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Date	May 2019	Marks available	3	Reference code	19M.1.SL.TZ1.S_3
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	S_3	Adapted from	N/A

Question

The following diagram shows a right triangle ABC. Point D lies on AB such that CD bisects AĈB.

AĈD = θ and AC = 14 cm

Given that ${\text{sin}}\,\theta = \frac{3}{5}$ , find the value of ${\text{cos}}\,\theta$ .

[3]

Find the value of ${\text{cos}}\,2\theta$ .

[3]

Hence or otherwise, find ${\text{BC}}$ .

[2]

Markscheme

valid approach (M1)

eg labelled sides on separate triangle, ${\text{si}}{{\text{n}}^2}\,x + {\text{co}}{{\text{s}}^2}\,x = 1$

correct working (A1)

eg missing side is 4, $\sqrt {1 - {{\left( {\frac{3}{5}} \right)}^2}}$

${\text{cos}}\,\theta = \frac{4}{5}$ A1 N3

[3 marks]

correct substitution into ${\text{cos}}\,2\theta$ (A1)

eg $2\left( {\frac{{16}}{{25}}} \right) - 1$ , $1 - 2{\left( {\frac{3}{5}} \right)^2}$ , $\frac{{16}}{{25}} - \frac{9}{{25}}$

${\text{cos}}\,2\theta = \frac{7}{{25}}$ A1 N2

[2 marks]

correct working (A1)

eg $\frac{7}{{25}} = \frac{{14}}{{{\text{BC}}}}$ , ${\text{BC}} = \frac{{14 \times 25}}{7}$

${\text{BC}} = 50$ (cm) A1 N2

[2 marks]

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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17N.1.SL.TZ0.S_4b:
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17M.1.SL.TZ1.T_13a:
Find the length of BC.
18N.2.SL.TZ0.T_5c:
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16N.2.SL.TZ0.T_5b:
the size of angle DAB;
19M.1.SL.TZ2.T_9a:
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:
Find the area of the postage stamp.
20N.2.SL.TZ0.T_3d:
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16N.2.SL.TZ0.T_5a:
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19M.2.SL.TZ2.T_2d:
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Find the area of the region ABCD.
16N.1.SL.TZ0.T_11a:
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19M.1.SL.TZ1.T_10a:
Find the size, in degrees, of angle BÂC.
18M.2.SL.TZ1.T_1e:
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17M.2.SL.TZ1.T_2d:
Find the length of the perimeter of ABCDE.
18N.2.SL.TZ0.T_5a:
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18M.2.SL.TZ2.T_5a.i:
Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
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18M.2.SL.TZ2.T_5a.ii:
Use Giovanni's diagram to calculate the length of AX.
18M.2.SL.TZ2.T_5a.iii:
Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.
18M.2.SL.TZ2.T_5b:
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17N.2.SL.TZ0.T_3b:
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17N.2.SL.TZ0.T_3d.i:
Calculate Abdallah’s estimate for the area.
17N.2.SL.TZ0.T_3d.ii:
Find the percentage error in Abdallah’s estimate.
19M.2.SL.TZ1.T_2a:
Find the length ${\text{AB}}$ , giving your answer to four significant figures.
19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
19M.2.SL.TZ1.T_2d:
Find the percentage error in her estimate.
19M.2.SL.TZ1.T_2e:
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State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
19M.2.SL.TZ2.T_2a:
Find the distance from A to C.
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 area of triangle EAD.
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Calculate the total volume of the barn.
18M.2.SL.TZ1.T_1f:
Calculate the total length of metal required for one support.
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17M.2.SL.TZ1.T_2b:
Find the length of CD.
17M.2.SL.TZ1.T_2e:
Calculate the length of CF.
18N.2.SL.TZ0.T_5d:
A vertical pole, TB, is constructed at point B and has height 25 m.

Calculate the angle of elevation of T from, M, the midpoint of the side AC.
17M.2.SL.TZ2.T_4a:
Find the length of the path BD.
17M.2.SL.TZ2.T_4c:
Find the area of the park.
16N.2.SL.TZ0.T_5c:
the area of triangle ABD;
16N.2.SL.TZ0.T_5e:
the length of AP;
20N.2.SL.TZ0.T_3a:
Calculate the length of $BD$ .
20N.2.SL.TZ0.T_3c:
Calculate the area of triangle $BDC$ .

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Topic 3— Geometry and trigonometry

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