Question

If the quadratic equation $m{x}^2-nx+12=0$, where $m$ and $n$ are positive integer not exceeding $10$ , has both roots greater than $2$ , then find the number of possible ordered pairs $(m,n)$.

Answer 1

Answer: 1

$1\le m,n\le 10$
(i) $D\ge 0,(ii)\frac{-b}{2a}>2,(iii)f(2)>0\}\cap$
(i) $D\ge 0$
$\Rightarrow n^2-48m\ge 0$ ... (1)
(ii) $\frac{-b}{2a}>2\Rightarrow n>4m$ ... (2)
(iii) $f(2)>0\Rightarrow 2m-n+6>0$
$\therefore$ From $(1)\cap (2)\cap (3)$
Since $m,n$ are positive integer which are less than or equal to $10$ .
Hence, $m=1$ and $n=7$ only possible.