Worksheet: Complex Number Conjugates

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In this worksheet, we will practice using the properties of conjugate numbers to evaluate an expression.

Q1:

Find the complex conjugate of −7−𝑖 and the sum of this number with its complex conjugate.

  • A − 7 + 𝑖 , 1 4 𝑖
  • B 7 − 𝑖 , − 2 𝑖
  • C 7 + 𝑖 , 0
  • D − 7 + 𝑖 , − 1 4

Q2:

Is it true that |𝑧|=|𝑧|∗ for all 𝑧?

  • Ayes
  • Bno

Q3:

What is the conjugate of the complex number 4+3𝑖?

  • A 3 + 4 𝑖
  • B − 4 − 3 𝑖
  • C − 4 + 3 𝑖
  • D 4 − 3 𝑖
  • E 3 − 4 𝑖

Q4:

What is the conjugate of the complex number 2−7𝑖?

  • A 7 + 2 𝑖
  • B − 7 + 2 𝑖
  • C − 2 − 7 𝑖
  • D − 2 + 7 𝑖
  • E 2 + 7 𝑖

Q5:

How do you find the conjugate of a complex number?

  • AChange the sign of both the real and imaginary parts.
  • BSwap the real and imaginary parts.
  • CChange the sign of its imaginary part.
  • DSwap the real and imaginary parts, then change the sign of them both.
  • EChange the sign of its real part.

Q6:

Is −9 the complex conjugate of the number −9?

  • Ano
  • Byes

Q7:

If 𝑧=−8𝑖,what is 𝑧∗?

  • A 8 𝑖
  • B − 𝑖 8
  • C − 8
  • D − 8 𝑖
  • E8

Q8:

Is 8𝑖+10 the complex conjugate of the number −8𝑖+10?

  • Ano
  • Byes

Q9:

Is the sum of a number and its complex conjugate always a real number?

  • Ayes
  • Bno

Q10:

Is the product of a number and its complex conjugate always a real number?

  • Ayes
  • Bno

Q11:

If 𝑧 is a real number, what will its conjugate be equal to?

  • A − 𝑧 𝑖
  • B − 𝑧
  • C 𝑧
  • D 𝑧 𝑖

Q12:

Is it true that 𝑧+𝑧=2(𝑧)∗Re for all 𝑧?

  • AYes
  • BNo

Q13:

Is it true that 𝑧−𝑧=2𝑖(𝑧)∗Im for all 𝑧?

  • AYes
  • BNo

Q14:

Find the complex conjugate of 1+𝑖 and the product of this number with its complex conjugate.

  • A 1 − 𝑖 , 1
  • B − 1 − 𝑖 , 0
  • C 1 − 𝑖 , 2
  • D 1 − 𝑖 , 0

Q15:

If 𝑠=8+2𝑖, what is 𝑠+𝑠∗?

Q16:

Given that |𝑍|=3, determine the value of 𝑍𝑍.

Q17:

Given that 𝑥=(2−𝑖) and 𝑦=(2+𝑖), determine the value of 𝑥−𝑥𝑦+𝑦.

Q18:

Given that |𝑍|+|𝑍|=12, determine the value of |𝑍𝑖|.

Q19:

Solve 2𝑧−𝑧=5 in ℂ.

Q20:

Simplify (1−𝑖)−(1+𝑖)(1−𝑖)+(1+𝑖).

Q21:

Simplify (1+2𝑖)−(1−2𝑖).

  • A − 1 4
  • B 1 + 2 𝑖
  • C0
  • D − 4 8 𝑖
  • E − 3 + 4 𝑖

Q22:

Consider 𝑧=5−𝑖√3 and 𝑤=√2+𝑖√5.

Calculate 𝑧∗ and 𝑤∗.

  • A 𝑧 = − 5 − 𝑖 √ 3 ∗ , 𝑤 = − √ 2 + 𝑖 √ 5 ∗
  • B 𝑧 = 5 + 𝑖 √ 3 ∗ , 𝑤 = √ 2 − 𝑖 √ 5 ∗
  • C 𝑧 = 5 − 𝑖 √ 3 ∗ , 𝑤 = √ 2 + 𝑖 √ 5 ∗
  • D 𝑧 = − 5 + 𝑖 √ 3 ∗ , 𝑤 = − √ 2 − 𝑖 √ 5 ∗
  • E 𝑧 = √ 3 − 5 𝑖 ∗ , 𝑤 = √ 5 + 𝑖 √ 2 ∗

Find 𝑧+𝑤∗∗ and (𝑧+𝑤)∗.

