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Complex Numbers, SPEC2 2024 VCAA 2

  1. Express the relation    in the form  , where  ,   and  .   (2 marks)

  2. The line segment from    to    is the diameter of a circle.
  3. Find the equation of this circle in the form , where is the centre of the circle and is the radius.   (2 marks)

  4. A second circle is given by  .
  5. Sketch this circle on the Argand diagram below, labelling the imaginary axis intercepts with their values.  (2 marks)
     

     

  1. A ray originating at the point    passes through the point  ,  cutting the second circle into two segments.
    1. Sketch the ray on the Argand diagram provided in part c.  (1 mark)

    2. Find the equation of the ray in the form   where  and is measured in radians in terms of .  (1 mark)

  3. Find the area of the minor segment formed by the intersection of the ray and the circle.  (2 marks)

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a.   

b.   

c.    

d.i.   

d.ii.   

e.   

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a.   

 

 

b.   


 


 


 

c.   


 

d.i.   

d.ii.   

Mean mark (d.ii.) 56%.

e.   

Complex Numbers, SPEC2 2024 VCAA 5 MC

If the point    is represented on an Argand diagram, the point representing    can be located by

  1. reflecting the point representing in the real axis.
  2. rotating the point representing anticlockwise about the origin by 90.
  3. reflecting the point representing in the imaginary axis.
  4. rotating the point representing clockwise about the origin by 90.
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Complex Numbers, SPEC2 2022 VCAA 2

Two complex numbers and are given by   and  , where .

  1.  i. Given that  , show that  .   (2 marks)

  2. ii. One set of possible values for and is    and  .
  3.     Hence, or otherwise, find the other set of possible values.   (1 mark)

  4. Plot and label the points representing    and    on the Argand diagram below.
     

  1. The ray given by    passes through the midpoint of the line interval that joins the points    and  .
  2. Find, in radians, the value of and plot this ray on the Argand diagram in part b.   (2 marks)

  3. The line interval that joins the points    and    cuts the circle    into a major and a minor segment.
  4. Find the area of the minor segment, giving your answer correct to two decimal places.   (2 marks)

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a.i. 

a.ii. 

b.   
       

c.   

d.   

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a.i. 

 

 

 
 
Mean mark (a) 54%.

 
a.ii.


 

b.   
       


c.    
♦ Mean mark (c) 39%.

d.   

♦ Mean mark (d) 40%.

Complex Numbers, SPEC2 2021 VCAA 2

The polynomial  , where    and  , can also be written as  , where    and  .

  1.  i. State the relationship between and .  (1 mark)

  2. ii. Determine the values of and , given that    and  .  (3 marks)

Consider the point  .

  1. Sketch the ray given by    on the Argand diagram below.  (2 marks)
     
  2. The ray    intersects the circle  , dividing it into a major and a minor segment.
  3.  i. Sketch the circle  on the Argand diagram in part b.  (1 mark)

  4. ii. Find the area of the minor segment.  (2 marks)

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  2.  
     

     

c.i.   

c.ii.   ²

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a.i.   

 

a.ii.   

♦♦ Mean mark part (a.ii.) 35%.


 

 

 
 

 

♦ Mean mark part (b) 45%.
MARKER’S COMMENT: Point of emanation is not part of required ray (see open circle).

 

b. 

 

c.i.  

 

c.ii.   

♦♦ Mean mark part (c.ii.) 30%.

 
  ²

Complex Numbers, SPEC2 2012 VCAA 2

  1.  Given that  , show that  .   (2 marks)

  2. Express    in polar form.   (1 mark)

  3. i.  Write down    in polar form.   (1 mark)

  4. ii. On the Argand diagram below, shade the region defined by

.   (2 marks)


 

  1.  Find the area of the shaded region in part c.   (2 marks) 
  2. i.  Find the value(s) of such that  , where  .   (3 marks)

  3. ii. Find    for the value(s) of found in part i.   (1 mark)

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  2. i. 

     

    ii. 

  5. i. 

     

    ii. 

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a.   
 
   

 

 

b.i.   
   

 

b.ii.   
   

 

c.   

 

d. 

Mean mark 51%.


  

 


 

♦ Mean mark (e)(i) 34%.

e.i.   
 
 
 
 

 

e.ii. 

♦ Mean mark (e)(ii) 36%.

 
   
   

 

Complex Numbers, SPEC2 2012 VCAA 6 MC

For any complex number , the location on an Argand diagram of the complex number    can be found by

A.   rotating through  in an anticlockwise direction about the origin

B.   reflecting about the -axis and then reflecting about the -axis

C.   reflecting about the -axis and then rotating anticlockwise through  about the origin

D.   reflecting about the -axis and then rotating anticlockwise through  about the origin

E.   rotating through in a clockwise direction about the origin

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♦♦ Mean mark 37%.

 
    ✔