Question

If $pϵW$ is chosen at random from the interval $[0,6]$, then the probability that the roots of the equation $x^2+px+\frac{1}{4}(p+2)=0$ will be real is

• A. $\frac{3}{5}$
• B. $\frac{1}{2}$
• C. $\frac{3}{4}$
• D. $\frac{5}{7}$

Answer 1

The correct option is D $\frac{5}{7}$

Discriminant of given quadratic is $D=p^2-p-2=(P+1)(P-2)$
For roots to be real $D\ge 0\Rightarrow P\le -1,P\ge 2$
Hence $n\left(E\right)=n\{2,3,4,5,6\}=5,n\left(S\right)=n\{0,1,2,3,4,5,6\}=7$
Hence required probability $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{5}{7}$