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 โˆ’7โˆ’๐‘– and the sum of this number with its complex conjugate.

  • Aโˆ’7+๐‘–, 14๐‘–
  • B7โˆ’๐‘–, โˆ’2๐‘–
  • C7+๐‘–, 0
  • Dโˆ’7+๐‘–, โˆ’14

Q2:

Is it true that |๐‘ง|=|๐‘ง|โˆ— for all ๐‘ง?

  • Ayes
  • Bno

Q3:

What is the conjugate of the complex number 4+3๐‘–?

  • A3+4๐‘–
  • Bโˆ’4โˆ’3๐‘–
  • Cโˆ’4+3๐‘–
  • D4โˆ’3๐‘–
  • E3โˆ’4๐‘–

Q4:

What is the conjugate of the complex number 2โˆ’7๐‘–?

  • A7+2๐‘–
  • Bโˆ’7+2๐‘–
  • Cโˆ’2โˆ’7๐‘–
  • Dโˆ’2+7๐‘–
  • E2+7๐‘–

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 โˆ’9 the complex conjugate of the number โˆ’9?

  • Ano
  • Byes

Q7:

If ๐‘ง=โˆ’8๐‘–,what is ๐‘งโˆ—?

  • A8๐‘–
  • Bโˆ’๐‘–8
  • Cโˆ’8
  • Dโˆ’8๐‘–
  • E8

Q8:

Is 8๐‘–+10 the complex conjugate of the number โˆ’8๐‘–+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 ๐‘ง is a real number, what will its conjugate be equal to?

  • Aโˆ’๐‘ง๐‘–
  • Bโˆ’๐‘ง
  • C๐‘ง
  • D๐‘ง๐‘–

Q12:

Is it true that ๐‘ง+๐‘ง=2(๐‘ง)โˆ—Re for all ๐‘ง?

  • AYes
  • BNo

Q13:

Is it true that ๐‘งโˆ’๐‘ง=2๐‘–(๐‘ง)โˆ—Im for all ๐‘ง?

  • AYes
  • BNo

Q14:

Find the complex conjugate of 1+๐‘– and the product of this number with its complex conjugate.

  • A1โˆ’๐‘–, 1
  • Bโˆ’1โˆ’๐‘–, 0
  • C1โˆ’๐‘–, 2
  • D1โˆ’๐‘–, 0

Q15:

If ๐‘ =8+2๐‘–, what is ๐‘ +๐‘ โˆ—?

Q16:

Given that |๐‘|=3, determine the value of ๐‘๐‘.

Q17:

Given that ๐‘ฅ=(2โˆ’๐‘–) and ๐‘ฆ=(2+๐‘–), determine the value of ๐‘ฅโˆ’๐‘ฅ๐‘ฆ+๐‘ฆ๏Šจ๏Šจ.

Q18:

Given that |๐‘|+|๐‘|=12, determine the value of |๐‘๐‘–|.

Q19:

Solve 2๐‘งโˆ’๐‘ง=5 in โ„‚.

  • A๐‘ง=1
  • B๐‘ง=5
  • C๐‘ง=5๐‘–
  • D๐‘ง=โˆ’5

Q20:

Simplify (1โˆ’๐‘–)โˆ’(1+๐‘–)(1โˆ’๐‘–)+(1+๐‘–)๏Šฉ๏Šฉ.

Q21:

Simplify (1+2๐‘–)โˆ’(1โˆ’2๐‘–)๏Šช๏Šช.

  • Aโˆ’14
  • B1+2๐‘–
  • C0
  • Dโˆ’48๐‘–
  • Eโˆ’3+4๐‘–

Q22:

Consider ๐‘ง=5โˆ’๐‘–โˆš3 and ๐‘ค=โˆš2+๐‘–โˆš5.

Calculate ๐‘งโˆ— and ๐‘คโˆ—.

  • A๐‘ง=โˆ’5+๐‘–โˆš3โˆ—, ๐‘ค=โˆ’โˆš2โˆ’๐‘–โˆš5โˆ—
  • B๐‘ง=5+๐‘–โˆš3โˆ—, ๐‘ค=โˆš2โˆ’๐‘–โˆš5โˆ—
  • C๐‘ง=5โˆ’๐‘–โˆš3โˆ—, ๐‘ค=โˆš2+๐‘–โˆš5โˆ—
  • D๐‘ง=โˆ’5โˆ’๐‘–โˆš3โˆ—, ๐‘ค=โˆ’โˆš2+๐‘–โˆš5โˆ—
  • E๐‘ง=โˆš3โˆ’5๐‘–โˆ—, ๐‘ค=โˆš5+๐‘–โˆš2โˆ—

Find ๐‘ง+๐‘คโˆ—โˆ— and (๐‘ง+๐‘ค)โˆ—.

