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Date November 2016 Marks available 1 Reference code 16N.2.sl.TZ0.3
Level SL only Paper 2 Time zone TZ0
Command term Write down Question number 3 Adapted from N/A

Question

The following diagram shows a circle, centre O and radius \(r\) mm. The circle is divided into five equal sectors.

N16/5/MATME/SP2/ENG/TZ0/03

One sector is OAB, and \({\rm{A\hat OB}} = \theta \).

The area of sector AOB is \(20\pi {\text{ m}}{{\text{m}}^2}\).

Write down the exact value of \(\theta \) in radians.

[1]
a.

Find the value of \(r\).

[3]
b.

Find AB.

[3]
c.

Markscheme

\(\theta  = \frac{{2\pi }}{5}\)     A1     N1

[1 mark]

a.

correct expression for area     (A1)

eg\(\,\,\,\,\,\)\(A = \frac{1}{2}{r^2}\left( {\frac{{2\pi }}{5}} \right),{\text{ }}\frac{{\pi {r^2}}}{5}\)

evidence of equating their expression to \(20\pi \)     (M1)

eg\(\,\,\,\,\,\)\(\frac{1}{2}{r^2}\left( {\frac{{2\pi }}{5}} \right) = 20\pi ,{\text{ }}{r^2} = 100,{\text{ }}r =  \pm 10\)

\(r = 10\)    A1     N2

[3 marks]

b.

METHOD 1

evidence of choosing cosine rule     (M1)

eg\(\,\,\,\,\,\)\({a^2} = {b^2} + {c^2} - 2bc\cos A\)

correct substitution of their \(r\) and \(\theta \) into RHS     (A1)

eg\(\,\,\,\,\,\)\({10^2} + {10^2} - 2 \times 10 \times 1{\text{0}}\cos \left( {\frac{{2\pi }}{5}} \right)\)

11.7557

\({\text{AB}} = 11.8{\text{ (mm)}}\)     A1     N2

METHOD 2

evidence of choosing sine rule     (M1)

eg\(\,\,\,\,\,\)\(\frac{{\sin A}}{a} = \frac{{\sin B}}{b}\)

correct substitution of their \(r\) and \(\theta \)     (A1)

eg\(\,\,\,\,\,\)\(\frac{{\sin \frac{{2\pi }}{5}}}{{{\text{AB}}}} = \frac{{\sin \left( {\frac{1}{2}\left( {\pi  - \frac{{2\pi }}{5}} \right)} \right)}}{{10}}\)

11.7557

\({\text{AB}} = 11.8{\text{ (mm)}}\)     A1     N2

[3 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

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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