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Date	November 2016	Marks available	1	Reference code	16N.2.SL.TZ0.S_3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Write down	Question number	S_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]

Find the value of $r$ .

[3]

Find AB.

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\theta = \frac{{2\pi }}{5}$ A1 N1

[1 mark]

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]

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]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.7—Radians

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20N.2.SL.TZ0.S_7b:
Find the value of $θ$ when the area of the shaded region is equal to the area of sector $OADB$ .
17M.2.AHL.TZ1.H_8b:
Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$ .
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19M.1.AHL.TZ1.H_3:
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The sector has an angle of 1 radian at the centre.

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16N.2.SL.TZ0.S_3c:
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19M.2.SL.TZ2.S_4b:
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16N.2.AHL.TZ0.H_9c:
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16N.2.AHL.TZ0.H_7b:
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16N.2.SL.TZ0.S_3b:
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17N.1.SL.TZ0.S_4a:
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17M.2.SL.TZ1.S_5:
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Find the area of the shaded segment.
17N.1.SL.TZ0.S_4b:
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Find the exact perimeter of this shape.
21N.1.AHL.TZ0.15a:
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Topic 3—Geometry and trigonometry

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