The number of integral values of $k$ for which the inequality $x^{2} + k x + 5 k > 16, \forall x \in R$ , is:

10

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11

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12

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13

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Solution

The correct option is B $13$
Given $x^{2} + k x + 5 k > 16$
or, $x^{2} + k x + (5 k - 16) > 0$ will occur only when the discriminant of the equation $x^{2} + k x + (5 k - 16) = 0$ is $\leq 0$ .
So, $k^{2} - 4 (5 k - 16) \leq 0$
or, $k^{2} - 20 k + 64 \leq 0$
or, $(k - 4) (k - 16) \leq 0$
$\Rightarrow 4 \leq k \leq 16$ .
Number of integral values of $k$ is $13$ .

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