Question

The set of values of $a$ for which $a{x}^2+(a-2)x-2$ is negative for exactly two integral values of $x$, is

• A. $(0,2)$
• B. $[1,2)$
• C. $(1,2]$
• D. $(0,2]$

Answer 1

The correct option is B $[1,2)$

$f\left(x\right)=a{x}^2+(a-2)x-2$
Let $f\left(x\right)=0$
Roots of the above equations are $-1$ and $\frac{2}{a}$
$\because f\left(0\right)=-2<0$
$\therefore$ For the above expression to be negative only for two integers then $a>0$,

1st integer is $0$ so second integer must be $1$

which implies the second root must $>1$ and $\le 2$
$\Rightarrow 1<\frac{2}{a}\le 2$
$\Rightarrow 1\le a<2$
$\therefore a\in [1,2)$.