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Date	November 2020	Marks available	6	Reference code	20N.2.SL.TZ0.S_7
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_7	Adapted from	N/A

Question

The following diagram shows a circle with centre $O$ and radius $1 cm$ . Points $A$ and $B$ lie on the circumference of the circle and $A \hat{O} B = 2 θ$ , where $0 < θ < \frac{π}{2}$ .

The tangents to the circle at $A$ and $B$ intersect at point $C$ .

Show that $AC = \tan θ$ .

[1]

Find the value of $θ$ when the area of the shaded region is equal to the area of sector $OADB$ .

[6]

Markscheme

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

correct working for $AC$ (seen anywhere) A1

eg $\tan θ = \frac{AC}{OA}, \tan θ = \frac{AC}{1}$

$AC = \tan θ$ AG N0

[1 mark]

METHOD 1 (working with half the areas)

area of triangle $OAC$ or triangle $OBC$ (A1)

eg $\frac{1}{2} \times 1 \times \tan θ$

correct sector area (A1)

eg $\frac{1}{2} \times θ \times (1^{2}), \frac{1}{2} θ$

correct approach using their areas to find the shaded area (seen anywhere) (A1)

eg $A_{their triangle} - A_{their sector}, \frac{1}{2} θ - \frac{1}{2} \tan θ$

correct equation A1

eg $\frac{1}{2} \tan θ - \frac{1}{2} θ = \frac{1}{2} θ, \tan θ = 2 θ$

$1.16556$

$1.17$ A2 N4

METHOD 2 (working with entire kite and entire sector)

area of kite $OACB$ (A1)

eg $2 \times \frac{1}{2} \times 1 \times \tan θ, \frac{1}{2} \times \frac{1}{\cos θ} \times 2 \sin θ$

correct sector area (A1)

eg $\frac{1}{2} \times 2 θ \times (1^{2}), θ$

correct approach using their areas to find the shaded area (seen anywhere) (A1)

eg $A_{kite OACB} - A_{sector OADB}, θ - \tan θ$

correct equation A1

eg $\tan θ - θ = θ, \tan θ = 2 θ$

$1.16556$

$1.17$ A2 N4

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.7—Radians

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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.
16N.2.SL.TZ0.S_3a:
Write down the exact value of $\theta$ in radians.
17M.2.AHL.TZ1.H_8b:
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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:
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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:
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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:
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16N.2.AHL.TZ0.H_9c:
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16N.2.AHL.TZ0.H_9b:
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SPM.1.AHL.TZ0.5b:
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18M.2.AHL.TZ2.H_4b:
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20N.2.SL.TZ0.S_7a:
Show that $AC = \tan θ$ .
17M.2.AHL.TZ1.H_8a:
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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.
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:
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:
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