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Complex Numbers, SPEC2 2020 VCAA 2

Two complex numbers, and , are defined as    and  .

  1. Express the relation    in the cartesian form  , where  .   (3 marks)

  2. Plot the points that represent and and the relation on the Argand diagram below.   (2 marks)

     
         
     

  3. State a geometrical interpretation of the graph of    in relation to the points that represent and .   (1 mark)

  4.  i. Sketch the ray given by    on the Argand diagram in part b.   (1 mark)

  5. ii. Write down the function that describes the ray  , giving the rule in cartesian form.   (1 mark)

  6. The points representing and and    lie on the circle given by  , where is the centre of the circle and is the radius.
  7. Find in the form  , where  , and find the radius .   (3 marks)

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  2.  
  4.  i.
  5. ii.
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a.   

 

b.   

 

c.   

 

d.i.   

♦♦ Mean mark (d.ii.) 25%.

 

d.ii.   

 

e.   

♦ Mean mark (e) 40%.

 


 

 
 

Complex Numbers, SPEC2-NHT 2019 VCAA 1

In the complex plane, is the with equation .

  1.  Verify that the point (0, 0) lies on .   (1 marks)

  2.  The line    can also be expressed in the form  , where  .

     

     Find  in cartesian form.   (2 marks)

  3.  Find, in cartesian form, the points(s) of intersection of    and the graph of  .   (2 marks)

  4.  Sketch    and the graph of    on the Argand diagram below.   (2 marks)

     
     
         
     

  5.  Find the area of the sector defined by the part of    where  , the graph of    where  , and imaginary axis where  .   (1 marks)

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  5.  

     
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a.   

 

b. 
 
 
 
 
 
 


c.


 

 

d.   

 

e.


 

f.    

 

Complex Numbers, SPEC2 2017 VCAA 4

  1. Express    in polar form.  (1 mark)

  2. Show that the roots of    are    and  .  (1 mark)

  3. Express the roots of    in terms of  .  (1 mark)

  4. Show that the cartesian form of the relation    is    (2 marks)

  5. Sketch the line represented by    and plot the roots of    on the Argand diagram below.  (2 marks)

     
       

  6. The equation of the line passing through the two roots of    can be expressed as  , where  .

     

    Find in terms of .  (1 mark)

  7. Find the area of the major segment bounded by the line passing through the roots of    and the major arc of the circle given by  .  (2 marks)

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  5.  
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a.   

 

 
 

 

 

b.   

 

♦♦ Mean mark part (c) 31%.

c.   
 

 

d.
    
   
 
 

 

 

e.   

 

f.   

♦♦♦ Mean mark 1%!


 

  

 
 

 

♦♦ Mean mark 31%.

g.   
   

 

 

 

Complex Numbers, SPEC2-NHT 2018 VCAA 2

In the complex plane, is the line given by  .

  1. Show that the cartesian equation of is given by  .   (2 marks)

  2.  Find the point(s) of intersection of and the graph of the relation    in cartesian form.   (2 marks)

  3. Sketch and the graph of the relation    on the Argand diagram below.   (2 marks)

 

 

The part of the line in the fourth quadrant can be expressed in the form  .

  1. State the value of .   (1 mark)

  2. Find the area enclosed by and the graphs of the relations    and  .   (2 marks)

  3. The straight line can be written in the form , where  .

     

    Find in the form  , where    is the principal argument of .   (2 marks)

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Show Worked Solution

a.   

 

b.  
 
 

 

 

 

c.   

 

d.  


 

e.   

²

 

f.