Let $m$ and $n$ be two integers such that $0 \leq m \leq 10$ and $0 \leq n \leq 10$. Then, the number of…
Let $m$ and $n$ be two integers such that $0 \leq m \leq 10$ and $0 \leq n \leq 10$. Then, the number of ordered pairs $(m, n)$ such that $x^2+m x+n=0$ has real roots is
71
73
75
72
Solution
Given, $x^2+m x+n=0$ has real roots.
$
\begin{array}{ll}
\therefore m^2-4 n \geq 0 \quad 0 \leq m, n \leq 10, \quad m^2 \geq 4 n \\
m=10, & n=\{0,1,2, \ldots, 10\}=11 \\
m=9, & n=\{0,1,2, \ldots, 10\}=11 \\
m=8, & n=\{0,1,2, \ldots, 10\}=11 \\
m=7, & n=\{0,1,2, \ldots, 10\}=11 \\
m=6, & n=\{0,1,2, \ldots, 9\}=10 \\
m=5, & n=\{0,1,2, \ldots, 6\}=7 \\
m=4, & n=\{0,1,2,3,4\}=5 \\
m=3, & n=\{0,1,2\}=3 \\
m=2, & n=\{0,1\}=2 \\
m=1, & n=\{0\}=1 \\
m=0, & n=\{0\}=1
\end{array}
$
$\therefore \quad$ Number of order pair of $(m, n)$ is
$
11+11+11+11+10+7+5+3+2+1+1=73
$