WAEC 2014 · Paper 2 · Q2

A binary operation $*$ is defined on the set of rational numbers by $m * n = \dfrac{m^2 - n^2}{2mn}$ , $n \ne 0$ and $m \ne 0$ .

(a)
Find $-3 * 2$ .
(b)
Show whether or not $*$ is associative.
Show the answer
$*$ is not associative

Worked solution (try it first)

$*$ is associative if $m * (n * p) = (m * n) * p$ for all $m$ , $n$ and $p$ .
Left side: $n * p = \dfrac{n^2 - p^2}{2np}$ , so $m * (n * p) = \dfrac{m^2 - \left(\frac{n^2 - p^2}{2np}\right)^2}{2m\left(\frac{n^2 - p^2}{2np}\right)}$ .
Right side: $(m * n) * p = \dfrac{\left(\frac{m^2 - n^2}{2mn}\right)^2 - p^2}{2p\left(\frac{m^2 - n^2}{2mn}\right)}$ .
These are not the same expression.
For example, with $m = 1$ , $n = 2$ , $p = 3$ the left side is $-\frac{119}{120}$ and the right side is $\frac{15}{8}$ .
So $*$ is not associative.