The probability of selecting integers a[][- 5, 30] such that x2 + 2(a + 4)x - 5a + 64 0,…

Question

The probability of selecting integers a$$\in$$[- 5, 30] such that x2 + 2(a + 4)x - 5a + 64 > 0, for all x$$\in$$R, is :

Accepted Answer

D < 0

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

$$\Rightarrow$$ a2 + 16 + 8a + 5a - 64 < 0

$$\Rightarrow$$ a2 + 13a - 48 < 0

$$\Rightarrow$$ (a + 16) (a - 3) < 0

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

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

$$\therefore$$ Required probability = $${8 \over {36}}$$ = $${2 \over {9}}$$

Solutions · 1