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Date November 2018 Marks available 2 Reference code 18N.2.SL.TZ0.S_7
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Show that Question number S_7 Adapted from N/A

Question

A communication tower, T, produces a signal that can reach cellular phones within a radius of 32 km. A straight road passes through the area covered by the tower’s signal.

The following diagram shows a line representing the road and a circle representing the area covered by the tower’s signal. Point R is on the circumference of the circle and points S and R are on the road. Point S is 38 km from the tower and RŜT = 43˚.

Let SR = x. Use the cosine rule to show that x2(76cos43)x+420=0.

[2]
a.

Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.

[4]
b.

Markscheme

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

recognizing TR =32  (seen anywhere, including diagram)      A1

correct working      A1

eg   322=x2+3822(x)(38)cos431024=1444+x276(x)cos43

x2(76cos43)x+420=0     AG N0

 

[2 marks]

a.

Note: There are many approaches to this question, depending on which triangle the candidate has used, and whether they used the cosine rule and/or the sine rule. Please check working carefully and award marks in line with the markscheme.

 

METHOD 1

correct values for x (seen anywhere)      A1A1

x = 9.02007,  46.5628

recognizing the need to find difference in values of x      (M1)

eg     46.5 − 9.02,   x1x2

37.5427

37.5 (km)     A1 N2

 

METHOD 2

correct use of sine rule in ΔSRT

eg     sinSRT38=sin4332SRT = 54.0835°     (A1)

recognizing isosceles triangle (seen anywhere)      (M1)

eg     ˆT=180254.0835, two sides of 32

correct working to find distance      A1

eg     322+32223232cos(180254.0835),

sin71.8329d=sin54.083532,  322=322+x2232xcos(0.944)

37.5427

37.5 (km)     A1 N2

 

[4 marks]

 

b.

Examiners report

[N/A]
a.
[N/A]
b.

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 » SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule
Topic 3— Geometry and trigonometry

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