A

Step 1: Understand the terms in the problem
- For the trajectory of a particle in the x-y plane, the equation of trajectory is typically described as a function of time, given as r = f(t).

Step 2: Matching terms from Column I to Column II
(i) The equation of trajectory is represented as r = f(t) which corresponds to option [A].

(ii) The slope of the path of the particle represents the change in y with respect to x and can be given as dy/dx. The corresponding representation is the slope indicated by [B].

(iii) The equation of motion describes the relationship of position, velocity, and acceleration with respect to time, matching with the mathematical description of motion, likely represented as a differentiation output represented by curves in the problem, corresponds with option [C].

(iv) The slope of a given curve, such as -t curve, would typically also indicate a change, hence related to the function of y = g(x), making it match with option [D].

Thus, based on the analysis, the links are:
(i) - [A], (ii) - [B], (iii) - [C], (iv) - [D].
Therefore, the correct answer is option [A].