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 $\alpha , \beta \in (1,5)$