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Complex Numbers, EXT2 N2 2024 HSC 16b

The number    is a complex cube root of unity. The number is a cube root of .

  1. Show that    is a real root of  .   (3 marks)

  2. By using part (i) to find the exact value of  , deduce the value(s) of    for all integers  . Justify your answer.   (3 marks)

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i.     

 
   
   
   
   
   

 


 

ii.   

 
 
 

 

 
 

 

 

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i.     

♦♦ Mean mark (i) 36%.
 
   
   
   
   
   

 


 

ii.   

♦♦♦ Mean mark (ii) 18%.

 
 
 

 

 
 

 

 

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 EQ-Bank 1

  is a root of the equation  .

  1. Express    in exponential form.  (2 marks)

  2. Find the value of .  (1 mark)

  3. Find the other 4 roots of the equation in exponential form.  (3 marks)

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i.   

 

ii.   
 
   
   

 

 

iii.   


 

 

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.