WACEhub

Mathematics Specialist Year 11

Student name: __________________________ Teacher name: ________________

Date: Monday 10 August 2020

Task type: Response + Investigation

Time allowed: 45 minutes (for the entire booklet)

Number of questions: 5

Materials required: Calculator with CAS capability (to be provided by the student)

Standard items: Pens (blue/black preferred), pencils (including coloured), sharpener, correction fluid/tape, eraser, ruler, highlighters

Special items: Drawing instruments, templates, notes on two unfolded sheets of
A4 paper, and up to three calculators approved for use in the WACE examinations

Marks available: 40 marks

Task weighting: 14% combined (8% for Test 2 and 6% for investigation 2)

Formula sheet provided: Yes

Note: All part questions worth more than 2 marks require working to obtain full marks.

Question 1 {1.3.4, 1.3.5} (4 marks)

Let x ϵ R. Prove that that x2>x is false by giving a counterexample. (1 mark)

Disprove the following statement: There exists x ϵ R such that 5+x2=1-x2 (3 marks)

Question 2 {2.1.1} (5 marks)

Solve 2cos2(x+π3)=-1 given that x∈0,2π. Show your working.

Question 3 {2.3.4, 2.3.6} (7 marks)

Use mathematical induction to prove that that 4n+6n-1 is divisible by 3 for all n∈N

Question 4 {2.1.2} (8 marks)

The function f(x)=a tan (b(x-c)) has been graphed below. Determine the values of the constants a, b and c. (4 marks)

Sketch the graph of y=6 cos12x+π4 . (4 marks)

Investigation Validation {2.1.3, 2.3.4, 2.3.5} (16 marks)

Use the identity

2sinAcosB=sinA+B+sinA-B

(or otherwise) to show that

2sin[x]cos[(2k+1)x]=sin2k+1x-sin⁡[2kx]

(4 marks)

Given that sinx≠0 prove, by mathematical induction, that for all positive integers n,

cos(x)+cos(3x)+…+cos2n-1x=sin⁡(2nx)2sin⁡(x)

You may find the identity sin2A=2sinAcosA useful.

(12 marks)

(additional working space)