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 $
$\Rightarrow 2 m -2<0 \Rightarrow m <1$
and
$f (2)<0 \Rightarrow 4+2 m -6+ m <0 \Rightarrow 3 m <2 \Rightarrow m <\frac{2}{3}$

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