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. $None$

Answer 1

The correct option is B $\frac{2}{3}$

Roots of the equation
$x^2+px+\frac{p}{4}+\frac{1}{2}=0$ are real if
$\Delta =P^2-4(\frac{p}{4}+\frac{1}{2})\ge 0$
i.e., $(p-2)(p+1)\ge 0$
i.e., $p\le -1$ or $p\ge 2.$
In $0\le p\le 5,$ possible value of p are 2, 3, 4, 5.
Thus probability=$\frac{4}{6}=\frac{2}{3}$.