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Date	May Specimen paper	Marks available	3	Reference code	SPM.2.SL.TZ0.2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	2	Adapted from	N/A

Question

The following diagram shows part of a circle with centre O and radius 4 cm.

Chord AB has a length of 5 cm and AÔB = θ.

Find the value of θ, giving your answer in radians.

[3]

Find the area of the shaded region.

[3]

Markscheme

METHOD 1

attempt to use the cosine rule (M1)

cos θ = $\frac{4^{2} + 4^{2} - 5^{2}}{2 \times 4 \times 4}$ $\frac{{{4^2} + {4^2} - {5^2}}}{{2 \times 4 \times 4}}$ (or equivalent) A1

θ = 1.35 A1

METHOD 2

attempt to split triangle AOB into two congruent right triangles (M1)

sin $(\frac{θ}{2}) = \frac{2.5}{4}$ $\left( {\frac{\theta }{2}} \right) = \frac{{2.5}}{4}$ A1

θ = 1.35 A1

[3 marks]

attempt to find the area of the shaded region (M1)

$\frac{1}{2} \times 4 \times 4 \times (2 π - 1.35 \dots)$ $\frac{1}{2} \times 4 \times 4 \times \left( {2\pi - 1.35 \ldots } \right)$ A1

= 39.5 (cm²) A1

[3 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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20N.2.SL.TZ0.S_7a:
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18M.2.AHL.TZ2.H_4b:
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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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19N.2.AHL.TZ0.H_4a:
the perimeter.
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Find the total distance travelled by $P$ $P$ in the first $1.5$ $1.5$ seconds of its motion.
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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 $OB = 9 cm$ ${\text{OB}} = 9{\text{ cm}}$ and four equal sectors of a smaller circle of radius $OA = 3 cm$ ${\text{OA}} = 3{\text{ cm}}$ .
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19N.2.SL.TZ0.S_4b:
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Find the area of sector OABC.
17N.1.SL.TZ0.S_4b:
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21M.1.SL.TZ2.1b:
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