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Date May 2019 Marks available 3 Reference code 19M.2.SL.TZ2.T_2
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number T_2 Adapted from N/A

Question

An archaeological site is to be made accessible for viewing by the public. To do this, archaeologists built two straight paths from point A to point B and from point B to point C as shown in the following diagram. The length of path AB is 185 m, the length of path BC is 250 m, and angle ABC is 125°.

The archaeologists plan to build two more straight paths, AD and DC. For the paths to go around the site, angle BAD is to be made equal to 85° and angle BCD is to be made equal to 70° as shown in the following diagram.

Find the distance from A to C.

[3]
a.

Find the size of angle BAC.

[3]
b.i.

Find the size of angle CAD.

[1]
b.ii.

Find the size of angle ACD.

[2]
c.

The length of path AD is 287 m.

Find the area of the region ABCD.

[4]
d.

Markscheme

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

AC2 = 1852 + 2502 − 2 × 185 × 250 × cos(125°)   (M1)(A1)

Note: Award (M1) for substitution in the cosine formula; (A1) for correct substitution.

387  (387.015…) (m)      (A1)(G2)

Note: If radians are used the answer is 154 (154.471…), award at most (M1)(A1)(A0).

[3 marks]

a.

250sinBAC=387.015sin(125)       (M1)(A1)(ft)

OR

cos1(1852+387.015225022×185×387.015)       (M1)(A1)(ft)

Note: Award (M1) for substitution in the sine or cosine formulas; (A1)(ft) for correct substitution.

BAC=31.9  (31.9478…°)       (A1)(ft)(G2)

Note: Follow through from part (a).

[3 marks]

b.i.

(CAD =) 53.1°  (53.0521…°)       (A1)(ft)

Note: Follow through from their part (b)(i) only if working seen.

[1 mark]

b.ii.

(ACD = ) 70° − (180° − 125° − 31.9478°…)      (M1)

Note: Award (M1) for subtracting their angle ACB from 70°.

OR

(ADC =) 360 − (85 + 70 + 125) = 80

(ACD =) 180 − 80 − 53.0521...      (M1)

46.9°  (46.9478…°)      (A1)(ft)(G2)

Note: Follow through from part (b)(i).

[2 marks]

c.

185×250×sin(125)2+287×387.015×sin(53.0521)2     (M1)(M1)(M1)

Note: Award (M1) for substitution in the area formula for either triangle; (M1) for correct substitution for both areas; (M1) for adding their two areas;

18942.8… + 44383.9…

63300 (m2)  (63326.8… (m2))      (A1)(ft)(G3)

Note: Follow through from parts (a) and (b)(ii).

OR

DC=287×sin(53.0521)sin(46.9478)=313.884

0.5×287×185×sin85+0.5×250×313.884×sin70      M1M1M1

Note: Award (M1) for substitution in the area formula for either triangle; (M1) for correct substitution for both areas; (M1) for adding their two areas;

26446.4… + 36869.3

63300  (63315.8…) (m2)     (A1)(ft)(G3)

[4 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.
[N/A]
d.

Syllabus sections

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

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