$\begin{array}{l} I f b o t h t h e r o o t s o f t h e q u a d r a t i c e q u a t i o n x^{2} - m x + 4 = 0 a r e r e a l a n d d i s t i n c t a n d t h e y l i e i n t h e int e r v a l \\ [1, 5], t h e n m l i e s i n t h e i t e r v a l \end{array}$

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$(5, 6)$

$(- 5, - 4)$

$(4, 5)$

$(3, 4)$

answer is C.

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Detailed Solution

$\begin{array}{l} G i v e n E q u a t i o n x^{2} - m x + 4 = 0 \\ c o m p a r i n g w i t h a x^{2} + b x + c = 0 t h e n \\ a = 1 > 0, b = - m, c = 4 \\ T h e n D > 0, 1 < \frac{- b}{2 a} < 5 a n d \\ f (1) > 0, f (5) > 0 \\ C a s e (i) : Δ > 0 \Rightarrow m^{2} - 4 (4) > 0 \Rightarrow (m + 4) (m - 4) > 0 \\ ∴ m < - 4 o r m > 4 \\ C a s e (i i) : 1 < \frac{- b}{2 a} < 5 \Rightarrow 1 < \frac{m}{2} < 5 \Rightarrow 2 < m < 10 \\ C a s e (i i i) : f (1) > 0 a n d f (5) > 0 \\ \Rightarrow 1 - m + 4 > 0 a n d 25 - 5 m + 4 > 0 \Rightarrow 29 - 5 m > 0 \\ \Rightarrow m < 5 \Rightarrow m < \frac{29}{5} \\ ∴ m \in (4, 5) \end{array}$