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} $