Question

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then $k$ lies in the interval:

• A. $[4,5]$
• B. $\left(-\infty ,4\right)$
• C. $\left(6,\infty \right)$
• D. $(5,6]$

Answer 1

Answer: (B)

Key Idea : If both roots of $a{x}^2+bx+c=0$ are less then $\alpha$, then $b^2-4ac\ge 0,f\left(\alpha \right)>0$ and $\frac{-b}{2a}<\alpha$
Let $f\left(x\right)=x^2-2kx+k^2+k-5$
$\therefore D=4k^2-4\left(k^2+k-5\right)$
$=-4k+20\ge 0$
$\Rightarrow k\le 5...\left(i\right)$
Also $-\frac{b}{2a}<5...\left(ii\right)$
and $f\left(5\right)>0...\left(iii\right)$
From $\left(ii\right),k<5...\left(iii\right)$
From $\left(iii\right),25-10k+k^2+k-5>0$
$\Rightarrow k^2-9k+20>0$
$\Rightarrow k^2-5k-4k+20>0$
$\Rightarrow \left(k-5\right)\left(fc-4\right)>0$
$\Rightarrow k<4andk>5...\left(v\right)$
From $\left(i\right),\left(iii\right)$ and $\left(iv\right)$
$\Rightarrow k<4$