Question

If $x^2+(m-3)x+m=0,(m\in R)$ be a quadratic equation, then the value of $m$ for which one root is greater than 2 and the other is smaller than 1 is

• A. $\left(-\infty ,\frac{2}{3}\right)$
• B. $(-\infty ,1]$
• C. $\left(\frac{2}{3},1\right)$
• D. $\left(-\infty ,\frac{2}{3}\right)\cup (1,\infty )$

Answer 1

Answer: (A)

Clearly 1 and 2 should lie between roots
$\therefore f(1)<0$
$\Rightarrow 1+(m-3)\times 1+m<0$
$\Rightarrow 2m-2<0\Rightarrow m<1$
and
$f(2)<0\Rightarrow 4+2m-6+m<0\Rightarrow 3m<2\Rightarrow m<\frac{2}{3}$

$\therefore m\in \left(-\infty ,\frac{2}{3}\right).$