Question

Statement 1 : If $f\left(x\right)=a{x}^2+bx+c$, where $a>0,c<0$ and $b\in R$, then roots of $f\left(x\right)=0$ must be real and distinct .
Statement 2 : If $f\left(x\right)=a{x}^2+bx+c,$ where $a>0,b\in R,b\ne 0$ and the roots of $f\left(x\right)=0$ are real and distinct, then $c$ is necessarily negative real number .

• 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 1

The correct option is C Statement-1 is True, Statement-2 is False

Statement -1
$b^2-4ac>0$ {Since $a>0,c<0$}
$\therefore$ Roots are real and distinct
$\therefore$ Statement -1 is true .
Statement -2
Since the roots are real and distinct
$\therefore b^2-4ac>0$ i.e. $c<\frac{b^2}{4a}$
Thus, c is not necessarily negative
$\therefore$ Statement -2 is false .