Question

If P(x) be a polynomial with real coefficients such that P(sin² x) = P(cos² x), for all x ∈ [0, π/2]. Consider the following statements :

I.  P(x) is an even function.

II.  P(x) can be expressed as a polynomial in (2x– 1)² 

I.  P(x) is a polynomial of even degree

Then.

(A)   all are false

(B)   only I and II are true

(C)  only II and III are true 

(D)  all are true 

Answer 1

Correct option (C) only II and III are true 

Explanation:

P(sin² x) = P(cos² x)

P(sin² x) = P(1 – sin² x)

P(x) = P(1 – x) ∀ x ∈ [0, 1]

Differentiable both sides w.r.t. x

P'(x) = – P'(1 – x)

So P '(x) is symmetric about point x = 1/2

So P '(x) has highest degree odd 

⇒ P(x) has highest degree even