If P(x) be a polynomial with real coefficients such that P(sin2 x) = P

1.	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 degreeThen.(A) all are false(B) only I and II are true(C) only II and III are true (D) all are true
Answer» 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

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 degreeThen.(A) all are false(B) only I and II are true(C) only II and III are true (D) all are true

Answer»

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

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