Question

If $p$ and $q$ are chosen randomly from the set $1,2,3,4,5,6,7,8,9,10$ with replacement, the probability that the roots of the equation $x^2+px+q$ are real is $\frac{30+k}{50}$. Find the value of $k$.

Answer 1

Root of $x^2+px+q=0$, will be real if ${\left(p\right)}^2-4\times 1\times q>0\Rightarrow p^2>4q$
Different ways for selection of $p$ and $q$ are

p	q	p	q
1	-	6	1,2,3,...8,9
2	1	7	1,2,3,...8,9,10
3	1,2	8	1,2,3,...8,9,10
4	1,2,3,4	9	1,2,3,...8,9,10
5	1,2,3,4,5	10	1,2,3,...8,9,10

Therefore favorable ways in which $p$ and $q$ can be selected = 62
Total number of ways in which $p$ and $q$ can be selected $=10\times 10=100$
Therefore required probability $=\frac{62}{100}=\frac{31}{50}$
$\therefore k=1$