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Date	November 2021	Marks available	1	Reference code	21N.2.SL.TZ0.5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Hence and Determine	Question number	5	Adapted from	N/A

Question

The following diagram shows a semicircle with centre $O$ and radius $r$ . Points $P, Q$ and $R$ lie on the circumference of the circle, such that $PQ = 2 r$ and $R \hat{O} Q = θ$ , where $0 < θ < π$ .

Given that the areas of the two shaded regions are equal, show that $θ = 2 \sin θ$ .

[5]

Hence determine the value of $θ$ .

[1]

Markscheme

attempt to find the area of either shaded region in terms of $r$ and $θ$ (M1)

Note: Do not award M1 if they have only copied from the booklet and not applied to the shaded area.

Area of segment $= \frac{1}{2} r^{2} θ - \frac{1}{2} r^{2} \sin θ$ A1

Area of triangle $= \frac{1}{2} r^{2} \sin (π - θ)$ A1

correct equation in terms of $θ$ only (A1)

$θ - \sin θ = \sin (π - θ)$

$θ - \sin θ = \sin θ$ A1

$θ = 2 \sin θ$ AG

Note: Award a maximum of M1A1A0A0A0 if a candidate uses degrees (i.e., $\frac{1}{2} r^{2} \sin (180 ° - θ)$ ), even if later work is correct.

Note: If a candidate directly states that the area of the triangle is $\frac{1}{2} r^{2} \sin θ$ , award a maximum of M1A1A0A1A1.

[5 marks]

$θ = 1.89549 \dots$

$θ = 1.90$ A1

Note: Award A0 if there is more than one solution. Award A0 for an answer in degrees.

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area

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18N.2.SL.TZ0.T_5c:
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16N.2.SL.TZ0.T_5b:
the size of angle DAB;
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Find the angle of elevation from B to C.
17M.2.SL.TZ2.T_4b:
Show that angle DBC is 48.7°, correct to three significant figures.
17M.1.SL.TZ1.T_13b:
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16N.2.SL.TZ0.T_5a:
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19M.2.SL.TZ2.T_2d:
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16N.1.SL.TZ0.T_11a:
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18M.2.SL.TZ1.T_1e:
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18N.2.SL.TZ0.T_5a:
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18M.2.SL.TZ2.T_5a.i:
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18M.2.SL.TZ2.T_5a.ii:
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18M.2.SL.TZ2.T_5a.iii:
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18M.2.SL.TZ2.T_5b:
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17N.2.SL.TZ0.T_3d.ii:
Find the percentage error in Abdallah’s estimate.
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19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
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Find the percentage error in her estimate.
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State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
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19M.2.SL.TZ2.T_2b.i:
Find the size of angle ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}$ .
19M.2.SL.TZ2.T_2b.ii:
Find the size of angle ${\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}}$ .
19M.2.SL.TZ2.T_2c:
Find the size of angle ${\text{A}}\mathop {\text{C}}\limits^ \wedge {\text{D}}$ .
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Calculate the angle of elevation of T from, M, the midpoint of the side AC.
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Find the area of the park.
16N.2.SL.TZ0.T_5c:
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16N.2.SL.TZ0.T_5e:
the length of AP;
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