$\begin{array}{l} C o n s i d e r t h e q u a d r a t i c e q u a t i o n (C - 5) x^{2} - 2 c x + (c - 4) = 0, C \neq 5. L e t S b e t h e s e t o f a l l int e g r a l v a l u e s o f C f o r w h i c h o n e r o o t o f t h e e q u a t i o n l i e s \\ i n t h e int e r v a l (0, 2) a n d i t s o t h e r r o o t l i e s i n \\ t h e int e r v a l (2, 3), t h e n t h e n u m b e r o f e l e m e n t s i n S i s : \end{array}$

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Detailed Solution

$\begin{array}{l} G i v e n E q u a t i o n (C - 5) x^{2} - 2 c x + (c - 4) = 0 \\ \Rightarrow x^{2} - (\frac{2 c}{c - 5}) x + (\frac{c - 4}{c - 5}) = 0 \\ ∴ f (0) . f (2) < 0 \\ \Rightarrow (\frac{c - 4}{c - 5}) \times (4 - \frac{4 c}{c - 5} + \frac{c - 4}{c - 5}) < 0 \\ \Rightarrow \frac{(c - 4) (c - 24)}{{(c - 5)}^{2}} < 0 \\ \Rightarrow C < (4, 25) - {5} - - (1) \\ a n d f (2) f (3) < 0 \\ \Rightarrow (\frac{c - 24}{c - 5}) (\frac{4 c - 49}{c - 5}) < 0 \\ \Rightarrow (c - 24) (4 c - 49) < 0 \\ c \in (\frac{49}{4}, 24) - - (2) \\ f r o m (1) & (2) C \in (\frac{49}{4}, 24) \\ ∴ S = {13, 14, - - - 23} \\ H e n c e n u m b e r o f e l e m e n t s i n t h e s e t s a r e' 11' \end{array}$