Q: If ac > b2 then the sum of the coefficients in the expansion of (a α2x2 + 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 α2x2 + 2b α x + c)n
the sum of the coefficients = (a α2 + 2b α + c)n
Let f(α) = (a α2 + 2b α + c)n
Its discriminant = 4b2 – 4ac = 4(b2 –ac ) < 0
Hence, f(α) < 0 or f(α) > 0 for all α ∈ R
If a > 0 then f(α) > 0 ⇒ (a α2 + 2b α + c)n > 0
If c > 0 i.e. f(0) > 0 ⇒ f(α) > 0 ⇒ (a α2 + 2b α + c)n > 0
If a < 0 then f(α) < 0 ⇒ (a α2 + 2b α + c)n < 0 if n is odd
If c < 0 i.e. f(0) < 0 ⇒ f(α) < 0 ⇒ (a α2 + 2b α + c)n > 0 if n is even.
Hence (A), (B), (C) and (D) are the correct answer.