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Complex Numbers, EXT2 N2 2021 HSC 14c

  1. Using de Moivre’s theorem and the binomial expansion of , or otherwise, show that
  2.    .  (3 marks)

  3. By using part (i), or otherwise, show that  .  (3 marks)

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i.   


 

 
 
 
 

 

ii.   

♦♦ Mean mark 31%.

  
 
 

 

Complex Numbers, EXT2 N2 2020 HSC 14a

Let be a complex number and let 

The diagram shows points and which represent and , respectively, in the Argand plane.
 


 

  1. Explain why triangle    is an equilateral triangle.   (2 marks)

  2. Prove that  .  (3 marks)

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i.

 

    
 
   

 

 

ii.    
 
 
 
 
 

COMMENT: The identity to prove suggests the addition of 2 cubes is a possible strategy.

Complex Numbers, EXT2 N2 2018 HSC 15b

  1. Use De Moivre's theorem and the expansion of to show that
     

     
                          (2 marks)

  2. Hence, show that
     
    .  (3 marks)

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i.   


 

 


 

ii.   
   

 

Complex Numbers, EXT2 N2 2016 HSC 12c

Let 

  1. By considering the real part of  , show that    is
     
      (2 marks)

     

  2. Hence, or otherwise, find an expression for    involving only powers of   (1 mark)

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i.  

 

 

 


 

 

ii.   
   
   

Complex Numbers, EXT2 N2 2007 HSC 8b

  1. Let    be a positive integer. Show that if    then
     
        .  (2 marks)

  2. By substituting    where  , into part (i), show that
     
       
     
              (3 marks)

  3. Suppose  .  Using part (ii), show that
     
          (2 marks)

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i.  

 
 
 

 

ii.   
   
 
   

 

 
 
 
   

 
 
 

 

iii. 

Complex Numbers, EXT2 N2 2009 HSC 7b

Let  

  1. Show that  , where    is a positive integer.  (2 marks)

  2. Let    be a positive integer. Show that
     

     
          (3 marks)

  3. Hence, or otherwise, prove that
     
       
     
    where    is a positive integer.  (2 marks)

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i.   
 
 
   
 
   

 

 

ii. 

 

iii. 

Complex Numbers, EXT2 N2 2011 HSC 2d

  1. Use the binomial theorem to expand    (1 mark)

  2. Use de Moivre’s theorem and your result from part (i) to prove that
     
          (3 marks)

  3. Hence, or otherwise, find the smallest positive solution of
     
          (2 marks)

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i.   

 

ii. 


 

 

 

iii.