The number of integral values of $m$ for which the equation $(1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0$ has no real root is :

2

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3

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infinitely many

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1

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Solution

The correct option is C infinitely many
Given quadratic equation has no real root.

$\Rightarrow D < 0$
$\Rightarrow 4 (1 + 3 m)^{2} - 4 (1 + m^{2}) (1 + 8 m) < 0$
$\Rightarrow 1 + 9 m^{2} + 6 m - 1 - 8 m - m^{2} - 8 m^{3} < 0$
$\Rightarrow 8 m^{3} - 8 m^{2} + 2 m > 0$
$\Rightarrow 2 m (2 m - 1)^{2} > 0$

$\Rightarrow m > 0 {m \neq \frac{1}{2}}$

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