JEE PYQ: Quadratic Equation Question 45
Question 45 - 2019 (10 Jan Shift 1)
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:
(1) 18
(2) 12
(3) 10
(4) 11
Show Answer
Answer: (4)
Solution
Consider the quadratic equation $(c-5)x^2 - 2cx + (c-4) = 0$
Now, $f(0) \cdot f(3) > 0$ and $f(0) \cdot f(2) < 0$
$\Rightarrow (c-4)(4c - 49) > 0$ and $(c-4)(c-24) < 0$
$\Rightarrow c \in (-\infty, 4) \cup \left(\frac{49}{4}, \infty\right)$ and $c \in (4, 24)$
$\Rightarrow c \in \left(\frac{49}{4}, 24\right)$
Integral values in the interval $\left(\frac{49}{4}, 24\right)$ are $13, 14, \ldots, 23$.
$\therefore S = {13, 14, \ldots, 23}$
BETA
