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Worksheet: Graphing Complex Numbers

Q1:

What does the modulus of a complex number represent?

  • A the angle it makes with the positive imaginary axis
  • B the angle it makes with the positive real axis
  • C its real coordinate in the complex plane
  • D its distance from the origin in the complex plane
  • E its imaginary coordinate in the complex plane

Q2:

Given that | 𝑍 | = | 𝑍 + 6 | , determine the real part of the complex number 𝑍 .

  • A6
  • B3
  • C βˆ’ 6
  • D βˆ’ 3

Q3:

Let us consider a complex number, 𝑧 , with nonzero real and imaginary parts.

If the real and imaginary parts of 𝑧 are of the same sign, in which quadrant(s) of the Argand diagram could 𝑧 appear?

  • A1st or 2nd
  • B2nd or 4th
  • C3rd or 4th
  • D1st or 3rd
  • E1st or 4th

If the real and imaginary parts of 𝑧 are of opposite signs, in which quadrant(s) of the Argand diagram could 𝑧 appear?

  • A2nd or 4th
  • B3rd or 4th
  • C1st or 2nd
  • D1st or 3rd
  • E1st or 4th

Q4:

Consider the complex number 𝑧 = 3 βˆ’ 𝑖 .

Find the modulus of 𝑧 .

  • A1
  • B3
  • C √ 8
  • D √ 1 0
  • E √ 2

Hence, find the modulus of 𝑧 5 .

  • A 1 0 0 √ 1 0
  • B10
  • C 1 0 √ 1 0
  • D243
  • E √ 1 0

Q5:

Find the complex number 𝑧 such that 4 + 3 𝑖 lies at the midpoint of 𝑧 and 3 βˆ’ 4 𝑖 when they are represented on a complex plane.

  • A 1 1 + 2 𝑖
  • B 1 + 7 𝑖
  • C 7 βˆ’ 𝑖
  • D 5 + 1 0 𝑖
  • E 5 + 1 3 𝑖

Q6:

Given that 𝑍 = 9 + 3 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q7:

Find the possible real values of 𝑏 such that the distance between the complex number βˆ’ 6 + 7 𝑖 and the complex number βˆ’ 3 + 𝑏 𝑖 is 5.

  • A 5 or 9
  • B 5
  • C 1.7 or 15.7
  • D 3 or 11
  • E 0.5 or 14.5

Q8:

Given that 𝑍 = βˆ’ 5 + 9 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q9:

Given that 𝑍 = βˆ’ 3 βˆ’ 7 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q10:

Given that 𝑍 = 6 βˆ’ 4 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q11:

Given that 𝑍 = βˆ’ 7 𝑖 , find the principal argument of 𝑍 .

  • A πœ‹
  • B πœ‹ 2
  • C0
  • D βˆ’ πœ‹ 2

Q12:

Given that 𝑍 = βˆ’ 1 2 + √ 3 2 𝑖 , find the principal argument of 𝑍 .

  • A πœ‹ 3
  • B βˆ’ 5 πœ‹ 6
  • C βˆ’ πœ‹ 3
  • D 2 πœ‹ 3

Q13:

What complex number lies at the midpoint of the complex numbers π‘Ž + 𝑏 𝑖 and π‘₯ + 𝑦 𝑖 , where π‘Ž , 𝑏 , π‘₯ , and 𝑦 are real, when they are represented on a complex plane?

  • A 𝑏 βˆ’ 𝑦 2 + π‘Ž βˆ’ π‘₯ 2 𝑖
  • B ( π‘Ž + π‘₯ ) + ( 𝑏 + 𝑦 ) 𝑖
  • C π‘Ž βˆ’ π‘₯ 2 + 𝑏 βˆ’ 𝑦 2 𝑖
  • D π‘Ž + π‘₯ 2 + 𝑏 + 𝑦 2 𝑖
  • E ( π‘Ž βˆ’ π‘₯ ) + ( 𝑏 βˆ’ 𝑦 ) 𝑖

Q14:

Simplify 𝑍 = βˆ’ 4 ( 2 + 7 𝑖 ) 𝑒 πœ‹ 𝑖 , giving your answer in algebraic form.

  • A 8 βˆ’ 2 8 𝑖
  • B βˆ’ 8 + 2 8 𝑖
  • C βˆ’ 8 βˆ’ 2 8 𝑖
  • D 8 + 2 8 𝑖

Q15:

If π‘Ÿ = 4 βˆ’ 5 𝑖 , find π‘Ÿ 2 .

  • A 1 6 βˆ’ 6 5 𝑖
  • B 4 1 βˆ’ 4 0 𝑖
  • C 1 6 βˆ’ 1 5 𝑖
  • D βˆ’ 9 βˆ’ 4 0 𝑖
  • E 1 6 + 2 5 𝑖