NIMCET 2015 Question Paper with solutions

We have $\left(\mathit{k}\mathit{–}2\right){\mathit{x}}^2+8\mathit{x}+\mathit{K}+4=0$ has both real, distinct and negative and negative roots.
There are three cases
(i) D > 0
(ii) Sum of the roots< 0 i.e.
(iii) Product of roots > 0i.e. $\frac{\mathit{c}}{\mathit{a}}>0$
Case I D > 0
$64-4\left(\mathit{k}-2\right)\left(\mathit{k}+4\right)>0$
$16-\left({\mathit{k}}^2+2\mathit{k}-8\right)>0$


....(i)
Case II Sum of roots < 0

$\Rightarrow \frac{8}{\mathit{k}-2}>0$
$\Rightarrow \mathit{k}>2$ ....(ii)
Case III Product of roots > 0
$\therefore \frac{\mathit{k}+4}{\mathit{k}-2}>0$
$\therefore \mathit{K}∊\left(-\mathit{\infty },-4\right)\cup \left(2,\mathit{\infty }\right)$ ....(iii)
From Eqs. (i), (ii) and (iii) We get
k = 3 is satisfied
Hence, value of k = 3