JEE PYQ: Quadratic Equation Question 18
Question 18 - 2020 (03 Sep Shift 2)
The set of all real values of $\lambda$ for which the quadratic equations, $(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0$ always have exactly one root in the interval $(0, 1)$ is:
(1) $(0, 2)$
(2) $(2, 4]$
(3) $(1, 3]$
(4) $(-3, -1)$
Show Answer
Answer: (3)
Solution
The given quadratic equation is $(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0$.
$\therefore$ One root is in the interval $(0, 1)$
$\therefore f(0) \cdot f(1) \leq 0$
$\Rightarrow 2(\lambda^2 + 1 - 4\lambda + 2) \leq 0$
$\Rightarrow 2(\lambda^2 - 4\lambda + 3) \leq 0$
$(\lambda - 1)(\lambda - 3) \leq 0 \Rightarrow \lambda \in [1, 3]$
But at $\lambda = 1$, both roots are 1 so $\lambda \neq 1$
$\therefore \lambda \in (1, 3]$
BETA
