The integer k for which the inequality $x^{2} - 2 (4 k - 1) x + 15 k^{2} - 2 k - 7 > 0$ is valid for any x is :

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Solution

The correct option is B 1
Let $f (x) = x^{2} - 2 (4 k - 1) x + 15 k^{2} - 2 k - 7.$ Then
$f (x) > 0 \Rightarrow D < 0 [∵ C o e f f i c i e n t o f x^{2} > 0]$
$\Rightarrow 4 (4 k - 1)^{2} - 4 (15 k^{2} - 2 k - 7) < 0$
$\Rightarrow k^{2} - 6 k + 8 < 0 \Rightarrow 2 < k < 4$

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