WAEC 2017 · Paper 2 · Q8

(a)
The probabilities that Akafi and Iniola will pass an examination are $\frac34$ and $\frac35$ respectively. Find the probability that only Iniola will pass the examination.
(b)
In the diagram, circle $XYZ$ is inscribed in an equilateral triangle $PQR$ , touching $PQ$ at $X$ , $PR$ at $Y$ and $QR$ at $Z$ . If $O$ is the centre of the circle and $|XY| = 10\text{ cm}$ , calculate, correct to the nearest whole number: (i) $\angle XOY$ ; (ii) the area of the major sector $XZY$ . $\left[\text{Take }\pi = \frac{22}{7}\right]$
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

"Only Iniola passes" means Iniola passes and Akafi fails.
$P(\text{Akafi fails}) = 1 - \frac34$
$= \frac14$ , so the probability is $\frac14 \times \frac35 = \frac{3}{20}$ .

A radius meets a tangent at $90^\circ$ , so $\angle OXP = \angle OYP = 90^\circ$ .
The triangle is equilateral, so $\angle XPY = 60^\circ$ .
The angles of quadrilateral $PXOY$ add up to $360^\circ$ : $\angle XOY = 360^\circ - 90^\circ - 90^\circ - 60^\circ$
$= 120^\circ$ .

Find the radius from the chord $XY = 10$ cm.
The perpendicular from $O$ to $XY$ halves both the chord and the angle, giving a right-angled triangle with half-chord 5 cm opposite an angle of $60^\circ$ : $r = \frac{5}{\sin 60^\circ}$
$\approx 5.7735$ cm.
The major sector $XZY$ has angle $360^\circ - 120^\circ = 240^\circ$ .
Area $= \frac{240}{360} \times \frac{22}{7} \times 5.7735^2$
$\approx 69.8$ , which is $70\text{ cm}^2$ to the nearest whole number.