WAEC 2009 · Paper 2 · Q4

(a)
Out of 30 candidates applying for a post, 17 have degrees, 15 have diplomas and 4 have neither a degree nor a diploma. How many of them have both?
(b)
In triangle $PQR$ , $M$ and $N$ are points on the sides $PQ$ and $PR$ respectively such that $MN$ is parallel to $QR$ . If $\angle PRQ = 75^\circ$ , $|PN| = |QN|$ and $\angle PNQ = 125^\circ$ , determine: (i) $\angle NQR$ ; (ii) $\angle NPM$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

Let $x$ have both.
Degree only is $17 - x$ and diploma only is $15 - x$ .
Everyone is in exactly one region: $(17 - x) + x + (15 - x) + 4 = 30$ .
Simplify: $36 - x = 30$ , so $x = 6$ .
Six applicants have both.

(b)

Sketch the triangle: $N$ is on $PR$ , $M$ on $PQ$ , and $NQ$ is joined.

(i)

$MN \parallel QR$ , so $\angle PNM = \angle PRQ = 75^\circ$ (corresponding angles).
$\angle MNQ = \angle PNQ - \angle PNM$
$= 125^\circ - 75^\circ$
$= 50^\circ$ .
$\angle NQR = \angle MNQ = 50^\circ$ (alternate angles, $MN \parallel QR$ ).

(ii)

$|PN| = |QN|$ , so triangle $PNQ$ is isosceles with equal base angles at $P$ and $Q$ .
$\angle NPM = \frac{180^\circ - 125^\circ}{2}$
$= 27.5^\circ$ .

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