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Complex Numbers, EXT2 N2 2022 HSC 13c

Consider the equation  , where is a complex number.

  1. Solve the equation    by finding the 5th roots of .  (2 marks)

  2. Show that if is a solution of    and  , then    is a solution of  .  (2 marks)

  3. Hence find the exact value of  .  (3 marks)

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i.   


  

ii.   


 

 
 
 
 

 


 


Mean mark (ii) 53%.

iii. 

 
   

♦ Mean mark (iii) 43%.
 
 
 
 

Complex Numbers, EXT2 N2 2021 HSC 11d

  1. Find the two square roots of  , giving the answers in the form  , where and are real numbers.  (2 marks)

  2. Hence, or otherwise, solve    giving your solutions in the form    where and are real numbers. (2 marks)

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i.     


 

 
 

 

 


 

 

ii.   

 
   
   

 

Complex Numbers, EXT2 N2 2007 HSC 2d

The points    and    on the Argand diagram represent the complex numbers    and    respectively.

The triangles    and    are equilateral with unit sides, so  

Let  ­­

  1. Explain why     (1 mark)

  2. Show that     (1 mark)

  3. Show that   and  are the roots of     (2 marks)

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i. 

 

ii. 

 

 

  ­­­­
  ­­
 

 

iii. 

 

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