question_answer

If both the roots of the quadratic equation \[{{x}^{2}}-5x+4=0\]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) None of these

Correct Answer: D

Solution :

\[{{x}^{2}}-mx+4=0\]

\[a,\beta \in [1,5]\]

[a] D < 0 \[\Rightarrow \]\[{{m}^{2}}-16>0\]

\[\Rightarrow \]\[m\in \,(-\infty ,-\,4)\cup \,(4,\infty )\]

[b] \[f(1)\ge 0\] \[\Rightarrow \]\[5-m\ge 0\]

\[\Rightarrow \] \[m\in \,(-\,\infty ,5)\]

[c] \[f\,(5)\ge 0\]\[\Rightarrow \] \[29-5m\ge 0\]

\[\Rightarrow \]   \[m\in \left( -\,\infty ,\left. \frac{29}{5} \right] \right.\]

[d] \[1<\frac{-\,b}{2a}<5\] \[\Rightarrow \] \[1<\frac{m}{2}<5\]

\[\Rightarrow \]   \[m\in \,(2,10)\]

\[\Rightarrow \]   \[m\in \,(4,5)\]

No option correct : Bonus