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Complex Numbers, EXT2 N2 2021 HSC 11d

  1. Find the two square roots of  `−i` , giving the answers in the form  `x + iy`, where `x` and `y` are real numbers.  (2 marks)

  2. Hence, or otherwise, solve  `z^2 + 2z + 1 + i = 0`  giving your solutions in the form  `a + i b`  where `a` and `b` are real numbers. (2 marks)

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  1. `z = 1/sqrt2 – 1/sqrt2 i \ \ text{or} \ \ z = – 1/sqrt2+ 1/sqrt2 i`
  2. `z = -1 + 1/sqrt2 – 1/sqrt2 i \ \ text{or} \ \ z = -1 – 1/sqrt2+ 1/sqrt2 i`
Show Worked Solution

i.     `text{Solution 1}`

`z = x + iy \ , \ z^2 = -i`

`z^2 = x^2 – y^2 + 2x yi = -i`

`x^2 – y^2 = 0 \ …\ (1)`

`2xy = -1 \ …\ (2)`

`x= ± y \ …\ (1)′`
 

`text{Substitute} \ \ x = y \ \ text{into} \ (2)`

`2x^2 = -1 \ -> \ text{no real solutions}`

`text{Substitute} \ \ x= -y \ \ text{into} \ (2)`

`-2x^2` `= -1`  
`x` `= ± 1/sqrt2`  

 
`:. \ z = 1/sqrt2 – 1/sqrt2 i \ \ text{or} \ \ z = – 1/sqrt2+ 1/sqrt2 i`
 

`text{Solution 2}`

`z = re^{i theta} \ , \ z^2 = -i`

`r^2 e^{i2 theta} = e^{i {3pi}/{2}} \ \ text{or} \ \ e^{-i pi/2}`

`=> \ r = 1 \ , \  theta = {3pi}/{4} \ \ text{or} \ \ – pi/4`
 

`z= text{cos} {3pi}/{4} + i \ text{sin} {3pi}/{4} = – 1/sqrt2 + 1/sqrt2 i`

`z= text{cos} (- pi/4) + i \ text{sin} (- pi/4) = 1/sqrt2 – 1/sqrt2 i`

 

ii.    `z^2 + 2z + 1 + i = 0`

`z` `= {2 ± sqrt{4 – 4 * 1 * (1 + i)}}/{2}`  
  `= {-2 ± sqrt{4 – 4 – 4 \ i}}/{2}`  
  `= -1 ± sqrt(-i)`  

 
`:. \ z = -1 + 1/sqrt2 – 1/sqrt2 i \ \ text{or} \ \ z = -1 – 1/sqrt2 + 1/sqrt2 i`

Complex Numbers, EXT2 N1 EQ-Bank 3

Find the values of `z`, in the form  `z = x + iy`, such that

`z = sqrt(-15 + 8i)`  (2 marks)

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`z= 1 + 4i\ \ text(or)\ \ z= -1 -4i`

Show Worked Solution
`z` `= sqrt(-15 + 8i)`
`z^2` `= -15 + 8i`
`-15 + 8i` `= (x + iy)^2`
  `= x^2-y^2 + 2xyi`

 

`x^2-y^2` `= -15\ …\ (1)`
`2xy` `= 8`
`xy` `= 4\ …\ (2)`

 
`x=1,\ \ y=4`

`x=-1,\ \ y=-4`
 

`:. z_1` `= 1 + 4i`
`z_2` `= -1-4i`

Complex Numbers, EXT2 N1 EQ-Bank 2

Find the values of `z`, in the form  `z = a + ib`, such that

`z = sqrt(7 + 24i)`  (2 marks)

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`z = 4 + 3i\ \ text(or)\ \ z=-4-3i`

Show Worked Solution
`z` `= sqrt(7 + 24i)`
`z^2` `= 7 + 24i`
`7 + 24i` `= (a + ib)^2`
  `= a^2 – b^2 + 2abi`

 

`a^2 – b^2` `= 7\ …\ (1)`
`2ab` `= 24`
`ab` `= 12\ …\ (2)`

 
`a=4,\ \ b=3`

`a=-4,\ \ b=-3`
 

`:. z_1` `= 4 + 3i`
`z_2` `= −4 – 3i`