WAEC 2019 · Paper 2 · Q15

(a)
Given that $\mathbf m = \mathbf i - \mathbf j$ , $\mathbf n = 2\mathbf i + 3\mathbf j$ and $2\mathbf m + \mathbf n - \mathbf r = \mathbf 0$ , find $|\mathbf r|$ .
(b)
The distance, $S$ metres, of a moving particle at any time $t$ seconds is given by $S = 3t - \dfrac{t^3}{3} + 9$ . Find the: (i) time; (ii) distance travelled, when the particle is momentarily at rest.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

$\mathbf r = 2\mathbf m + \mathbf n$
$= 2(\mathbf i - \mathbf j) + (2\mathbf i + 3\mathbf j)$
$= 4\mathbf i + \mathbf j$ .
$|\mathbf r| = \sqrt{4^2 + 1^2}$
$= \sqrt{17}$
$\approx 4.123$ .

(b)(i)

Momentarily at rest when $\dfrac{dS}{dt} = 3 - t^2 = 0$ , so $t = \sqrt3 \approx 1.732$ s (time is positive).

(ii)

$S = 3\sqrt3 - \dfrac{(\sqrt3)^3}{3} + 9$
$= 3\sqrt3 - \sqrt3 + 9$
$= 9 + 2\sqrt3$
$\approx 12.46$ m.

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