Question

Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0$, $c\ne 5$. 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)

$11$

Let $f\left(x\right)=(c-5)c^2-2cx+c-4$
$\therefore f\left(0\right)f\left(2\right)<0$ .................$\left(1\right)$
& $f\left(2\right)f\left(3\right)<0$ ...........................$\left(2\right)$
from $\left(1\right)$ & $\left(2\right)$
$(c-4)(c-24)<0$
& $(c-24)(4c-49)<0$
$\Rightarrow \frac{49}{4}<c<24$
$\therefore s=\{13,14,15,.....23\}$
Number of elements in set S$=11$.