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  4. The number of integral values of m for which the quadratic expression (1 + 2m)x2 - 2(1 + 3m)x + 4(1 + m), x ∈ R, is always positive, is :

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Q. The number of integral values of m for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m), x \in R$, is always positive, is :

JEE MainJEE Main 2019Linear Inequalities

Solution:

Exprsssion is always positve it
$2m \, + \, 1 \, > \, 0 \, \Rightarrow \, m \, > \, - \frac{1}{2}$
&
$D \, < \, 0 \, \Rightarrow \, m^2 \, - \, 6m \, - 3 < \, 0$
$ \, \, \, \, \, \, \, \, \, 3 - \sqrt{12} < m < 3 +\sqrt{12} \, \, \, \, ...(iii)$
$\therefore $Common interval is
$ \, \, \, \, \, \, \, \, 3- \sqrt{1} < m < 3 + \sqrt{12}$
$\therefore $Integral value of m {0,1,2,3,4,5,6}