WAEC 2022 · Paper 2 · Q8

(a)
The points $M$ , $N$ , $Q$ and $R$ are in the $xy$ plane with position vectors $\mathbf m = \mathbf i + \mathbf j$ , $\mathbf n = 2\mathbf i - \mathbf j$ , $\mathbf q = x\mathbf i + \mathbf j$ and $\mathbf r = (x + 1)\mathbf i - 3\mathbf j$ respectively. If $\overrightarrow{NQ}$ is perpendicular to $\overrightarrow{MR}$ , find the values of $x$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

$\overrightarrow{NQ} = \mathbf q - \mathbf n$
$= (x - 2)\mathbf i + 2\mathbf j$ .
$\overrightarrow{MR} = \mathbf r - \mathbf m$
$= x\mathbf i - 4\mathbf j$ .
Perpendicular: $x(x - 2) + (2)(-4) = 0$ , so $x^2 - 2x - 8 = 0$ .
Factorise: $(x - 4)(x + 2) = 0$ , so $x = 4$ or $x = -2$ .