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Date	November 2016	Marks available	3	Reference code	16N.2.SL.TZ0.S_3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	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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18N.2.AHL.TZ0.H_7:
Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.
16N.2.SL.TZ0.S_3a:
Write down the exact value of $\theta$ in radians.
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 $\theta = \frac{\pi }{3}$ .
21N.1.AHL.TZ0.15b.i:
Find an expression for $V$ in terms of $θ$ .
21N.1.AHL.TZ0.15b.ii:
Find the expression $\frac{d V}{d θ}$ .
21N.1.AHL.TZ0.15b.iii:
Solve algebraically $\frac{d V}{d θ} = 0$ to find the value of $θ$ that will maximize the volume, $V$ .
19M.1.AHL.TZ1.H_3:
A sector of a circle with radius $r$ cm , where $r$ > 0, is shown on the following diagram.
The sector has an angle of 1 radian at the centre.

Let the area of the sector be $A$ cm² and the perimeter be $P$ cm. Given that $A = P$ , find the value of $r$ .
16N.2.SL.TZ0.S_3c:
Find AB.
19M.2.SL.TZ2.S_4b:
Find the area of triangle OBC in terms of $r$ and θ.
SPM.1.AHL.TZ0.5a:
Find 50° in radians.
17M.2.SL.TZ2.S_1a:
Find the length of arc ABC.
18M.2.AHL.TZ2.H_4a.i:
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17M.2.SL.TZ2.S_1b:
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17M.2.SL.TZ2.S_1c:
Find the area of sector OABC.
16N.2.AHL.TZ0.H_9a:
Find an expression for the shaded area in terms of $\alpha$ , $\theta$ and $r$ .
16N.2.AHL.TZ0.H_9c:
Hence find the value of $r$ given that the shaded area is equal to 4.
16N.2.AHL.TZ0.H_9b:
Show that $\alpha = 4\arcsin \frac{1}{4}$ .
SPM.1.AHL.TZ0.5b:
Find the volume of this log.
18M.2.AHL.TZ2.H_4b:
Find the area of the shaded region.
20N.2.SL.TZ0.S_7a:
Show that $AC = \tan θ$ .
17M.2.AHL.TZ1.H_8a:
Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms of $\theta$ .
16N.2.AHL.TZ0.H_7a:
Use the cosine rule to find the two possible values for AC.
16N.2.AHL.TZ0.H_7b:
Find the difference between the areas of the two possible triangles ABC.
17N.1.SL.TZ0.S_4a:
Show that ${\text{AC}} = 7{\text{ cm}}$ .
17M.2.SL.TZ1.S_5:
The following diagram shows the chord [AB] in a circle of radius 8 cm, where ${\text{AB}} = 12{\text{ cm}}$ .

Find the area of the shaded segment.
17N.1.SL.TZ0.S_4b:
The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.

Find the exact perimeter of this shape.
21N.1.AHL.TZ0.15a:
Show that $r = \frac{6}{2 + θ}$ .

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Topic 3—Geometry and trigonometry

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