of emergence e and the rates of change of refractive indices n_(1) and n_(2).
Substituting the values of dn_(1)/dt and dn_(2)/dt in the equation obtained in step 3, we get:

n_(1)cos(I)dI/dt = n_(2)cos(e)de/dt
N_(1)cos(I) = n _(2)cos(e)de/dt / dI/dt

Step 6: Choose the correct option(s).
We can see that the rate of change of angle of emergence e with respect to time is dependent on the rate of change of angle of incidence I with respect to time and the refractive indices of the prism and the liquid. Therefore, option (a) is correct. Option (b) is incorrect as the angle of incidence I is constant. Option (c) is incorrect as the angle of incidence I is not dependent on the rate of change of refractive indices. Option (d) is incorrect as the rate of change of angle of emergence e is dependent on the rate of change of angle of incidence I and the refractive indices of the prism and the liquid. Therefore, the correct option is (a).

Answer:
Option (a) is correct.