Derivatives Of Functions In Parametric Forms

We are given the equation:

Multiply through to simplify:

Multiply both sides by 2:

Rewriting this equation:

This is the equation of a straight line in polar form:

Hence, the given equation represents a straight line.

Correct option: (D) a pair of straight lines

Definition: A function f(x) is called periodic with period T if: for all

Part 1: About the integral We consider the integral:
Since is periodic with period , this integral over one full period is the same no matter where it starts:
Therefore, I is independent of the choice of a So, Option (B) is correct

Part 2: Consider the function This is the Dirichlet function.

Let us examine when it is periodic: - Suppose is rational, then for any , is rational if is rational and irrational if is irrational. - Hence, So, function is periodic if T is rational
If is irrational, this property fails.

Therefore, Option (C) is also correct

Final Answer: (B) and (C)