Find the value of \(\theta \) .

Find the area of the shaded region.

correct substitution     (A1)

e.g. \(8.5 = \theta (6.8)\) , \(\theta  = \frac{{8.5}}{{6.8}}\)

\(\theta  = 1.25\) (accept \({71.6^ \circ }\) )     A1     N2

[2 marks]

METHOD 1

correct substitution into area formula (seen anywhere)     (A1)

e.g. \(A = \pi {(6.8)^2}\) , \(145.267 \ldots \)

correct substitution into area formula (seen anywhere)     (A1)

e.g. \(A = \frac{1}{2}(1.25)({6.8^2})\) , 28.9

valid approach     M1

e.g. \(\pi {(6.8)^2} - \frac{1}{2}(1.25)({6.8^2})\) ; \(145.267 \ldots - 28.9\) ; \(\pi {r^2} - \frac{1}{2}{r^2}\sin \theta \)

\(A = 116\) (\({\text{c}}{{\text{m}}^2}\))     A1     N2

METHOD 2

attempt to find reflex angle     (M1)

e.g. \(2\pi - \theta \) , \(360 - 1.25\)

correct reflex angle     (A1)

\({\rm{A}}\widehat {\rm{O}}{\rm{B}} = 2\pi - 1.25\) (\( = 5.03318 \ldots \))

correct substitution into area formula     A1

e.g. \(A = \frac{1}{2}(5.03318 \ldots )({6.8^2})\)

\(A = 116\) (\({\text{c}}{{\text{m}}^2}\))     A1     N2

[4 marks]

Part (a) was almost universally done correctly.

Many also had little trouble in part (b), with most subtracting from the circle's area, and a minority using the reflex angle. A few candidates worked in degrees, although some of these did so incorrectly by using the radian area formula. Some candidates only found the area of the unshaded sector.