smc-7431-10-Lines

Complex Numbers, EXT2 N2 2022 HSC 1 MC

Let `R` be the region in the complex plane defined by `1 < text{Re}(z) <= 3` and `(pi)/(6) <= text{Arg}(z) < (pi)/(3)`.

Which diagram best represents the region `R`?

Show Answers Only

`A`

Show Worked Solution

`1 < text{Re}(z) <= 3\ \ =>\ \ text{Eliminate}\ B and D`

`(pi)/(6) <= text{Arg}(z) < (pi)/(3)\ \ =>\ \ text{Eliminate}\ C`

`=>A`

Complex Numbers, EXT2 N2 2004 HSC 2c

Sketch the region in the complex plane where the inequalities

`| z + overset_z | ≤ 1` and `| z-i | ≤ 1`

hold simultaneously. (3 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

`| z + overset_z | ≤ 1 `

`\| x + i y + x-iy \|`	`≤ 1`
`\| 2x \|`	`≤ 1`
`\| x \|`	`≤ frac{1}{2}`

`| z-i | ≤ 1 \ => \ text{Circle, radius = 1 , centre (0, 1)`

Complex Numbers, EXT2 N2 2005 HSC 2c

Sketch the region on the Argand diagram where the inequalities

`| z-overset_z | < 2` and `| z-1 | >=1`

hold simultaneously. (3 marks)

--- 12 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

`\| z-overset_z \|`	`< 2`
`\| x + i y-(x-i y) \|`	`< 2`
`\| 2 i y \|`	`< 2`
`\| y \|`	`< 1`

`| z-1 | = 1 \ => \ text{Circle, radius = 1, centre (1, 0)}`

`:.\ text(Graph:)\ | z-overset_z |<2 \ ∩ \ | z-1 | >= 1`

Complex Numbers, EXT2 N2 2019 HSC 12a

Sketch the region defined by `pi/4 <= text(arg)(z) <= pi/2` and `text(Im)(z) <= 1`. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

`text(Shaded Area): pi/4 <= text(arg)(z) <= pi/2 and text(Im)(z) <= 1`

Complex Numbers, EXT2 N2 2010 HSC 2c

Sketch the region in the complex plane where the inequalities `1 ≤ |\ z\ | ≤ 2` and `0 ≤ z + bar z ≤ 3` hold simultaneously. (2 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(See Worked Solutions.)`

Show Worked Solution

`text(Consider)\ \ \ \ `	`1≤\|\ z\ \|≤2`
	`1≤x^2+y^2≤4`

`z + bar z = (x+iy)+(x-iy)=2x`

`text(Consider)\ \ \ \ `	`0≤z + bar z≤3`
	`0≤2x≤3`
	`0≤x≤3/2`

Complex Numbers, EXT2 N2 2011 HSC 6c

On an Argand diagram, sketch the region described by the inequality

`|\ 1 + 1/z\ | <= 1.` (2 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: Substituting `z=x+iy` immediately was common and caused major algebraic problems.

`\|\ 1 + 1/z\ \|<= `	`1`
`\|\ (z + 1)/z\ \|<= `	`1`
`\|\ z + 1\ \|/\|\ z\ \|<= `	`1`
`\|\ z + 1\ \|<= `	`\|\ z\ \|`
`sqrt ((x + 1)^2 + y^2) <=`	`sqrt (x^2 + y^2)`
`(x + 1)^2 + y^2 <=`	`x^2 + y^2`
`x^2 + 2x + 1 <=`	` x^2`
`:.x <=`	`-1/2`

Complex Numbers, EXT2 N2 2011 HSC 4a

Let `a` and `b` be real numbers with `a != b`. Let `z = x + iy` be a complex number such that

`|\ z-a\ |^2-|\ z-b\ |^2 = 1.`

Prove that `x = (a + b)/2 + 1/(2 (b-a)).` (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Hence, describe the locus of all complex numbers `z` such that
`|\ z-a\ |^2-|\ z-b\ |^2 = 1.` (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `text(Proof)\ \ text{(See Worked Solutions)`

b. `x = (a + b)/2 + 1/(2(b-a))`

Show Worked Solution

a.	`\|\ z-a\ \|^2-\|\ z-b\ \|^2`	`= 1`
	`\|\ (x-a)+iy\ \|^2-\|\ (x-b)+iy\ \|^2`	`=1`
	`(x-a)^2 + y^2-((x-b)^2 + y^2)`	`=1`
	`(x-a)^2-(x-b)^2`	`=1`
	`(x-a-(x-b)) (x-a + x-b)`	`=1`
	`(b-a) (2x-a-b)`	`=1`

`2x-a-b`	`= 1/(b-a)`
`2x`	`= a + b + 1/(b-a)`
`:. x`	`= (a + b)/2 + 1/(2(b-a))`

♦ Mean mark part (ii) 44%.

b. `text(The locus is the vertical line:)`

`x = (a + b)/2 + 1/(2(b-a)).`