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, 14𝑖
  • B7−𝑖, −2𝑖
  • C7+𝑖, 0
  • D−7+𝑖, −14

Q2:

Is it true that |𝑧|=|𝑧|∗ for all 𝑧?

  • Ayes
  • Bno

Q3:

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

  • A3+4𝑖
  • B−4−3𝑖
  • C−4+3𝑖
  • D4−3𝑖
  • E3−4𝑖

Q4:

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

  • A7+2𝑖
  • B−7+2𝑖
  • C−2−7𝑖
  • D−2+7𝑖
  • E2+7𝑖

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∗?

  • A8𝑖
  • B−𝑖8
  • C−8
  • D−8𝑖
  • 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−𝑧𝑖
  • B−𝑧
  • C𝑧
  • D𝑧𝑖

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.

  • A1−𝑖, 1
  • B−1−𝑖, 0
  • C1−𝑖, 2
  • D1−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.

  • A𝑧=1
  • B𝑧=5
  • C𝑧=5𝑖
  • D𝑧=−5

Q20:

Simplify (1−𝑖)−(1+𝑖)(1−𝑖)+(1+𝑖).

Q21:

Simplify (1+2𝑖)−(1−2𝑖).

  • A−14
  • B1+2𝑖
  • C0
  • D−48𝑖
  • E−3+4𝑖

Q22:

Consider 𝑧=5−𝑖√3 and 𝑤=√2+𝑖√5.

Calculate 𝑧∗ and 𝑤∗.

  • A𝑧=−5+𝑖√3∗, 𝑤=−√2−𝑖√5∗
  • B𝑧=5+𝑖√3∗, 𝑤=√2−𝑖√5∗
  • C𝑧=5−𝑖√3∗, 𝑤=√2+𝑖√5∗
  • D𝑧=−5−𝑖√3∗, 𝑤=−√2+𝑖√5∗
  • E𝑧=√3−5𝑖∗, 𝑤=√5+𝑖√2∗

Find 𝑧+𝑤∗∗ and (𝑧+𝑤)∗.

  • A𝑧+𝑤=5+√2+√3−√5𝑖∗∗, (𝑧+𝑤)=5+√2+√3−√5𝑖∗
  • B𝑧+𝑤=5+√2+√3−√5𝑖∗∗, (𝑧+𝑤)=5+√2−√3−√5𝑖∗
  • C𝑧+𝑤=5+√2+√3+√5𝑖∗∗, (𝑧+𝑤)=5+√2−√3+√5𝑖∗
  • D𝑧+𝑤=5+√2−√3−√5𝑖∗∗, (𝑧+𝑤)=5+√2−√3−√5𝑖∗
  • E𝑧+𝑤=√3−√5+5+√2𝑖∗∗, (𝑧+𝑤)=√3−√5+5+√2𝑖∗

Find 𝑧𝑤∗∗ and (𝑧𝑤)∗.

  • A𝑧𝑤=5√2+2√15+5√5−√6𝑖∗∗, (𝑧𝑤)=5√2+2√15−5√5+√6𝑖∗
  • B𝑧𝑤=5√2+√15−5√5−√6𝑖∗∗, (𝑧𝑤)=5√2+√15−5√5−√6𝑖∗
  • C𝑧𝑤=5√2+2√15−5√5+√6𝑖∗∗, (𝑧𝑤)=5√2+2√15+5√5+√6𝑖∗
  • D𝑧𝑤=5√5−√6−5√2+2√15𝑖∗∗, (𝑧𝑤)=5√5−√6+5√2+2√15𝑖∗
  • E𝑧𝑤=5√2+√15+5√5−√6𝑖∗∗, (𝑧𝑤)=5√2+√15+5√5−√6𝑖∗

Q23:

Solve 𝑧𝑧+𝑧−𝑧=4+2𝑖∗∗.

  • A𝑧=√3−𝑖, 𝑧=−√3−𝑖
  • B𝑧=1+𝑖√3, 𝑧=1−𝑖√3
  • C𝑧=−√3+𝑖, 𝑧=√3−𝑖
  • D𝑧=1+𝑖√3, 𝑧=−1+𝑖√3
  • E𝑧=√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
  • B8+𝑖
  • C8−𝑖
  • D1+8𝑖
  • E−8−𝑖

Q25:

If 𝑟=𝑎+𝑏𝑖 and 𝑠=𝑎−𝑏𝑖, find 𝑟×𝑠.

  • A𝑎+𝑏𝑖
  • B𝑎−𝑏𝑖
  • C𝑎+𝑏
  • D(𝑎+𝑏)
  • E𝑎−𝑏

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