Question

Let $S$ be the set of values of $a$ for which $2$ lies between the roots of the quadratic equation $x^2+(a+2)x-(a+3)=0$, then $S$ is given by

• A. $(-\infty ,-5]$
• B. $(5,\infty )$
• C. $(-\infty ,-5)$
• D. $[5,\infty )$

Answer 1

The correct option is C $(-\infty ,-5)$

$\Delta =a^2+8a+16$ $=(a+4{)}^2$

Hence, the roots of the equation are always real.

The parabola opens upward since coefficient of $x^2$ is positive.

Now, $2$ must lie between the roots of the quadratic equations.

$\Rightarrow f\left(2\right)<0$.

$\Rightarrow 4+2a+4-a-3<0$

$\Rightarrow a+5<0$

$\Rightarrow a\in (-\infty ,-5)$