Question

If $p$ is chosen at random in the closed interval $[0,5]$. The probability of the equation $x^2+px+\frac{1}{4}(p+2)=0$ to have the real roots is

• A. $\frac{3}{4}$
• B. $\frac{3}{5}$
• C. $\frac{2}{5}$
• D. None of these

Answer 1

The correct option is B $\frac{3}{5}$

For real roots $D\ge 0\Rightarrow p^2-4.\frac{1}{4}(p+2)\ge 0\Rightarrow p^2-p-2\ge 0$
$\Rightarrow p^2-2p+p-2\ge 0\Rightarrow p\ge 2$ or $p\le -1$
But $0\le p\le 5$, so favorable values of $p$ are in $[2,5]$
Required probability $=\frac{lengthof[2,5]}{lengthof[0,5]}=\frac{3}{5}$