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Date	May 2018	Marks available	7	Reference code	18M.1.SL.TZ2.S_4
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Find	Question number	S_4	Adapted from	N/A

Question

The following diagram shows a circle with centre O and radius r cm.

The points A and B lie on the circumference of the circle, and $A \land O B$ ${\text{A}}\mathop {\text{O}}\limits^ \wedge {\text{B}}$ = θ. The area of the shaded sector AOB is 12 cm² and the length of arc AB is 6 cm.

Find the value of r.

Markscheme

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

evidence of correctly substituting into circle formula (may be seen later) A1A1
eg $\frac{1}{2}\theta {r^2} = 12,\,\,r\theta = 6$

attempt to eliminate one variable (M1)
eg $r = \frac{6}{\theta },\,\,\theta = \frac{1}{r},\,\,\frac{{\frac{1}{2}\theta {r^2}}}{{r\theta }} = \frac{{12}}{6}$

correct elimination (A1)
eg $\frac{1}{2} \times \frac{6}{r} \times {r^2} = 12,\,\,\frac{1}{2}\theta \times {\left( {\frac{6}{\theta }} \right)^2} = 12,\,\,A = \frac{1}{2} \times {r^2} \times \frac{l}{r},\,\,\frac{{{r^2}}}{{2r}} = 2$

correct equation (A1)
eg $\frac{1}{2} \times 6r = 12,\,\,\frac{1}{2} \times \frac{{36}}{\theta } = 12,\,\,12 = \frac{1}{2} \times {r^2} \times \frac{6}{r}$

correct working (A1)
eg $3r = 12,\,\,\frac{{18}}{\theta } = 12,\,\,\frac{r}{2} = 2,\,\,24 = 6r$

r = 4 (cm) A1 N2

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.4—Circle: radians, arcs, sectors

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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.
18M.2.SL.TZ1.S_3b:
The area of the shaded region is 12 cm². Find the value of θ.
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.

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17M.2.SL.TZ1.S_5:
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16N.2.SL.TZ0.S_3b:
Find the value of $r$ .
16N.2.AHL.TZ0.H_9c:
Hence find the value of $r$ given that the shaded area is equal to 4.
18M.2.AHL.TZ2.H_4a.ii:
Find ${\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}$ in radians.
18M.2.AHL.TZ2.H_4a.i:
Find AM.
21M.1.SL.TZ2.1a:
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16N.2.SL.TZ0.S_3a:
Write down the exact value of $\theta$ in radians.
19M.2.SL.TZ2.S_4c:
Given that the area of triangle OBC is $\frac{3}{5}$ of the area of sector OAB, find θ.
22M.2.SL.TZ1.3b:
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20N.2.AHL.TZ0.H_11e.i:
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21M.2.SL.TZ1.8e.ii:
Find the length of new fence required.
20N.2.AHL.TZ0.H_11c:
Find the maximum displacement of $P$ , in metres, from its initial position.
20N.2.SL.TZ0.S_7a:
Show that $AC = \tan θ$ .
18M.2.AHL.TZ2.H_4b:
Find the area of the shaded region.
17M.2.AHL.TZ1.H_8b:
Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$ .
21M.2.SL.TZ1.8a:
Write down an expression for $r$ in terms of $θ$ .
21M.2.SL.TZ1.8e.i:
Find the size of $A \hat{B} C$ .
16N.2.AHL.TZ0.H_9a:
Find an expression for the shaded area in terms of $\alpha$ , $\theta$ and $r$ .
20N.2.AHL.TZ0.H_11b:
Find an expression for $s$ in terms of $t$ .
18M.2.SL.TZ1.S_3a:
Find the area of the shaded region, in terms of θ.
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 $OADB$ .
21M.3.AHL.TZ1.2c:
Show that $y = 2 x \sin \frac{π}{n}$ .
21M.3.AHL.TZ1.2d:
Use the results from parts (b) and (c) to show that $A = P = 4 n \tan \frac{π}{n}$ .
16N.2.AHL.TZ0.H_9b:
Show that $\alpha = 4\arcsin \frac{1}{4}$ .
19N.2.SL.TZ0.S_4a:
Find ${\text{B}}\widehat {\text{A}}{\text{C}}$ .
21M.2.SL.TZ1.8b:
Show that the area of the field that the horse can reach is $\frac{392}{θ^{2}} (θ + \sin θ)$ .
17M.2.SL.TZ2.S_1a:
Find the length of arc ABC.
19M.2.SL.TZ2.S_4a:
Show that ${\text{OC}} = r\,{\text{cos}}\,\theta$ .
EXN.2.SL.TZ0.2b:
Show that $L = \sqrt{18 - 18 \cos θ}$ .
21M.2.SL.TZ1.8d:
Hence, find the size of $D \hat{A} E$ .
19N.2.AHL.TZ0.H_4b:
the area.
22M.2.SL.TZ2.1a:
Find the length of the chord $[AB]$ .
21N.2.SL.TZ0.5b:
Hence determine the value of $θ$ .
22M.2.SL.TZ1.3a:
Find the area of one of the shaded segments in terms of $θ$ .
21N.2.SL.TZ0.5a:
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17M.2.SL.TZ2.S_1b:
Find the perimeter of sector OABC.
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.SL.TZ0.S_3c:
Find AB.
17N.1.SL.TZ0.S_4a:
Show that ${\text{AC}} = 7{\text{ cm}}$ .
19M.2.SL.TZ2.S_4b:
Find the area of triangle OBC in terms of $r$ and θ.
SPM.2.SL.TZ0.2b:
Find the area of the shaded region.
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Find the times when $P$ comes to instantaneous rest.
20N.2.AHL.TZ0.H_11e.ii:
Hence show that $\frac{v_{2}}{v_{1}} = \frac{v_{3}}{v_{2}} = - e^{- \frac{π}{2}}$ .
19N.2.AHL.TZ0.H_4a:
the perimeter.
20N.2.AHL.TZ0.H_11d:
Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.
SPM.2.SL.TZ0.2a:
Find the value of θ, giving your answer in radians.
17N.2.AHL.TZ0.H_3:
This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius ${\text{OB}} = 9{\text{ cm}}$ and four equal sectors of a smaller circle of radius ${\text{OA}} = 3{\text{ cm}}$ .
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Find the area of the pendant.
19N.2.SL.TZ0.S_4b:
Find the area of $R$ .
17M.2.SL.TZ2.S_1c:
Find the area of sector OABC.
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.
EXN.2.SL.TZ0.2a:
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21M.2.SL.TZ1.8c:
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21M.1.SL.TZ2.1b:
Hence, find the exact area of the non-shaded region.

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Topic 3— Geometry and trigonometry

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