Question

Let $A$ be the set of values of $k$ for which $2$ lies between the roots of the quadratic equation $x^{2}+\left(k + 2\right)x-\left(k + 3\right)=0,$ then $A$ is given by

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

Answer 1

Answer: (A)

Opening upward parabola as the coefficient of $x^{2}$ is $+ve$
$2$ lies between the roots $\Rightarrow f\left(2\right) < 0$
$f\left(2\right)=4+2\left(k + 2\right)-\left(k + 3\right) < 0$
$4+2k+4-k-3 < 0$
$k+5 < 0\Rightarrow k < -5$