JEE PYQ: Quadratic Equation Question 35
Question 35 - 2019 (08 Apr Shift 2)
The number of integral values of $m$ for which the equation $(1 + m^2)x^2 - 2(1 + 3m)x + (1 + 8m) = 0$ has no real root is:
(1) 1
(2) 2
(3) infinitely many
(4) 3
Show Answer
Answer: (3)
Solution
Given equation is $(1 + m^2)x^2 - 2(1 + 3m)x + (1 + 8m) = 0$
$\therefore$ equation has no real solution
$\therefore D < 0$
$\Rightarrow 4(1 + 3m)^2 < 4(1 + m^2)(1 + 8m)$
$\Rightarrow 1 + 9m^2 + 6m < 1 + 8m + m^2 + 8m^3$
$\Rightarrow 8m^3 - 8m^2 + 2m > 0$
$\Rightarrow 2m(4m^2 - 4m + 1) > 0 \Rightarrow 2m(2m - 1)^2 > 0$
$\Rightarrow m > 0$ and $m \neq \frac{1}{2}$ [$\therefore \frac{1}{2}$ is not an integer]
$\Rightarrow$ number of integral values of m are infinitely many.
BETA
