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Worksheet: Representing Complex Numbers Using Argand Diagrams

Q1:

What complex number lies at the midpoint of 𝑧 1 and 𝑧 2 on the given complex plane?

  • A 5 + 4 𝑖
  • B 4 + 4 𝑖
  • C 4 + 5 𝑖
  • D 2 + 2 𝑖
  • E 8 + 1 0 𝑖

Q2:

Describe the geometric transformation that maps every complex number 𝑧 to its conjugate 𝑧 βˆ— .

  • Areflection in the line R e I m ( 𝑧 ) = ( 𝑧 )
  • Breflection in the imaginary axis
  • Creflection in the line R e I m ( 𝑧 ) = βˆ’ ( 𝑧 )
  • Dreflection in the real axis
  • Erotation by 1 8 0 ∘ about the origin

Q3:

In what quadrant does 𝑧 βˆ— lie?

  • Athe second quadrant
  • Bthe first quadrant
  • Cthe fourth quadrant
  • Dthe third quadrant

Q4:

Find the value of 𝑍 given 𝑍 on the Argand diagram below.

  • A 𝑍 = 3 βˆ’ 5 𝑖
  • B 𝑍 = 3 + 5 𝑖
  • C 𝑍 = βˆ’ 3 + 5 𝑖
  • D 𝑍 = βˆ’ 3 βˆ’ 5 𝑖

Q5:

Find the value of 𝑍 given 𝑍 on the Argand diagram below.

  • A 𝑍 = 2 𝑖
  • B 𝑍 = βˆ’ 2 𝑖
  • C 𝑍 = 2
  • D 𝑍 = βˆ’ 2

Q6:

Find the value of 𝑍 given 𝑍 on the Argand diagram below.

  • A 𝑍 = βˆ’ 7 + 7 𝑖
  • B 𝑍 = βˆ’ 7 βˆ’ 7 𝑖
  • C 𝑍 = 7 βˆ’ 7 𝑖
  • D 𝑍 = 7 + 7 𝑖

Q7:

Given that the complex number 𝑍 is represented by the point ( βˆ’ 4 , βˆ’ 4 ) on the Argand diagram below, find | 𝑍 | .

  • A | 𝑍 | = 4
  • B | 𝑍 | = 3 2
  • C | 𝑍 | = 1 6
  • D | 𝑍 | = 4 √ 2
  • E | 𝑍 | = 0

Q8:

Given that the complex number 𝑍 is represented by the point ( βˆ’ 4 , βˆ’ 3 ) on the Argand diagram below, find | 𝑍 | .

  • A | 𝑍 | = 2 √ 3
  • B | 𝑍 | = 2 5
  • C | 𝑍 | = 1 2
  • D | 𝑍 | = 5
  • E | 𝑍 | = √ 7

Q9:

Find the value of Μ„ 𝑍 given 𝑍 on the Argand diagram below.

  • A Μ„ 𝑍 = 3 βˆ’ 3 𝑖
  • B Μ„ 𝑍 = 3 + 3 𝑖
  • C Μ„ 𝑍 = βˆ’ 3 + 3 𝑖
  • D Μ„ 𝑍 = βˆ’ 3 βˆ’ 3 𝑖

Q10:

If the number 𝑍 = 8 + 𝑖 is represented on Argand diagram by the point 𝐴 , determine the Cartesian coordinates of that point.

  • A ( βˆ’ 8 , 1 )
  • B ( βˆ’ 8 , βˆ’ 1 )
  • C ( 8 , βˆ’ 1 )
  • D ( 8 , 1 )

Q11:

If the number 𝑍 = 1 1 + 3 𝑖 is represented on Argand diagram by the point 𝐴 , determine the Cartesian coordinates of that point.

  • A ( βˆ’ 1 1 , 3 )
  • B ( βˆ’ 1 1 , βˆ’ 3 )
  • C ( 1 1 , βˆ’ 3 )
  • D ( 1 1 , 3 )

Q12:

If the number 𝑍 = 1 0 + 1 8 𝑖 is represented on Argand diagram by the point 𝐴 , determine the Cartesian coordinates of that point.

  • A ( βˆ’ 1 0 , 1 8 )
  • B ( βˆ’ 1 0 , βˆ’ 1 8 )
  • C ( 1 0 , βˆ’ 1 8 )
  • D ( 1 0 , 1 8 )

Q13:

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their sum with 3 βˆ’ 2 𝑖 .

  • Aa translation by  βˆ’ 3 2 
  • Ba translation by  βˆ’ 2 3 
  • Ca translation by  2 βˆ’ 3 
  • Da translation by  3 βˆ’ 2 
  • Ea translation by  βˆ’ 3 βˆ’ 2 

Q14:

Using the Argand diagram shown, find the value of 𝑧 + 𝑧 1 2 .

Q15:

The numbers in the complex plane are mapped to their product with a particular complex number 𝑧 . Given that this transforms the complex plane by a dilation with center the origin followed by a rotation by πœ‹ radians about the origin, what kind of number is 𝑧 ?

  • Aa positive imaginary number
  • Ba positive real number
  • Ca negative imaginary number
  • Da negative real number