  • A 𝑧 + 𝑤 = √ 3 − √ 5 + 5 + √ 2 𝑖 ∗ ∗ , ( 𝑧 + 𝑤 ) = √ 3 − √ 5 + 5 + √ 2 𝑖 ∗
  • B 𝑧 + 𝑤 = 5 + √ 2 + √ 3 − √ 5 𝑖 ∗ ∗ , ( 𝑧 + 𝑤 ) = 5 + √ 2 − √ 3 − √ 5 𝑖 ∗
  • C 𝑧 + 𝑤 = 5 + √ 2 − √ 3 − √ 5 𝑖 ∗ ∗ , ( 𝑧 + 𝑤 ) = 5 + √ 2 − √ 3 − √ 5 𝑖 ∗
  • D 𝑧 + 𝑤 = 5 + √ 2 + √ 3 − √ 5 𝑖 ∗ ∗ , ( 𝑧 + 𝑤 ) = 5 + √ 2 + √ 3 − √ 5 𝑖 ∗
  • E 𝑧 + 𝑤 = 5 + √ 2 + √ 3 + √ 5 𝑖 ∗ ∗ , ( 𝑧 + 𝑤 ) = 5 + √ 2 − √ 3 + √ 5 𝑖 ∗

Find 𝑧𝑤∗∗ and (𝑧𝑤)∗.

  • A 𝑧 𝑤 = 5 √ 2 + 2 √ 1 5 − 5 √ 5 + √ 6 𝑖 ∗ ∗ , ( 𝑧 𝑤 ) = 5 √ 2 + 2 √ 1 5 + 5 √ 5 + √ 6 𝑖 ∗
  • B 𝑧 𝑤 = 5 √ 2 + √ 1 5 − 5 √ 5 − √ 6 𝑖 ∗ ∗ , ( 𝑧 𝑤 ) = 5 √ 2 + √ 1 5 − 5 √ 5 − √ 6 𝑖 ∗
  • C 𝑧 𝑤 = 5 √ 2 + 2 √ 1 5 + 5 √ 5 − √ 6 𝑖 ∗ ∗ , ( 𝑧 𝑤 ) = 5 √ 2 + 2 √ 1 5 − 5 √ 5 + √ 6 𝑖 ∗
  • D 𝑧 𝑤 = 5 √ 5 − √ 6 − 5 √ 2 + 2 √ 1 5 𝑖 ∗ ∗ , ( 𝑧 𝑤 ) = 5 √ 5 − √ 6 + 5 √ 2 + 2 √ 1 5 𝑖 ∗
  • E 𝑧 𝑤 = 5 √ 2 + √ 1 5 + 5 √ 5 − √ 6 𝑖 ∗ ∗ , ( 𝑧 𝑤 ) = 5 √ 2 + √ 1 5 + 5 √ 5 − √ 6 𝑖 ∗

Q23:

Solve 𝑧𝑧+𝑧−𝑧=4+2𝑖∗∗.

  • A 𝑧 = √ 3 − 𝑖 , 𝑧 = − √ 3 − 𝑖
  • B 𝑧 = 1 + 𝑖 √ 3 , 𝑧 = 1 − 𝑖 √ 3
  • C 𝑧 = − √ 3 + 𝑖 , 𝑧 = √ 3 − 𝑖
  • D 𝑧 = 1 + 𝑖 √ 3 , 𝑧 = − 1 + 𝑖 √ 3
  • E 𝑧 = √ 3 + 𝑖 , 𝑧 = − √ 3 + 𝑖

Q24:

Given that 𝑟=5+𝑖 and 𝑠=3−2𝑖, express the complex conjugate of (𝑟+𝑠) in the form 𝑎+𝑏𝑖.

  • A − 8 + 𝑖
  • B 8 + 𝑖
  • C 8 − 𝑖
  • D 1 + 8 𝑖
  • E − 8 − 𝑖

Q25:

If 𝑟=𝑎+𝑏𝑖 and 𝑠=𝑎−𝑏𝑖, find 𝑟×𝑠.

  • A 𝑎 + 𝑏 𝑖
  • B 𝑎 − 𝑏 𝑖
  • C 𝑎 + 𝑏
  • D ( 𝑎 + 𝑏 )
  • E 𝑎 − 𝑏

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