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Complex Numbers, EXT2 N2 2023 HSC 16c

The complex numbers \(w\) and \(z\) both have modulus 1, and  \(\dfrac{\pi}{2} \lt \text{Arg} \Big{(}\dfrac{z}{w}\Big{)} \lt \pi\), where \(\text{Arg}\) denotes the principal argument.

For real numbers \(x\) and \(y\), consider the complex number \(\dfrac{xz+yw}{z}\).

On an \(xy\)-plane, clearly sketch the region that contains all points \((x,y)\) for which \(\dfrac{\pi}{2} \lt \text{Arg} \Big{(}\dfrac{xz+yw}{z}\Big{)} \lt \pi\).

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\(\text{One strategy:}\)

\(\text{Let}\ \ u=\dfrac{z}{w},\ \text{where}\ \abs{z} = \abs{w} = 1\ \ \Rightarrow \ \ \abs{u}=1. \)

\(\dfrac{\pi}{2} \lt \text{Arg} (u) \lt \pi\ \text{(given)}\ \ \Rightarrow \text{Re}(u) \lt 0\ \ \text{and}\ \ \text{Im}(u) \gt 0 \)

\(\text{Using}\ \ \dfrac{z}{w} = \dfrac{1}{u} = \dfrac{\bar u}{\abs{u}} = \bar u, \)

\(\text{find all real numbers}\ x\ \text{and}\ y\ \text{such that:} \)

\(\dfrac{\pi}{2} \lt \text{Arg} \Big{(}\dfrac{xz+yw}{z}\Big{)} \lt \pi\)

\(\dfrac{\pi}{2} \lt \text{Arg} (x+ \bar y u) \lt \pi\)

\(\text{i.e.}\ \ \text{Re}(x+y \bar u) = x + y \times \text{Re}(u) \lt 0\ \ \text{and}\ \ \text{Im}(y \bar u) \gt 0 \)

  \(\therefore y \lt 0\ \ \text{and} \)

\(x \lt -y \times \text{Re}(u) = -y \times \cos (\text{Arg}(u)) = -y \times \cos \Bigg{(}\text{Arg}\Big{(} \dfrac{z}{w} \Big{)} \Bigg{)} \)

\(\text{Region is all real}\ (x,y)\ \text{that lie below the}\ x\text{-axis and to the} \)

\(\text{left of the line}\ \ y=-\sec \Bigg{(}\text{Arg}\Big{(} \dfrac{z}{w} \Big{)} \Bigg{)}x \)

♦♦♦ Mean mark 3%.
COMMENT: On the podium for the toughest questions in Ext2 in the last decade.

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 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 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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`| 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 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 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 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.)`

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`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