Let `f(x)=a x^2+bx +c `a ,b ,c` in R`. If `f(x)` takes real values for real values of `x` and non-real values for non-real values of `x` , then (a)`a=0` (b) `b=0` (c) `c=0` (d) nothing can be said about `a ,b ,c`.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

C

Explanation

If `a ne 0`, then `f(x) = {(x+(b)/(2a))^(2)+(4ac-b^(2))/(4a^(2))}`
Clearly, `f((-b)/(2a)+i)=a(-1+(4ac-b^(2))/(4a^(2)))`, which is real.
This contradicts that f(x) assumes only non-real values for non-real values of x. Therefore, a = 0.
So, Statement-1 is true.
Statement-2 is false, because - i is a root of `x^(2) + x + 1 + i = 0` but i is not its root.