2013澳大利亚高考数学试题答案(7)

Section II

90 marks

Attempt Questions 11–16

Allow about 2 hours and 45 minutes for this section

Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11–16, your responses should include relevant mathematical reasoning and/or calculations.

Question 11 (15 marks) Use a SEPARATE writing booklet.

w =1 +i3.(a) z =2 i3 (i) z+ .

(ii) Express w in modulus–argument form. (iii) Write w24 in its simplest form.

12 2

(b) Find numbers A, B and C such that 2

= A x 3

+ Bx +C x+2

2

x 3 x+2 2

x 2+8x +11

.

(c)

Factorise z2 + 4iz + 5.

2

32

(d) Evaluate x 1 xdx.

0

1

3

(e) Sketch the region on the Argand diagram defined by z2 +2 ≤8.

3