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Complex Numbers, EXT2 N1 2025 HSC 7 MC

The complex number \(z\) lies on the unit circle.
 

What is the range of \(\operatorname{Arg}(z-2 i)\) ?

  1. \(\dfrac{\pi}{6} \leq \operatorname{Arg}(z-2 i) \leq \dfrac{5 \pi}{6}\)
  2. \(\dfrac{\pi}{3} \leq \operatorname{Arg}(z-2 i) \leq \dfrac{2 \pi}{3}\)
  3. \(-\dfrac{5 \pi}{6} \leq \operatorname{Arg}(z-2 i) \leq-\dfrac{\pi}{6}\)
  4. \(-\dfrac{2 \pi}{3} \leq \operatorname{Arg}(z-2 i) \leq-\dfrac{\pi}{3}\)
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\(D\)

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\(\text{The limits of Arg}(z-2i)\ \where it intersects the unit circle are:}\)
 

\(\sin \theta=\dfrac{1}{2} \ \Rightarrow \ \theta=\dfrac{\pi}{6}\)

\(\text {Range Arg}(z-2 i):\)

\(-\dfrac{\pi}{2}-\dfrac{\pi}{6} \leqslant \operatorname{Arg}(z-2 i) \leqslant-\dfrac{\pi}{2}+\dfrac{\pi}{6}\)

\(-\dfrac{2 \pi}{3} \leqslant \operatorname{Arg}(2-2 i) \leqslant-\dfrac{\pi}{3}\)

\(\Rightarrow D\)

Complex Numbers, EXT2 N1 2024 VCAA 5 MC

If the point  \(z=1+\sqrt{3} i\)  is represented on an Argand diagram, the point representing  \(-\bar{z}\)  can be located by

  1. reflecting the point representing \(z\) in the real axis.
  2. rotating the point representing \(z\) anticlockwise about the origin by 90\(^{\circ}\).
  3. reflecting the point representing \(z\) in the imaginary axis.
  4. rotating the point representing \(z\) clockwise about the origin by 90\(^{\circ}\).
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\(C\)

Show Worked Solution

\(z=1+\sqrt{3}i\ \ \Rightarrow\ \ \bar{z} = 1-\sqrt{3}i\)

\(-\bar{z} = -1+\sqrt{3}i\)

\(\therefore \text{It is a reflection of}\ z\ \text{in the imaginary axis.}\)

\(\Rightarrow C\)

Complex Numbers, EXT2 N1 2020 HSC 4 MC

The diagram shows the complex number `z` on the Argand diagram.
 


 

Which of the following diagrams best shows the position of  `frac{z^2}{|z|}`?
 

 

 

  
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`A`

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`text{Let} \ \ z = r\ text(cis)\ theta`

`z^2` `= r^2  text(cis)\ (2 theta)`
`|z|` `= r`
`therefore  frac{z^2}{|z|}` `= frac{r^2 \ text(cis)\ (2 theta)}{r}`
  `= r\ text(cis)\ (2 theta)`

 
`text{On Argand diagram, it lies on the dotted line`

`text{(modulus the same) with an argument that is}`

`text{doubled.}`
  

`=> \ A`

Complex Numbers, EXT2 N1 2016 HSC 4 MC

The Argand diagram shows the complex numbers `z` and `w`, where `z` lies in the first quadrant and `w` lies in the second quadrant.
  

ext2-hsc-2016-4mc
 

Which complex number could lie in the 3rd quadrant?

  1. `-w`
  2. `2 iz`
  3. `bar z`
  4. `w - z`
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`=> D`

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`text(Using the parallelogram method:)`
 

ext2-hsc-2016-4mc-answer1
 

`text(From the graph above,)`

`w – z\ \ text(could lie in 3rd quadrant.)`

`=> D`

Complex Numbers, EXT2 N1 2014 HSC 8 MC

The Argand diagram shows the complex numbers  `w`, `z`  and  `u`, where  `w`  lies in the first quadrant, `z`  lies in the second quadrant and  `u`  lies on the negative real axis.
 

Complex Numbers, EXT2 2014 HSC 8 MC
 

Which statement could be true?

  1. `u = zw`  and  `u = z + w`
  2. `u = zw`  and  `u = z − w`
  3. `z = uw`  and  `u = z + w`
  4. `z = uw`  and  `u = z − w` 
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`B`

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`text(Using the parallelogram method, it could be true that)`

`u = z − w`

`=>\ text{Eliminate (A) and (C)}`
 

Complex Numbers, EXT2 2014 HSC 8 MC Answer
 

`text{arg}(u)` `=\ text{arg}(w) + text{arg}(z)`
`u` ` = zw`

 
`=> B`

Complex Numbers, EXT2 N1 2009 HSC 2c

The points  `P`  and  `Q`  on the Argand diagram represent the complex numbers  `z`  and  `w` respectively.
 


 

Copy the diagram into your writing booklet, and mark on it the following points:

  1. the point  `R`  representing  `iz.`   (1 mark)
  2. the point  `S`  representing  `bar w.`   (1 mark)
  3. the point  `T`  representing  `z + w.`   (1 mark)

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i, ii, and iii.

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i, ii, and iii.