Question

If 'a' is an integer lying in [- 5, 30] and the probability that the graph of $y=x^2+2(a+4)x-5a+64$ is strictly above x-axis is $\frac{\lambda }{9}$, then find the value of $\lambda$.

Answer 1

'a' is an integer lying in [- 5, 30]

Probability that the graph of y=x2+2(a+4)x−5a+64y=x2+2(a+4)x−5a+64 is strictly above x-axis -

$y=x^2+2(a+4)-5a+64y={(x+a+4)}^2-5a+64-{(a+4)}^{2}y={(x+a+4)}^2-(a^2+13a-48)$

For y to be always greater then zero,

${(x+a+4)}^2>0-(a^2+13a-48)>0i.e(a^2+13a-48)<0(a-3)(a+16)<0so,-16<a<3$

Total possible values of a are -5,-4...0,1,2...30=36

Favourable outcomes are -5,-4,-3,-2,-1,0,1,2 = 8

Probability=8/36=2/9

Therefore, value of λ=2

λ