Question

If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval :

• A. (4,5)
• B. (3,4)
• C. (5,6)
• D. (-5,-4)

Answer 1

Answer: (A)

$x^2-mx+4=0$
$\alpha ,\beta \in [1,5]$
(1) $D>0$ $\Rightarrow m^2-16>0$
$\Rightarrow m\in (-\infty ,-4)\cup (4,\infty )$
(2) $f(1)\ge 0\Rightarrow 5-m\ge 0\Rightarrow m\in (-\infty ,5]$
(3) $f(5)\ge 0\Rightarrow 29-5m\ge 0\Rightarrow m\in (-\infty ,\frac{29}{5}]$
(4) $1<\frac{-b}{2a}<5\Rightarrow 1<\frac{m}{2}<5\Rightarrow m\in (2,10)$
$\Rightarrow m\in (4,5)$
$\ast$ If we consider $\alpha ,\beta \in (1,5)$