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Complex Numbers, EXT2 N1 2015 HSC 12a

The complex number `z` is such that `|\ z\ |=2`  and  `text(arg)(z) = pi/4.`

Plot each of the following complex numbers on the same half-page Argand diagram.

  1.  `z`   (1 mark)

  2.  `u = z^2`   (1 mark)

  3.  `v = z^2-bar z`   (1 mark)

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a.    `text(See Worked Solutions)`

b.    `text(See Worked Solutions)`

c.    `text(See Worked Solutions)`

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a.    `z ­=2 text(cis) pi/4=sqrt 2 (1 + i)`
 

b.    `u=z^2=4 text(cis)\ pi/2=4i`
 

COMMENT: Mean mark (c) 55%.
c.    `v ­` `=z^2-bar z`
  `=4i-sqrt 2 (1-i)`
  `=- sqrt 2 + (4 + sqrt 2) i`

Complex Numbers, EXT2 N1 2014 HSC 4 MC

Given  `z = 2(cos\ pi/3 + i sin\ pi/3)`, which expression is equal to  `(bar {:z:})^(−1)`?

  1. `1/2(cos\ pi/3 − i sin\ pi/3)`
  2. `2(cos\ pi/3 − i sin\ pi/3)`
  3. `1/2(cos\ pi/3 + i sin\ pi/3)`
  4. `2(cos\ pi/3 + i sin\ pi/3)` 
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`C`

Show Worked Solution
`z` `= 2text(cis)(pi/3)`
`barz` `= 2text(cis)(−pi/3)`
`(barz)^(−1)` `= (2text(cis)(−pi/3))^(−1)`
  `= 2^(−1)text(cis)(pi/3)`
  `= 1/2text(cis)(pi/3)`
  `= 1/2(cos\ pi/3 + i sin\ pi/3)`

 
`=> C`

Complex Numbers, EXT2 N1 2013 HSC 3 MC

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


 

Which diagram best represents  `z^2?`

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`D`

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`text(Consider)\ \ z\ \ text(in polar form:)`

`|\ z\ |` `< 1`
`:.|\ z^2\ |` `< |\ z\ |`

 

`text(arg) (z^2) = 2text(arg) (z)`

`=>  D`