The probability of selecting


The probability of selecting integers $a \in[-5,30]$ such that $x^{2}+2(a+4) x-5 a+64>0$, for all $x \in \mathbf{R}$, is:

  1. $\frac{7}{36}$

  2. $\frac{2}{9}$

  3. $\frac{1}{6}$

  4. $\frac{1}{4}$

Correct Option: , 2



$\Rightarrow 4(a+4)^{2}-4(-5 a+64)<0$

$\Rightarrow a^{2}+16+8 a+5 a-64<0$

$\Rightarrow a^{2}+13 a-48<0$


$\Rightarrow a \in(-16,3)$

$\therefore$ Possible a : $\{-5,-4, \ldots \ldots . ., 3\}$

$\therefore$ Required probability $=\frac{8}{36}$


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