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Date	November 2018	Marks available	3	Reference code	18N.2.SL.TZ0.T_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	T_5	Adapted from	N/A

Question

A flat horizontal area, ABC, is such that AB = 100 m , BC = 50 m and angle AĈB = 43.7° as shown in the diagram.

Show that the size of angle BÂC is 20.2°, correct to 3 significant figures.

[3]

Calculate the area of triangle ABC.

[4]

Find the length of AC.

[3]

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.

[5]

Markscheme

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

$\frac{{{\text{sin}}\,43.7^\circ }}{{100}} = \frac{{{\text{sin}}\,{\text{BAC}}}}{{50}}$ (M1)(A1)

Note: Award (M1) for substitution into sine rule formula, (A1) for correct substitution.

BAC = 20.2087… = 20.2° (A1)(AG)

Note: Award (A1) only if both the correct unrounded and rounded answers are seen.

[3 marks]

units are required in part (b)

$\frac{1}{2}\left( {100} \right)\left( {50} \right){\text{sin}}\,\left( {116.1} \right)$ (A1)(M1)(A1)

Note: Award (A1) for 116.1 or unrounded value or 116 seen, (M1) for substitution into area of triangle formula, (A1) for correct substitution.

= 2250 m² (2245.06… m²) (A1)(G3)

Note: The answer is 2250 m²; the units are required. Use of 20.2087… gives 2245.23….

[4 marks]

$\frac{{100}}{{{\text{sin}}\,43.7}} = \frac{{{\text{AC}}}}{{{\text{sin}}\,\left( {116.1} \right)}}$ (M1)(A1)(ft)

Note: Award (M1) for substitution into sine rule formula, (A1)(ft) for their correct substitution. Follow through from their 116.1.

AC = 130 (m) (129.982… (m)) (A1)(ft)(G2)

Note: Use of 20.2087… gives 129.992….

AC² = 100² + 50² −2(100)(50) cos (116.1) (M1)(A1)(ft)

Note: Award (M1) for substitution into cosine rule formula, (A1)(ft) for their correct substitution. Follow through from their 116.1.

AC = 130 (m) (129.997… (m)) (A1)(ft)(G2)

Note: Award (M1) for substitution into cosine rule formula, (A1)(ft) for their correct substitution.

[3 marks]

BM² = 100² + 65² − 2(100)(65) cos (20.2°) (M1)(A1)(ft)

BM² = 50² + 65² − 2(50)(65) cos (43.7°) (M1)(A1)(ft)

Note: Award (M1) for substitution into cosine rule formula, (A1)(ft) for correct substitution, including half their AC.

BM = 45.0 (44.9954… OR 45.0079…) (A1)(ft)

Note: Use of 20.2052… gives 45. Award (G2) for 45.0 seen without working.

${\text{tan}}\,\left( {{\text{TMB}}} \right) = \frac{{25}}{{{\text{their BM}}}}$ (M1)

Note: Award (M1) for correct substitution into tangent formula.

T $\mathop {\text{M}}\limits^ \wedge$ B = 29.1° (29.0546…°) (A1)(ft)(G4)

Note: Follow through within part (d) provided their BM is seen. Use of 44.9954 gives 29.0570… and use of 45.0079… gives 29.0503…. Follow through from their AC in part (c).

[5 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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16N.2.SL.TZ0.T_5b:
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Find the angle of elevation from B to C.
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Show that angle ${\rm{B\hat AE}} = 67.4^\circ$ , correct to one decimal place.
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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:
Calculate CD, the height of the observation deck above the ground.
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Find the size, in degrees, of angle BÂC.
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18N.2.SL.TZ0.T_5a:
Show that the size of angle BÂC is 20.2°, correct to 3 significant figures.
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:
Find the percentage error on Giovanni’s diagram.
17N.2.SL.TZ0.T_3a:
Show that ${\text{BD}} = 93{\text{ m}}$ correct to the nearest metre.
17N.2.SL.TZ0.T_3b:
Calculate angle ${\rm{B\hat CD}}$ .
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.
17M.2.SL.TZ1.T_2a.ii:
Show that the equation in part (a)(i) simplifies to $3{x^2} + 19x - 414 = 0$ .
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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