  • A๐‘ง+๐‘ค=5+โˆš2+๏€ปโˆš3โˆ’โˆš5๏‡๐‘–โˆ—โˆ—, (๐‘ง+๐‘ค)=5+โˆš2+๏€ปโˆš3โˆ’โˆš5๏‡๐‘–โˆ—
  • B๐‘ง+๐‘ค=5+โˆš2+๏€ปโˆš3โˆ’โˆš5๏‡๐‘–โˆ—โˆ—, (๐‘ง+๐‘ค)=5+โˆš2โˆ’๏€ปโˆš3โˆ’โˆš5๏‡๐‘–โˆ—
  • C๐‘ง+๐‘ค=5+โˆš2+๏€ปโˆš3+โˆš5๏‡๐‘–โˆ—โˆ—, (๐‘ง+๐‘ค)=5+โˆš2โˆ’๏€ปโˆš3+โˆš5๏‡๐‘–โˆ—
  • D๐‘ง+๐‘ค=5+โˆš2โˆ’๏€ปโˆš3โˆ’โˆš5๏‡๐‘–โˆ—โˆ—, (๐‘ง+๐‘ค)=5+โˆš2โˆ’๏€ปโˆš3โˆ’โˆš5๏‡๐‘–โˆ—
  • E๐‘ง+๐‘ค=โˆš3โˆ’โˆš5+๏€ป5+โˆš2๏‡๐‘–โˆ—โˆ—, (๐‘ง+๐‘ค)=โˆš3โˆ’โˆš5+๏€ป5+โˆš2๏‡๐‘–โˆ—

Find ๐‘ง๐‘คโˆ—โˆ— and (๐‘ง๐‘ค)โˆ—.

  • A๐‘ง๐‘ค=5โˆš2+2โˆš15+๏€ป5โˆš5โˆ’โˆš6๏‡๐‘–โˆ—โˆ—, (๐‘ง๐‘ค)=5โˆš2+2โˆš15โˆ’๏€ป5โˆš5+โˆš6๏‡๐‘–โˆ—
  • B๐‘ง๐‘ค=5โˆš2+โˆš15โˆ’๏€ป5โˆš5โˆ’โˆš6๏‡๐‘–โˆ—โˆ—, (๐‘ง๐‘ค)=5โˆš2+โˆš15โˆ’๏€ป5โˆš5โˆ’โˆš6๏‡๐‘–โˆ—
  • C๐‘ง๐‘ค=5โˆš2+2โˆš15โˆ’๏€ป5โˆš5+โˆš6๏‡๐‘–โˆ—โˆ—, (๐‘ง๐‘ค)=5โˆš2+2โˆš15+๏€ป5โˆš5+โˆš6๏‡๐‘–โˆ—
  • D๐‘ง๐‘ค=5โˆš5โˆ’โˆš6โˆ’๏€ป5โˆš2+2โˆš15๏‡๐‘–โˆ—โˆ—, (๐‘ง๐‘ค)=5โˆš5โˆ’โˆš6+๏€ป5โˆš2+2โˆš15๏‡๐‘–โˆ—
  • E๐‘ง๐‘ค=5โˆš2+โˆš15+๏€ป5โˆš5โˆ’โˆš6๏‡๐‘–โˆ—โˆ—, (๐‘ง๐‘ค)=5โˆš2+โˆš15+๏€ป5โˆš5โˆ’โˆš6๏‡๐‘–โˆ—

Q23:

Solve ๐‘ง๐‘ง+๐‘งโˆ’๐‘ง=4+2๐‘–โˆ—โˆ—.

  • A๐‘ง=โˆš3โˆ’๐‘–, ๐‘ง=โˆ’โˆš3โˆ’๐‘–
  • B๐‘ง=1+๐‘–โˆš3, ๐‘ง=1โˆ’๐‘–โˆš3
  • C๐‘ง=โˆ’โˆš3+๐‘–, ๐‘ง=โˆš3โˆ’๐‘–
  • D๐‘ง=1+๐‘–โˆš3, ๐‘ง=โˆ’1+๐‘–โˆš3
  • E๐‘ง=โˆš3+๐‘–, ๐‘ง=โˆ’โˆš3+๐‘–

Q24:

Given that ๐‘Ÿ=5+๐‘– and ๐‘ =3โˆ’2๐‘–, express the complex conjugate of (๐‘Ÿ+๐‘ ) in the form ๐‘Ž+๐‘๐‘–.

  • Aโˆ’8+๐‘–
  • B8+๐‘–
  • C8โˆ’๐‘–
  • D1+8๐‘–
  • Eโˆ’8โˆ’๐‘–

Q25:

If ๐‘Ÿ=๐‘Ž+๐‘๐‘– and ๐‘ =๐‘Žโˆ’๐‘๐‘–, find ๐‘Ÿร—๐‘ .

  • A๐‘Ž+๐‘๐‘–๏Šจ๏Šจ
  • B๐‘Žโˆ’๐‘๐‘–๏Šจ๏Šจ
  • C๐‘Ž+๐‘๏Šจ๏Šจ
  • D(๐‘Ž+๐‘)๏Šจ
  • E๐‘Žโˆ’๐‘๏Šจ๏Šจ

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