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 \(\frac{\pi }{4}\).