Q: If ac > b^{2} then the sum of the coefficients in the expansion of (a α^{2}x^{2} + 2b α x + c)^{n}; (a, b, c, α ∈ R, n ∈ N ) is

(A) positive if a > 0

(B) positive if c > 0

(C) negative if a < 0, n is odd

(D) positive if c < 0, n is even

Sol. In the expansion of (a α^{2}x^{2} + 2b α x + c)^{n}

the sum of the coefficients = (a α^{2} + 2b α + c)^{n}

Let f(α) = (a α^{2} + 2b α + c)^{n}

Its discriminant = 4b^{2} – 4ac = 4(b^{2} –ac ) < 0

Hence, f(α) < 0 or f(α) > 0 for all α ∈ R

If a > 0 then f(α) > 0 ⇒ (a α^{2} + 2b α + c)^{n} > 0

If c > 0 i.e. f(0) > 0 ⇒ f(α) > 0 ⇒ (a α^{2} + 2b α + c)^{n} > 0

If a < 0 then f(α) < 0 ⇒ (a α^{2} + 2b α + c)^{n} < 0 if n is odd

If c < 0 i.e. f(0) < 0 ⇒ f(α) < 0 ⇒ (a α^{2} + 2b α + c)^{n} > 0 if n is even.

Hence (A), (B), (C) and (D) are the correct answer.