WAEC 2020 · Paper 2 · Q12

(a)
In the diagram, $\angle QMN = 34^\circ$ , $|MN| = |NQ| = |QO|$ , $M$ , $N$ , $O$ , $P$ lie on a straight line and $\angle QOP = x$ . Find the value of $x$ .
(b)
A box contains 3 red balls and 4 white balls. Two balls are picked at random one after the other without replacement. Find the probability that the two balls picked are: (i) both red; (ii) both white; (iii) of the same colour; (iv) of different colours.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

$|MN| = |NQ|$ , so triangle $MNQ$ is isosceles and $\angle NQM = \angle NMQ = 34^\circ$ .
The exterior angle at $N$ equals the sum of the two opposite interior angles: $\angle QNO = 34^\circ + 34^\circ$
$= 68^\circ$ .
$|NQ| = |QO|$ , so triangle $NQO$ is isosceles and $\angle QON = \angle QNO = 68^\circ$ .
$N$ , $O$ and $P$ are on a straight line, so $x = \angle QOP$
$= 180^\circ - 68^\circ$
$= 112^\circ$ .

Different colours is everything else: $1 - \frac37 = \frac47$ .
(Check: red then white $\frac37 \times \frac46 = \frac{12}{42}$ , plus white then red $\frac47 \times \frac36 = \frac{12}{42}$ , gives $\frac{24}{42} = \frac47$ .)