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Date	May 2017	Marks available	7	Reference code	17M.2.SL.TZ1.S_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	S_5	Adapted from	N/A

Question

The following diagram shows the chord [AB] in a circle of radius 8 cm, where ${\text{AB}} = 12{\text{ cm}}$ .

M17/5/MATME/SP2/ENG/TZ1/05

Find the area of the shaded segment.

Markscheme

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

attempt to find the central angle or half central angle (M1)

eg $\,\,\,\,\,$ M17/5/MATME/SP2/ENG/TZ1/05/M , cosine rule, right triangle

correct working (A1)

eg $\,\,\,\,\,$ $\cos \theta = \frac{{{8^2} + {8^2} - {{12}^2}}}{{2 \bullet 8 \bullet 8}},{\text{ }}{\sin ^{ - 1}}\left( {\frac{6}{8}} \right),{\text{ }}0.722734,{\text{ }}41.4096^\circ ,{\text{ }}\frac{\pi }{2} - {\sin ^{ - 1}}\left( {\frac{6}{8}} \right)$

correct angle ${\rm{A\hat OB}}$ (seen anywhere)

eg $\,\,\,\,\,$ ${\text{1.69612, }}97.1807^\circ ,{\text{ 2}} \times {\text{si}}{{\text{n}}^{ - 1}}\left( {\frac{6}{8}} \right)$ (A1)

correct sector area

eg $\,\,\,\,\,$ $\frac{1}{2}(8)(8)(1.70),{\text{ }}\frac{{97.1807}}{{360}}(64\pi ),{\text{ }}54.2759$ (A1)

area of triangle (seen anywhere) (A1)

eg $\,\,\,\,\,$ $\frac{1}{2}(8)(8)\sin 1.70,{\text{ }}\frac{1}{2}(8)(12)\sin 0.722,{\text{ }}\frac{1}{2} \times \sqrt {64 - 36} \times 12,{\text{ }}31.7490$

appropriate approach (seen anywhere) (M1)

eg $\,\,\,\,\,$ ${A_{{\text{triangle}}}} - {A_{{\text{sector}}}}$ , their sector-their triangle

22.5269

area of shaded region $= 22.5{\text{ }}({\text{c}}{{\text{m}}^2})$ A1 N4

Note: Award M0A0A0A0A1 then M1A0 (if appropriate) for correct triangle area without any attempt to find an angle in triangle OAB.

[7 marks]

Examiners report

[N/A]

Syllabus sections

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

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18M.2.SL.TZ1.S_3b:
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19M.1.AHL.TZ1.H_3:
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16N.2.AHL.TZ0.H_9c:
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18M.2.AHL.TZ2.H_4a.ii:
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18M.2.AHL.TZ2.H_4a.i:
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21M.1.SL.TZ2.1a:
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16N.2.SL.TZ0.S_3a:
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19M.2.SL.TZ2.S_4c:
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22M.2.SL.TZ1.3b:
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20N.2.AHL.TZ0.H_11e.i:
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17M.2.AHL.TZ1.H_8b:
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21M.2.SL.TZ1.8a:
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21M.2.SL.TZ1.8e.i:
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16N.2.AHL.TZ0.H_9a:
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20N.2.AHL.TZ0.H_11b:
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18M.2.SL.TZ1.S_3a:
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20N.2.SL.TZ0.S_7b:
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21M.3.AHL.TZ1.2c:
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21M.3.AHL.TZ1.2d:
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16N.2.AHL.TZ0.H_9b:
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19N.2.SL.TZ0.S_4a:
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18M.1.SL.TZ2.S_4:
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21M.2.SL.TZ1.8d:
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19N.2.AHL.TZ0.H_4b:
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22M.2.SL.TZ2.1a:
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21N.2.SL.TZ0.5b:
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22M.2.SL.TZ1.3a:
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21N.2.SL.TZ0.5a:
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17M.2.SL.TZ2.S_1b:
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17M.2.AHL.TZ1.H_8a:
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16N.2.SL.TZ0.S_3c:
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17N.1.SL.TZ0.S_4a:
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19M.2.SL.TZ2.S_4b:
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20N.2.AHL.TZ0.H_11e.ii:
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19N.2.AHL.TZ0.H_4a:
the perimeter.
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SPM.2.SL.TZ0.2a:
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17N.2.AHL.TZ0.H_3:
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19N.2.SL.TZ0.S_4b:
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17M.2.SL.TZ2.S_1c:
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17N.1.SL.TZ0.S_4b:
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Topic 3— Geometry and trigonometry