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Date May 2010 Marks available 11 Reference code 10M.2.hl.TZ1.13
Level HL only Paper 2 Time zone TZ1
Command term Determine, Hence, and Show that Question number 13 Adapted from N/A

Question

Points A, B and C are on the circumference of a circle, centre O and radius r . A trapezium OABC is formed such that AB is parallel to OC, and the angle AˆOC is θ , π2θπ .

 

 

(a)     Show that angle BˆOC is πθ.

(b)     Show that the area, T, of the trapezium can be expressed as

T=12r2sinθ12r2sin2θ.

(c)     (i)     Show that when the area is maximum, the value of θ satisfies

cosθ=2cos2θ.

  (ii)     Hence determine the maximum area of the trapezium when r = 1.

    (Note: It is not required to prove that it is a maximum.)

Markscheme

(a)     OˆAB=πθ(allied)     A1

recognizing OAB as an isosceles triangle     M1

so AˆBO=πθ

BˆOC=πθ(alternate)     AG

Note: This can be done in many ways, including a clear diagram.

 

[3 marks]

 

(b)     area of trapezium is T=areaΔBOC+areaΔAOB     (M1)

=12r2sin(πθ)+12r2sin(2θπ)     M1A1

=12r2sinθ12r2sin2θ     AG

[3 marks]

 

(c)     (i)     dTdθ=12r2cosθr2cos2θ     M1A1

for maximum area 12r2cosθr2cos2θ=0     M1

cosθ=2cos2θ     AG

 

(ii)     θmax=2.205     (A1)

12sinθmax12sin2θmax=0.880     A1

[5 marks]

 

Total [11 marks]

Examiners report

In part (a) students had difficulties supporting their statements and were consequently unable to gain all the marks here. There were some good attempts at parts (b) and (c) although many students failed to recognise r as a constant and hence differentiated it, often incorrectly.

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.7 » Area of a triangle as 12absinC .

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