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Date November 2019 Marks available 1 Reference code 19N.1.AHL.TZ0.H_9
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Show that Question number H_9 Adapted from N/A

Question

In the following diagram, the points ABC and D are on the circumference of a circle with centre O and radius r[AC] is a diameter of the circle. BC=rAD=CD and ABC=ADC=90.

Given that cos75=q, show that cos105=q.

[1]
a.

Show that BAD=75.

[3]
b.

By considering triangle ABD, show that BD2=5r22r2q6.

[4]
c.i.

By considering triangle CBD, find another expression for BD2 in terms of r and q.

[3]
c.ii.

Use your answers to part (c) to show that cos75=16+2.

[3]
d.

Markscheme

cos105=cos(18075)=cos75      R1

=q       AG

Note: Accept arguments using the unit circle or graphical/diagrammatical considerations.

[1 mark]

a.

AD=CDCAD=45      A1

valid method to find BAC        (M1)

for example: BC=rBCA=60

BAC=30      A1

hence BAD=45+30=75      AG

[3 marks]

b.

AB=r3AD=(CD)=r2       A1A1

applying cosine rule        (M1)

BD2=(r3)2+(r2)22(r3)(r2)cos75       A1

=3r2+2r22r26cos75

=5r22r2q6       AG

[4 marks]

c.i.

BCD=105        (A1)

attempt to use cosine rule on ΔBCD        (M1)

BD2=r2+(r2)22r(r2)cos105

=3r2+2r2q2       A1

[3 marks]

c.ii.

5r22r2q6=3r2+2r2q2        (M1)(A1)

2r2=2r2q(6+2)       A1

Note: Award A1 for any correct intermediate step seen using only two terms.

q=16+2       AG

Note: Do not award the final A1 if follow through is being applied.

[3 marks]

d.

Examiners report

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

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area
Show 133 related questions
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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