The diagram below shows two straight lines intersecting at O and two circles, each with centre O. The outer circle has radius R and the inner circle has radius r .

Consider the shaded regions with areas A and B . Given that $A:B = 2:1$, find the exact value of the ratio $R:r$ .

$A = \frac{\theta }{2}({R^2} - {r^2})$     A1

$B = \frac{\theta }{2}{r^2}$     A1

from $A:B = 2:1$ , we have ${R^2} - {r^2} = 2{r^2}$     M1

$R = \sqrt 3 r$     (A1)

hence exact value of the ratio $R:r{\text{ is }}\sqrt 3 :1$     A1     N0

[5 marks]

This question was successfully answered by most candidates using a variety of correct approaches. A few candidates, however, did not use a parameter for the angle, but instead substituted an angle directly, e.g., $\frac{\pi }{2}$ or $\frac{\pi }{4}$.