Question

If an integer p is chosen at random in the interval $0\le p\le 5,$ the probability that the roots of the equation $x^2+px+\frac{p}{4}+\frac{1}{2}=0$ are real, is:

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

Answer 1

Answer: (B)

If roots of the quadratic equation are real, then discriminant is always be greater than equal to zero.
Given equation is $x^2+px+\frac{p}{4}+\frac{1}{2}=0$
Since roots are real, therefore discriminant $\ge 0$
$\Rightarrow$ $p^2-4\left(\frac{p}{4}+\frac{1}{2}\right)\ge 0$
$\Rightarrow$ $p^2-p-2\ge 0$
$\Rightarrow$ $(p-2)(p+1)\ge 0$
$\Rightarrow$ $p\ge 2$ or $p\le -1$
Since, it is given $0\le p\le 5,$ so we neglect $p\le -1.$
The possible values of p are 2, 3, 4, 5
$\therefore$ Required probability $=\frac{4}{6}=\frac{2}{3}$