Worksheet: Complex Number Conjugates

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

Q1:

Find the complex conjugate of 7𝑖 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 27𝑖?

  • 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 𝑧𝑧=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=(2𝑖) and 𝑦=(2+𝑖), determine the value of 𝑥𝑥𝑦+𝑦.

Q18:

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

Q19:

Solve 2𝑧𝑧=5 in .

Q20:

Simplify (1𝑖)(1+𝑖)(1𝑖)+(1+𝑖).

Q21:

Simplify (1+2𝑖)(12𝑖).

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

Q22:

Consider 𝑧=5𝑖3 and 𝑤=2+𝑖5.

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+𝑧𝑧=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=32𝑖, 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=𝑎𝑏𝑖, find 𝑟×𝑠.

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

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