Worksheet: Complex 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 , − 2 𝑖
  • B 7 + 𝑖 , 0
  • C − 7 + 𝑖 , 1 4 𝑖
  • 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 + 3 𝑖 ?

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

Q4:

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

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

Q5:

How do you find the conjugate of a complex number?

  • AChange the sign of both the real and imaginary parts.
  • BChange the sign of its real part.
  • CSwap the real and imaginary parts.
  • DChange the sign of its imaginary part.
  • ESwap the real and imaginary parts, then change the sign of them both.

Q6:

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

  • Ayes
  • Bno

Q7:

If 𝑧 = − 8 𝑖 ,what is 𝑧 ∗ ?

  • A8
  • B − 8 𝑖
  • C − 8
  • D 8 𝑖
  • E − 𝑖 8

Q8:

Is 8 𝑖 + 1 0 the complex conjugate of the number − 8 𝑖 + 1 0 ?

  • Ayes
  • Bno

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 𝑖
  • D 𝑧

Q12:

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

  • AYes
  • BNo

Q13:

Is it true that 𝑧 − 𝑧 ∗ = 2 𝑖 I m ( 𝑧 ) 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 , 0
  • B 1 − 𝑖 , 0
  • C 1 − 𝑖 , 1
  • D 1 − 𝑖 , 2

Q15:

If 𝑠 = 8 + 2 𝑖 , what is 𝑠 + 𝑠 ∗ ?

Q16:

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

Q17:

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

Q18:

Given that | 𝑍 | + | 𝑍 | = 1 2 , determine the value of | 𝑍 𝑖 | .

Q19:

If 𝜃 is the principal argument of the complex number 𝑍 then what is the principal argument of 𝑍 ?

  • A 𝜋 − 𝜃
  • B 𝜃
  • C − 𝜋 + 𝜃
  • D − 𝜃

Q20:

Put 1 + 2 𝑖 9 + 7 𝑖 in the form 𝑥 + 𝑦 𝑖 .

  • A 2 3 3 2 + 1 1 3 2 𝑖
  • B − 2 3 1 3 0 − 1 1 1 3 0 𝑖
  • C − 2 3 3 2 − 1 1 3 2 𝑖
  • D 2 3 1 3 0 + 1 1 1 3 0 𝑖

Q21:

Put − 1 2 𝑖 ( − 1 + 𝑖 ) in the form 𝑥 + 𝑦 𝑖 .

Q22:

Simplify 2 − 2 𝑖 3 − 𝑖 .

  • A 2 − 4 𝑖 5
  • B 2 − 2 𝑖 5
  • C 2 3 − 2 𝑖
  • D 4 − 2 𝑖 5
  • E 1 2 + 4 𝑖 5

Q23:

Express 1 ( − 3 + 𝑖 ) 2 in the form 𝑎 + 𝑏 𝑖 .

  • A 4 5 + 3 5 𝑖
  • B 2 2 5 − 3 5 0 𝑖
  • C 4 5 − 3 5 𝑖
  • D 2 2 5 + 3 5 0 𝑖
  • E − 2 2 5 + 3 5 0 𝑖

Q24:

Put − 9 + 𝑖 + 3 𝑖 − 5 𝑖 5 − 𝑖 + 5 𝑖 + 2 𝑖 in the form 𝑎 + 𝑏 𝑖 .

  • A − 2 + 4 𝑖
  • B − 4 + 1 2 𝑖
  • C − 4 − 1 2 𝑖
  • D − 2 − 4 𝑖

Q25:

Solve 2 𝑧 − 𝑧 = 5 in ℂ .

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