WAEC 2022 · Paper 2 · Q1

A binary operation $*$ is defined on the set $T = \{-2, -1, 1, 2\}$ by $p * q = p^2 + 2pq - q^2$ , where $p, q \in T$ .

(a)

Copy and complete the table.

Model answer

Each entry is $p^2 + 2pq - q^2$ with $p$ from the row and $q$ from the column; for example $2 * 1 = 4 + 4 - 1 = 7$ .

(b)
Using the table in (a), find the value of $p$ such that $(-2 * p) * 2 = -7$ .

Worked solution (try it first)

Put the row heading in for $p$ and the column heading in for $q$ in $p^2 + 2pq - q^2$ .
Row $-2$ : $8$ , $7$ , $-1$ , $-8$ .
For example $(-2) * (-2) = 4 + 8 - 4 = 8$ .
Row $-1$ : $1$ , $2$ , $-2$ , $-7$ .
For example $(-1) * 2 = 1 - 4 - 4 = -7$ .
Row $1$ : $-7$ , $-2$ , $2$ , $1$ .
Row $2$ : $-8$ , $-1$ , $7$ , $8$ .

Let $x = -2 * p$ .
Then $x * 2 = -7$ : in the column under 2, $-7$ is in the row of $-1$ , so $x = -1$ .
Now $-2 * p = -1$ : in the row of $-2$ , $-1$ is under $1$ , so $p = 1$ .