WAEC 2022 · Paper 2 · Q1

Binary operations

A binary operation $\Delta$ is defined on the set of real numbers, $\mathbb R$ , by $p\,\Delta\,q = pq + p + q$ for $p, q \in \mathbb R$ .

(a)
Find the: (i) identity element; (ii) inverse element.
Show the answer
(i) $e = 0$ ; (ii) $p^{-1} = -\dfrac{p}{p + 1}$ , $p \ne -1$
(b)
Given that $m\,\Delta\,8 = 35$ , find the value of $m$ .

Worked solution (try it first)

(a)(i)

The identity satisfies $p\,\Delta\,e = p$ : $pe + p + e = p$ , so $e(p + 1) = 0$ .
This must hold for every $p$ , so $e = 0$ .

(ii)

The inverse satisfies $p\,\Delta\,p^{-1} = 0$ : $pp^{-1} + p + p^{-1} = 0$ .
So $p^{-1}(p + 1) = -p$ and $p^{-1} = -\dfrac{p}{p + 1}$ , for $p \ne -1$ .

(b)

$m\,\Delta\,8 = 8m + m + 8 = 35$ , so $9m = 27$ and $m = 3$ .

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