Question

Consider the quadratic equation $(c - 5)x^2-2cx + (c-4) = 0, c \neq 5$. Let S be the set of all integral values of $c$ 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 :

• A. 11
• B. 18
• C. 10
• D. 12

Answer 1

Answer: (A)

Let $f(x)=(c-5) x^{2}-2 c x+c-4$
$\therefore f(0) f(2)<0$
$f(2) f(3)<0$
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 $S=11$