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Complex Numbers, EXT2 N1 2023 HSC 7 MC

Which of the following statements about complex numbers is true?

  1. For all real numbers   with  ,

, for some real number .

  1. For all non-zero complex numbers and ,

  and 

where denotes the principal argument.

  1. For all real numbers   with  ,

  and 

  1. For all real numbers   with    and  ,

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♦♦♦ Mean mark 18%.


 

Complex Numbers, EXT2 N1 2004 HSC 2b

Let    and  .

  1. Find  , in the form  .   (1 mark)

  2. Express in modulus-argument form.   (3 marks)

  3. Given that has the modulus-argument form
     
         .
     
    find the modulus-argument form of  .   (1 mark)

  4. Hence find the exact value of     (1 mark)

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i.    
   
   
   

 

ii.  
 

 

iii.  
 
   

 

iv. 

 

Complex Numbers, EXT2 N1 2008 HSC 2b

  1. Write    in the form  , where and are real.  (2 marks)

  2. By expressing both    and    in  modulus-argument form, write    in modulus-argument form.   (3 marks)

  3. Hence find    in surd form.  (1 mark)

  4. By using the result of part (ii), or otherwise, calculate  .   (1 mark)

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i.     
   
   

 

ii.   


  


 


 

 
 

 

iii. 

 

 

iv.     
   
   

Complex Numbers, EXT2 N1 2015 HSC 12a

The complex number is such that  and  

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

  1.     (1 mark)

  2.     (1 mark)

  3.     (1 mark)

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i.   ­
­

 

ii.   ­
­
­

COMMENT: 12a(iii) had the lowest mean mark (55%) of any part within Q12 and deserves attention.
iii.  ­
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Complex Numbers, EXT2 N1 2015 HSC 11b

Consider the complex numbers    and  

  1. Evaluate     (1 mark)

  2. Evaluate     (1 mark)

  3. Find the argument of     (1 mark)

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i.   
 

 

ii.   ­
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iii.   ­
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