WACEhub

Mathematics Specialist Year 11

Student name: __________________________ Teacher name: ________________

Date: Friday 24 September 2021

Task type: Response

Time allowed: 40 mins

Number of questions: 7

Materials required: Notes on two unfolded sheets of paper (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 and up to three calculators approved for use in the WACE examinations

Marks available: 40 marks

Task weighting: 10%

Formula sheet provided: Yes

Scientific Calculator and CAS: Not Permitted

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

Question 1 (2.2.1, 2.2.2) (6 marks)

Given that A, B and C are 2×2 matrices, X= 21, Y=34, Z=3-102 and I is the 2×2 identity matrix, find the following where possible

XY (1 mark)

YX (1 mark)

Matrix W given that 3Z-W=I (2 marks)

vivi

An expression for matrix V in terms of other matrices given that V-ABV=C (2 marks)

Question 2 (2.2.3, 2.2.11) (5 marks)

For what values of a is the matrix a53a singular? (2 marks)

Use matrices to find the point of intersection of the lines given by the equations

3x+y=2 and 5x+2y=1 . (3 marks)

Question 3 (2.1.4) (5 marks)

Using the same scale, sketch the graphs of y=sin2x and y=cosec2x+π on the grid below for 0≤x≤ 2π

Question 4 (2.1.5, 2.1.6, 2.1.8) (5 marks)

Prove the identity below

1-sin⁡(2θ)sinθ-cosθ=sinθ-cosθ

Question 5 (2.2.5, 2.2.7, 2.2.10) (5 marks)

Find the matrices that produce each of the transformations described below

A reflection in the line y=x (1 mark)

A rotation clockwise about the origin by 90° (2 mark)

Find and describe a single transformation matrix T that is a result of a reflection in the line y=x followed by a 90° clockwise rotation about the origin. (2 marks)

Question 6 (2.2.6, 2.2.9) (9 marks)

Find the matrix of the linear transformation such that (1,2)→(12,7) and (-3,1)→(-1,0) (4 marks)

The matrix tt1t maps the unit square into a parallelogram of area 2 square units. Find the possible value(s) of t (5 marks)

Question 7 (2.1.7) (5 marks)

Find the general solution of 3cosx- 3 sinx=3