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Date November 2016 Marks available 3 Reference code 16N.2.AHL.TZ0.H_9
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number H_9 Adapted from N/A

Question

The diagram shows two circles with centres at the points A and B and radii 2 r and r , respectively. The point B lies on the circle with centre A. The circles intersect at the points C and D.

N16/5/MATHL/HP2/ENG/TZ0/09

Let α be the measure of the angle CAD and θ be the measure of the angle CBD in radians.

Find an expression for the shaded area in terms of α , θ and r .

[3]
a.

Show that α = 4 arcsin 1 4 .

[2]
b.

Hence find the value of r given that the shaded area is equal to 4.

[3]
c.

Markscheme

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

A = 2 ( α sin α ) r 2 + 1 2 ( θ sin θ ) r 2    M1A1A1

 

Note: Award M1A1A1 for alternative correct expressions eg. A = 4 ( α 2 sin α 2 ) r 2 + 1 2 θ r 2 .

 

[3 marks]

a.

METHOD 1

consider for example triangle ADM where M is the midpoint of BD     M1

sin α 4 = 1 4    A1

α 4 = arcsin 1 4

α = 4 arcsin 1 4    AG

METHOD 2

attempting to use the cosine rule (to obtain 1 cos α 2 = 1 8 )     M1

sin α 4 = 1 4 (obtained from sin α 4 = 1 cos α 2 2 )     A1

α 4 = arcsin 1 4

α = 4 arcsin 1 4    AG

METHOD 3

sin ( π 2 α 4 ) = 2 sin α 2 where  θ 2 = π 2 α 4

cos α 4 = 4 sin α 4 cos α 4    M1

 

Note: Award M1 either for use of the double angle formula or the conversion from sine to cosine.

 

1 4 = sin α 4    A1

α 4 = arcsin 1 4

α = 4 arcsin 1 4    AG

[2 marks]

b.

(from triangle ADM), θ = π α 2   ( = π 2 arcsin 1 4 = 2 arcsin 1 4 = 2.6362 )      A1

attempting to solve  2 ( α sin α ) r 2 + 1 2 ( θ sin θ ) r 2 = 4

with α = 4 arcsin 1 4  and θ = π α 2   ( = 2 arccos 1 4 )  for r     (M1)

r = 1.69    A1

[3 marks]

c.

Examiners report

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

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.7—Radians
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Topic 3—Geometry and trigonometry

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