Question

If an integer p is chosen at random in the closed interval [0, 5]. The probability of the equation $4x^2+4px+p+2=0$ to have real roots is:

• A. $\frac{3}{4}$
• B. $\frac{1}{2}$
• C. $\frac{2}{5}$
• D. $None$

Answer 1

The correct option is B $\frac{1}{2}$

$4x^2+4px+p+2=0$ has real roots if
$D=16p^2-16(p+2)\ge 0$
$\Rightarrow p^2-p-2\ge 0$
$\Rightarrow p^2-2p+p-2\ge 0$
$\Rightarrow p(p-2)+1(p-2)\ge 0$
$\Rightarrow (p+1)(p-2)\ge 0$
$\Rightarrow p\le -1$ or $p\ge 2$.
Since $p\in [0,5]=\{0,1,2,3,4,5\}$, for real roots $p\in [2,5]=\{2,3,4,5\}$.
So, total elements in sample space is $=6$ and
Number of favourable elements$=4$