Complex Numbers 2 (Specialist Maths QCAA Unit 3)

Cartesian Forms

100

Determine Re(z) when z = 3 - 2i

3

200

Determine the imaginary part of the conjugate of 4z if z = 3 - 2i

Im(4(3 + 2i)) 

Im(12+8i) 

8

300

Determine 3u + v² if u = 3 - 2i and v = -6 + 2i

3(3-2i) + (-6+2i)^2

9 - 6i +36 - 24i -4

41 - 30i

400

Double Points: 

Simplify z³ if z = 5 - i

(5 - i) ^3

5^3 +3 * 5^2 *(- i) + 3 * 5 * (-i)^2 + (-i)^3
125 - 75i - 15 + i

110 -74i

500

Determine the conjugate of z, where z = (4+2i)^2 +(-1 - i)(-1+i)

(4+2i)^2 +(-1 - i)(-1+i)

16 +16i - 4 + 1 + 1

14 +16i 

Conjugate = 14 - 16i

Complex ArithmeticPolar Form

100

Convert z = 3 - 3i into polar form.

3sqrt(2) cis (-pi/4)

200

Double Points:

Determine the polar form of -5i

5cis(-pi/2)

300

Convert 6cis(2pi/3) into Cartesian Form.

-3 + 3sqrt(3)i

400

Determine z^3 where z = 3cis(-pi/2).

27cis(-3pi/2)

27cis(pi/2)

500

Determine z⁵ for z = -2 - 2i

z = sqrt(8)cis(-3pi/4)

z^5 = 64sqrt(8)cis(-15pi/4)

=128sqrt(2)cis(pi/4)

Complex Plane

100

Determine the equation of the Re(z) = 5 in Cartesian Form.

x = 5

200

Determine the equation of the Re(5z) = 20 in Cartesian Form.

5x=20

x=4

300

Double Points: Determine the complex subset defined by 

Im(z+2i) + Re(z-2)= 6

Im(x+yi+2i) + Re(x+yi-2)= 6

y+2 +x - 2 = 6
y = -x + 6

y int at (0,6). x int at (6,0).

400

Sketch the complex subset defined by |z| = 5.

x^2 + y^2 = 5^2

Circle with radius 5, centre (0,0).

500

Sketch the graph described by:

|z-3+2i| =2

(x-3)^2 + (y+2)^2 = 2^2

Circle with Radius = 2 and centre at (3,-2). y intercept at (3,0).

Roots of Complex Numbers

100

Determine the square root of 25cis(pi)

5cis(pi/2 + 2pin/ 2), n integer

5cis(pi/2), 5cis(3pi/2)=5cis(-pi/2)

200

Double Points:

Determine the cube root of 8cis(pi/2)

2cis(pi/6 + 2pin/3), n integer

2cis(pi/6), 2cis(5pi/6), 2cis(9pi/6) = 2cis(-pi/2)

300

Determine the square root of (4i)

4i = 4cis(pi/2)

sqrt(4i) = 2cis(pi/4 +2pin/2) , n interger

2cis(pi/4), 2cis(5pi/4) = 2cis(-3pi/4)

400

Determine the square root of 3 + 3i

3sqrt(2)cis(pi/4)

sqrt(3sqrt(2))cis(pi/8 + 2pin/2), n integer

sqrt(3)forthrt(2)cis(pi/8), sqrt(3)forthrt(2)cis(-7pi/8)

500

Determine the cube root of z = -sqrt(2) + sqrt(2)i

2cis(3pi/4)

cbrt(2)cis(3pi/12 + 2pin/3), n integer

cbrt(2)cis(3pi/12) = cbrt(2)cis(pi/4)

cbrt(2)cis(11pi/12)

cbrt(2)cis(19pi/12) = cbrt(2)cis(-5pi/12)

Factorisation of Polynomials

100

Factorise x² +9

(x+3i)(x-3i)

200

Factorise z² -4x + 8

(z +2 -2i)(z +2 +2i)

300

Double Points:

Factorise z² -6z + 20

(z - 3 + sqrt(11)i)(z - 3 - sqrt(11)i)

400

Factorise u^4 - 9

(u^2 +3)(u^2 - 3)

(u+sqrt(3)i)(u-sqrt(3)i)(u+sqrt(3))(u-sqrt(3))

500

Factorise z³ -3z² -5z + 39

(z+3)(z-3-2i)(z-3+2i)

Mr. Kettle

100

Mr Kettle's Primary Instrument

Trumpet

200

The Name of Mr Kettle's Partner

Ms Lal

300

Mr Kettle's Favourite Colour

Blue

400

Double Points:

The Name of Mr Kettle's Brother

Gough

500

The 6 subjects Mr Kettle did in year 12

English, Maths B/Methods, Maths C/Specialist, Music, Music Extension, Japanese