First slide

Theory of expressions

Question

$The greatest integral value of c so that both the roots of the equation (c - 5) x^{2} - 2 c x + (c - 4) = 0 are positive, one$
$root is less than 2 and other root is lying between 2 and 3 is$

Difficult

A

22
B

23
C

24
D

25

Solution

$We have, x^{2} - (\frac{2 c}{c - 5}) x + (\frac{c - 4}{c - 5}) = 0$
$\begin{array}{l} Let f (x) = x^{2} - (\frac{2 c}{c - 5}) x + (\frac{c - 4}{c - 5}) \\ So, f (0) > 0, f (2) < 0, f (3) > 0 must be satisfied simultaneously. \end{array}$
$Now, f (0) > 0 \Rightarrow \frac{c - 4}{c - 5} > 0 - - - - - (1)$
$f (2) < 0 \Rightarrow \frac{c - 24}{c - 5} < 0 - - - - - (2)$
$and f (3) > 0 \Rightarrow \frac{4 c - 49}{c - 5} > 0 - - - - - (3)$
$\begin{aligned} Hence, (1) \cap (2) \cap (3) \\ \Rightarrow c \in (\frac{49}{4}, 24) \end{aligned}$
So, the greatest integral value of c is 23.

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