The following diagram shows a circle with centre O and radius r cm.

The points A and B lie on the circumference of the circle, and \({\text{A}}\mathop {\text{O}}\limits^ \wedge  {\text{B}}\) = θ. The area of the shaded sector AOB is 12 cm² and the length of arc AB is 6 cm.

Find the value of r.

evidence of correctly substituting into circle formula (may be seen later)      A1A1
eg  \(\frac{1}{2}\theta {r^2} = 12,\,\,r\theta  = 6\)

attempt to eliminate one variable      (M1)
eg  \(r = \frac{6}{\theta },\,\,\theta  = \frac{1}{r},\,\,\frac{{\frac{1}{2}\theta {r^2}}}{{r\theta }} = \frac{{12}}{6}\)

correct elimination      (A1)
eg  \(\frac{1}{2} \times \frac{6}{r} \times {r^2} = 12,\,\,\frac{1}{2}\theta  \times {\left( {\frac{6}{\theta }} \right)^2} = 12,\,\,A = \frac{1}{2} \times {r^2} \times \frac{l}{r},\,\,\frac{{{r^2}}}{{2r}} = 2\)

correct equation     (A1)
eg  \(\frac{1}{2} \times 6r = 12,\,\,\frac{1}{2} \times \frac{{36}}{\theta } = 12,\,\,12 = \frac{1}{2} \times {r^2} \times \frac{6}{r}\)

correct working      (A1)
eg  \(3r = 12,\,\,\frac{{18}}{\theta } = 12,\,\,\frac{r}{2} = 2,\,\,24 = 6r\)

r = 4 (cm)      A1 N2

[7 marks]