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Complex Numbers, EXT2 N2 2025 HSC 12c

Sketch the region of the complex plane defined by  \(\abs{z+5-i}>\abs{z-3+3 i}\).   (3 marks)

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\(\abs{z+5-i}=\abs{z-(-5+i)}\)

\(\abs{z-3+3 i}=\abs{z-(3-3 i)}\)

\(\text{Find equation of} \ \perp \ \text{bisector between}\ \ (-5,1) \ \ \text{and}\ \ (3,-3):\)

\(\text{Midpoint} \equiv \left(\dfrac{-5+3}{2}, \dfrac{1-3}{2}\right) \equiv (-1,-1)\)

\(m=\dfrac{-3-1}{3+5}=-\dfrac{1}{2} \ \Rightarrow \ m_{\perp}=2\)
 

\(\text{Equation of line} \ \ m=2 \ \ \text{through}\ \ (-1,-1):\)

\(y+1\) \(=2(x+1)\)
\(y\) \(=2 x+1\)

 
\(\text{Sketch:}\ \abs{z+5-i}>\abs{z-3+3 i}\)

Complex Numbers, EXT2 N2 2024 HSC 7 MC

It is given that \(\abs{z-1+i}=2\).

What is the maximum possible value of \(\abs{z}\)?

  1. \(\sqrt{2}\)
  2. \(\sqrt{10}\)
  3. \(2+\sqrt{2}\)
  4. \(2-\sqrt{2}\)
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\(C\)

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\(\abs{z-1+i}=2 \ \Rightarrow \ \text{circle centre} \ \ (1,-i), \ \ \text {radius}=2\)
 
 

\(OA=\sqrt{1^2+1^2}=\sqrt{2}\)

\(AB=2 \ \ \text{(radius)}\)

\(\therefore \abs{z}_{\text{max}}=2+\sqrt{2}\)

\(\Rightarrow C\)

Complex Numbers, EXT2 N2 2024 HSC 11f

Sketch the region defined by  \(|z|<3\)  and  \(0 \leq \arg (z-i) \leq \dfrac{\pi}{2}\).   (3 marks)

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\(\text {Region: }\abs{z}<3\ \ \text{and}\ \ 0 \leqslant \arg (z-i) \leqslant \dfrac{\pi}{2}\)
 

Complex Numbers, EXT2 N2 2023 HSC 8 MC

A shaded region on a complex plane is shown.
 

Which relation best describes the region shaded on the complex plane?

  1. \(|z-i|>2|z-1|\)
  2. \(|z-i|<2|z-1|\)
  3. \(|z-1|>2|z-i|\)
  4. \(|z-1|<2|z-i|\)
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\(D\)

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\(\text{By elimination:}\)

\(\text{Let}\ \ z=x+iy \ \ (x,y \in \mathbb{R}) \)

\(\text{Since shaded area is outside the circle, coefficients of}\ x^2\ \text{and} \)

\( y^2\ \text{must be positive (eliminate A and C).}\)

♦♦ Mean mark 34%.

\(\text{Consider option D:}\)

\(|z-1|\) \(<2|z-i|\)  
\((x-1)^2+y^2\) \(<4(x^2+(y-1)^2) \)  
\(x^2-2x+1+y^2\) \(<4x^2+4y^2-8y+4\)  
\(0\) \(<3x^2+2x+3y^2-8y+3\)  

 
\(\text{Circle equation has centre where}\ \ x<0, y>0. \)

\(\Rightarrow D\)

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`?
 


 

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`A`

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`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 2021 SPEC2 5

The graph of the circle given by  `|z-2-sqrt3i| = 1`, where  `z ∈ C`, is shown below.
 

For points on this circle, find the maximum value of  `|z|`.   (2 marks)

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`sqrt7+1`

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`text(Centre of circle at)\ (2, sqrt3)`

`text(Radius = 1)`

`text(By Pythagoras, line from origin to the centre of the circle)`

`d = sqrt(2^2 + sqrt(3)^2) = sqrt7`

`:. |z|_text(max) = sqrt7 + 1`

Complex Numbers, EXT2 N2 2021 HSC 16c

Sketch the region of the complex plane defined by  `text{Re}(z) ≥ text{Arg}(z)`  where `text{Arg}(z)` is the principal argument of `z`.   (3 marks)

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`text{Find region where} \ \ text{Re}(z) ≥ text{Arg}(z).`

