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

Answer 1

The correct option is C $(-\infty ,4)$

$f\left(x\right)=x^2-2kx+k^2+k-5$
both roots are less than 5, then
(i) Discriminant $\ge 0$
$\left(ii\right)f\left(5\right)>0\left(iii\right)\frac{Sumofroots}{2}<5$
Hence$i\left)\right(4k^2-4(k^2+k-5)\ge 04k^2-4k^2-4k+20\ge 04k\le 20\Rightarrow k\le 5$

$\left(ii\right)f\left(5\right)>0;25-10k+k^2+k-5>0or{k}^2-9k+20>0ork(k-4)-5(k-4)>0or(k-5)(k-4)>0\Rightarrow kϵ(-\infty ,4)\bigcup (5,\infty )\left(iii\right)\frac{Sumofroots}{2}=-\frac{b}{2a}=\frac{2k}{2}<5$
The intersection of (i), (ii) & (iii) gives k $ϵ(-\infty ,4).$