Question

If $p\in Z$ is chosen at random in $[0,5]$ and the probability that the equation $x^2+px+\frac{p+2}{4}=0$ has real roots is $\frac{\lambda }{15}$, then $\lambda =$

Answer 1

Roots of $x^2+px+\left(\frac{p+2}{4}\right)=0$ are real if $\Delta \ge 0$

$\Rightarrow p^2-p-2\ge 0\Rightarrow p\le -1\text{or}p\ge 2$
Favourable interval {$2,3,4,5$} =4 cases
where $p\in [0,5]$
Total interval {$0,1,2,3,4,5$} = $6$ cases

$\therefore$ Required probability =$\left[\frac{favourableinterval}{totalinterval}\right]=\frac{4}{6}=\frac{2}{3}=\frac{\lambda }{15}\text{(given)}$
$\Rightarrow \lambda =10$