♦♦♦ Mean mark 15%.
`text{In quadrant 1:} \ x ≥ 0 \ , \ y ≥ 0 \ , \ 0 ≤ text{Arg}(z)≤ pi/2`
`x` `≥ tan^-1 (y/x)`  
`x tan (x)` `≥ y`  

  
`text(If)\ \ x>=pi/2, \ x>=tan^-1(y/x)\ \ text(as)\ \ tan^-1(y/x)<pi/2` 

`text{Arg}(0)\ text(is not defined.)`
 

`text{In quadrant 2:} \ \ x ≤ 0 \ , \  y ≤ 0 \ , \ pi/2 ≤ text{Arg}(z ) ≤ pi`

`text{Re}(z) < text{Arg}(z ) \ => \ text{no points satisfy}`
 

`text{In quadrant 3:} \ \ x ≤ 0 \ , \  y ≤ 0 \ , \ -pi ≤ text{Arg}(z) ≤ -pi/2`

`text{Arg}(z)` `= -pi + tan^-1 (y/x)`
`x` `≥ -pi + tan^-1 (y/x)`
`tan^-1 (y/x)` `≤ pi + x`
`y/x` `≤ tan (pi + x)`
`y` `≥ xtan (pi + x)`

 

`text{In the domain} \ \ -pi/2 <= x <= 0,`

`text{Arg}(z) = -pi+ tan^-1(y/x) < -pi/2\ \ \ (text{s}text{ince}\ \ tan^-1(y/x)<pi/2)`

`=>\ text(all points satisfy for)\ \ \ -pi/2 <= x <= 0`
 

`text{In quadrant 4:} \ \ x ≥ 0 \ , \  y ≤ 0 \ , \ -pi/2 ≤ text{Arg}(z) ≤ 0`

`text{Re}(z) > text{Arg}(z) \ => \ text{all points satisfy}`

 

Complex Numbers, EXT2 N2 2020 SPEC2 2

Two complex numbers, `u` and `v`, are defined as  `u = −2-i`  and  `v = −4-3i`.

  1. Express the relation  `|z-u| = |z-v|`  in the cartesian form  `y = mx + c`, where  `m, c ∈ R`.   (3 marks)

  2. Plot the points that represent `u` and `v` and the relation `|z-u| = |z-v|` on the Argand diagram below.   (2 marks)
     
         
     
  3. State a geometrical interpretation of the graph of  `|z-u| = |z-v|`  in relation to the points that represent `u` and `v`.   (1 mark)

  4.  Sketch the ray given by  `text(Arg)(z-u) = pi/4`  on the Argand diagram in part b.   (1 mark)

  5. In Cartesian form, write down the function that describes the ray  `text(Arg)(z-u) = pi/4`.   (1 mark)
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a.    `y = −x-5`

b.    
       

c.   `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`

`text(line joining)\ u and v.`

d.    

       

e.    `f: (−2, ∞) ->, f(x) = x + 1`

Show Worked Solution

a.   `text(Let)\ \ z = x + iy`

`z-u = x + 2 + iy + i`

`z-v = x + 4 + iy + 3i`

`|z-u| = |z-v|`

`(x + 2)^2 + (y + 1)^2` `= (x + 4)^2 + (y + 3)^2`
`x^2 + 4x + 4 + y^2 + 2y + 1` `= x^2 + 8x + 16 + y^2 + 6y + 9`
`-4y` `= 4x + 20`
`y` `= −x-5`

 

b.   

 

c.   `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`

`text(line joining)\ u and v.`

 

d.   

 

e.    `text(Arg)(z-u) = pi/4 =>\ text(gradient) = 1, ytext(-intercept at)\ (0, 1)`

`:. f: (−2, ∞) -> RR, \ f(x) = x + 1`

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)

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`| 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)

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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 2018 HSC 7 MC

Which diagram best represents the solutions to the equation  `text(arg)(z) = text(arg)(z + 1-i)`?

A. B.
C. D.
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`D`

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`text(arg)(z)` `= text(arg)(z + 1-i)`
  `=text(arg)(z-(−1 + i))`

 
`=>\ text(arg)(z-(−1 + i))\ \ text(is the argument of)\ z\ text(from)\ (-1+i).`

 
`text(Plot)\ (-1 + i)\ text(on the argand diagram and then test different)`

`text(positions of)\ z\ text(along the solutions for each option.)`

`=>D`

Complex Numbers, EXT2 N2 2017 HSC 11c

Sketch the region in the Argand diagram where

    `-pi/4 <= text(arg)(z) <= 0 and |z-1 + i| <= 1`.  (2 marks)

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Show Worked Solution

`|z-1 + i| = 1\ \ text{is a circle with centre (1, –1 )}`

`text{and radius 1.}`

`text(Shaded area:)\ -pi/4 <= text(arg)(z) <= 0\  ∩\  |z-1 + i| <= 1`

Complex Numbers, EXT2 N2 2017 HSC 3 MC

Which complex number lies in the region  `2 < |z-1| < 3`?

