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Date November 2011 Marks available 2 Reference code 11N.2.sl.TZ0.3
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 3 Adapted from N/A

Question

Consider the following circle with centre O and radius 6.8 cm.


The length of the arc PQR is 8.5 cm.

Find the value of \(\theta \) .

[2]
a.

Find the area of the shaded region.

[4]
b.

Markscheme

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]

a.

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]

b.

Examiners report

Part (a) was almost universally done correctly.

a.

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.

b.

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.1 » The circle: radian measure of angles; length of an arc; area of a sector.
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