If both the roots of the quadratic equation

Question:

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

  1. $(4,5)$

  2. $(3,4)$

  3. $(5,6)$

  4. $(-5,-4)$


Correct Option: 1

Solution:

$x^{2}-m x+4=0$

$\alpha, \beta \in[1,5]$

(1) $\begin{aligned} \mathrm{D} &>0 \Rightarrow \mathrm{m}^{2}-16>0 \\ \Rightarrow & \mathrm{m} \in(-\infty,-4) \cup(4, \infty) \end{aligned}$

(2) $f(1) \geq 0 \Rightarrow 5-\mathrm{m} \geq 0 \Rightarrow \mathrm{m} \in(-\infty, 5]$

(3) $f(5) \geq 0 \Rightarrow 29-5 \mathrm{~m} \geq 0 \Rightarrow \mathrm{m} \in\left(-\infty, \frac{29}{5}\right]$

(4) $1<\frac{-b}{2 a}<5 \Rightarrow 1<\frac{m}{2}<5 \Rightarrow m \in(2,10)$

$\Rightarrow \mathrm{m} \in(4,5)$

No option correct : Bonus

$*$ If we consider $\alpha, \beta \in(1,5)$ then option (1) is correct.

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