  1. `1 + sqrt 3 i`
  2. `1 + 3i`
  3. `2 + i`
  4. `3-i`
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`D`

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`|z-1| = 2\ \ text(is a circle with centre)\ \ (1, 0),\ text(radius 2)`

`|z-1| = 3\ \ text(is a circle with centre)\ \ (1, 0),\ text(radius 3)`

`:.\ text(Only)\ \ 3-i\ \ text(lies within the two circles,)`

`text(i.e. satisfies)\ \ 2 < |z-1| < 3`

`=>  D`

Complex Numbers, EXT2 N2 2007 HSC 2c

The point  `P`  on the Argand diagram represents the complex number `z`, where  `z`  satisfies

    `1/z + 1/bar z = 1.`

Give a geometrical description of the locus of `P` as `z` varies.   (3 marks)

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`text(Locus is a circle, centre)\ (1, 0), text(radius 1,)`

`text(excluding the point)\ (0, 0).`

Show Worked Solution
`1/z + 1/bar z` `=1/(x + iy) +1/(x-iy)`
  `=(x-iy + x+iy)/(x^2+y^2)`
  `=(2x)/(x^2+y^2)`

 

`(2x)/(x^2+y^2)` `=1`
`x^2 + y^2` `=2x`
`x^2-2x + 1 + y^2` `=1`
`(x-1)^2 + y^2` `=1`

 

`:.\ text(Locus is a circle, centre)\ (1, 0), text(radius 1,)`

`text(excluding the point)\ (0, 0).`

Complex Numbers, EXT2 N2 2015 HSC 9 MC

The complex number `z` satisfies  `| z-1 | = 1.`

What is the greatest distance that `z` can be from the point `i` on the Argand diagram?

  1. `1`
  2. `sqrt 5`
  3. `2 sqrt 2`
  4. `sqrt 2 + 1`
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`D`

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HSC 2015 9MC

`| z-1 | = 1\ \ text{is a circle, centre (1,0), radius 1.}`

`text(Consider the graph:)`

`text(Distance from)\ \(0, i)\ \ text(to the centre)=sqrt2`

`:.\ text(Greatest distance)=sqrt2 + text(radius)=sqrt2+1`

`=>  D`

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)

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`text(See Worked Solutions.)`

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`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 2010 HSC 2c

Complex Numbers, EXT2 N2 2011 HSC 6c

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

    `|\ 1 + 1/z\ | <= 1.`   (2 marks)

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♦♦♦ 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.` 

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

  2. Hence, describe the locus of all complex numbers  `z`  such that  
  3.     `|\ z-a\ |^2-|\ z-b\ |^2 = 1.`   (1 mark)

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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)).`

Complex Numbers, EXT2 N2 2012 HSC 11b

Shade the region on the Argand diagram where the two inequalities

    `|\ z + 2\ | ≥ 2`  and  `|\ z-i\ | ≤ 1`  both hold.   (2 marks)

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Complex Numbers, EXT2 2012 HSC 11b Answer

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`|\ z + 2\ | ≥ 2  and  |\ z-i\ | ≤ 1`

`(x + 2)^2 + y^2` `≥ 4`
`x^2 + (y-1)^2` `≤ 1`

Complex Numbers, EXT2 2012 HSC 11b Answer 

Complex Numbers, EXT2 N2 2014 HSC 11c

Sketch the region in the Argand diagram where `|\ z\ | ≤ |\ z-2\ |`  and  `−pi/4 ≤ text(arg)\ z ≤ pi/4`.   (3 marks)

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`text(See Worked Solutions.)`

Show Worked Solution

`text(Let)\ \ z=x+iy`

`|\ x + iy\ |` `≤ |\ x-2 + iy\ |`
`sqrt(x^2 + y^2)` `≤ sqrt((x-2)^2 + y^2)`
`x^2 + y^2` `≤ x^2-4x + 4 + y^2`
`4x-4` `≤ 0`
`x` `≤ 1`

 

`−pi/4 ≤ text(arg)\ z ≤ pi/4`
 

Complex Numbers, EXT2 2014 HSC 11c Answer4