Question

If the quadratic equation $mx ^{2}- nx +12=0$, 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

$1 \leq m , n \leq 10$
(i) $D \ge 0, (ii) \frac{-b}{2a} > 2, (iii) f(2) > 0 \} \cap$
(i) $D \geq 0$
$\Rightarrow n ^{2}-48 m \geq 0$ ... (1)
(ii) $\frac{-b}{2 a}>2 \Rightarrow n>4 m$ ... (2)
(iii) $f (2)>0 \Rightarrow 2 m - n +6>0$
$\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.