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Complex Numbers, EXT2 N1 2025 SPEC2 6*

Let  \(z \in C\).

Given that  \(|z|=1\)  and  \(z \neq 1,\) express  \(\operatorname{Re}\left(\dfrac{1}{1-z}\right)\) in its simplest form.   (3 marks)

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\(\operatorname{Re}\left(\dfrac{1}{1-z}\right) = \dfrac{1}{2}\)

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\(z=a+b i\)

\(\abs{z}=a^2+b^2=1\)

\(\dfrac{1}{1-z}=\dfrac{1}{1-(a+bi)}=\dfrac{1}{(1-a)-bi} \times \dfrac{(1-a)+bi}{(1-a)+bi}=\dfrac{(1-a)+bi}{(1-a)^2+b^2}\)

\(\operatorname{Re}\left(\dfrac{1}{1-z}\right)\) \(=\dfrac{-a+1}{a^2-2 a+b^2+1}\)
  \(=\dfrac{-a+1}{1-2 a+1}\)
  \(=\dfrac{-(a-1)}{-2(a-1)}\)
  \(=\dfrac{1}{2}\)

Complex Numbers, EXT2 N1 2023 HSC 1 MC

Which of the following is equal to \((a+i b)^3\)?

  1. \( (a^3-3 a b^2)+i (3 a^2 b+b^3) \)
  2. \( (a^3+3 a b^2)+i (3 a^2 b+b^3) \)
  3. \( (a^3-3 a b^2)+i (3 a^2 b-b^3) \)
  4. \( (a^3+3 a b^2)+i(3 a^2 b-b^3)\)
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\(C\)

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\((a+i b)^3\) \(=a^3+3a^2ib+3a(ib)^2+(ib)^3\)  
  \(=a^3+3a^2ib-3ab^2-ib^3\)  
  \(=(a^3-3ab^2)+i(3a^2b-b^3) \)  

 
\(\Rightarrow C\)

Complex Numbers, EXT2 N1 2020 SPEC2 8 MC

Given that  `(x + iy)^14 = a + ib`, where  `x, y, a, b ∈ R, \ (y - ix)^14`  for all values of `x` and `y` is equal to

  1. `−a - ib`
  2. `−b + ia`
  3. `−a + ib`
  4. `b + ia`
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`A`

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`(y – ix)^14` `= (−i(x + iy))^14`
  `= −i^14(x + iy)^14`
  `= −(x + iy)^14`
  `= −a – ib`

 
`=>A`

Complex Numbers, EXT2 N1 2019 HSC 8 MC

Let `z` be a complex number such that  `z^2 = -i bar z`.

Which of the following is a possible value for `z`?

  1. `1/2-sqrt 3/2 i`
  2. `1/2 + sqrt 3/2 i`
  3. `sqrt 3/2-1/2 i`
  4. `sqrt 3/2 + 1/2 i`
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`C`

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`text(Solution 1)`

`text(Consider)\ C:`

`z` `=sqrt3/2-1/2 i`  
`barz` `=sqrt3/2 + 1/2 i`  
`-i barz` `=1/2-sqrt3/2 i`  

 

`z^2` `=(sqrt3/2-1/2 i)^2`  
  `=3/4-2 sqrt3/2 * 1/2 i -1/4`  
  `=1/2-sqrt3/2 i`  

 
`=>C`

 

`text(Solution 2)`

`text(Let)\ \ text(arg)(z)` `= theta`
`text(arg)(z^2)` `= 2 theta`

 

`text(arg)(-i bar z)` `=text(arg)(i bar z)-pi`
  `= text(arg) (bar z)-pi + pi/2`
  `= -theta-pi/2`

 

`:. 2 theta` `= -theta-pi/2`
`3 theta` `= -pi/2`
`theta` `= -pi/6`

 
`=>   C`