Consider the quadratic equation
Question:

Consider the quadratic equation

$(c-5) x^{2}-2 c x+(c-4)=0, c \neq 5$. Let $S$ be the set

of all integral values of $\mathrm{c}$ for which one root of the equation lies in the interval $(0,2)$ and its other root lies in the interval $(2,3)$. Then the number of elements in $S$ is :

 

  1. 11

  2. 18

  3. 10

  4. 12


Correct Option: 1

Solution:

Let $f(x)=(c-5) x^{2}-2 c x+c-4$

$\therefore f(0) f(2)<0$   ……………….(1)

$\& f(2) f(3)<0$   …………..(2)

from (1) \& (2)

$(c-4)(c-24)<0$

$\&(c-24)(4 c-49)<0$

$\Rightarrow \frac{49}{4}<c<24$

$\therefore s=\{13,14,15, \ldots .23\}$

Number of elements in set $S=11$

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