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Lesson: Operations on Complex Numbers in Different Forms

Worksheet • 14 Questions

Q1:

Express βˆ’ 6 ο€» ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   Γ— βˆ’ 1 0 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n c o s s i n in the form π‘₯ + 𝑦 𝑖 .

  • A βˆ’ 3 0 + 3 0 √ 3 𝑖
  • B 3 0 βˆ’ 3 0 √ 3 𝑖
  • C βˆ’ 3 0 βˆ’ 3 0 √ 3 𝑖
  • D 3 0 + 3 0 √ 3 𝑖

Q2:

Given that 𝑍 = ο€Ό 2 πœ‹ 3  + 𝑖 ο€Ό 2 πœ‹ 3  c o s s i n , express 𝑍 βˆ’ 1 in exponential form.

  • A √ 3 𝑒 5 πœ‹ 6 𝑖
  • B 𝑒 5 πœ‹ 6 𝑖
  • C 𝑒 πœ‹ 6 𝑖
  • D √ 3 𝑒 πœ‹ 6 𝑖

Q3:

Given that 𝑧 = 8 ( 2 4 0 + 𝑖 2 4 0 ) 1 ∘ ∘ c o s s i n , 𝑧 = 4 ο€Ό 5 πœ‹ 4 + 𝑖 5 πœ‹ 4  2 c o s s i n , and 𝑧 = 8 ( 4 5 + 𝑖 4 5 ) 3 ∘ ∘ c o s s i n , find 𝑧 𝑧 𝑧 1 6 2 4 3 , giving your answer in exponential form.

  • A 8 𝑒 1 1 πœ‹ 6 𝑖
  • B 8 𝑒 5 πœ‹ 6 𝑖
  • C 4 𝑒 1 1 πœ‹ 6 𝑖
  • D 3 2 7 6 8 𝑒 1 1 πœ‹ 6 𝑖
  • E 8 𝑒 πœ‹ 3 𝑖

Q4:

Given that and , determine .

  • A
  • B
  • C
  • D
  • E

Q5:

If 𝑧 = 3 ( 4 5 + 𝑖 4 5 ) c o s s i n ∘ ∘ , what is 𝑧 2 ?

  • A 9 ( 9 0 + 𝑖 9 0 ) c o s s i n ∘ ∘
  • B 6 ο€Ί 4 5 + 𝑖 4 5  c o s s i n 2 ∘ 2 ∘
  • C 6 ( 9 0 + 𝑖 9 0 ) c o s s i n ∘ ∘
  • D 3 ο€Ί 4 5 + 𝑖 4 5  c o s s i n 2 ∘ 2 ∘
  • E 9 ( 4 5 + 𝑖 4 5 ) c o s s i n ∘ ∘

Q6:

Given that 𝑍 = 8 ο€Ό ο€Ό 1 9 πœ‹ 1 2  βˆ’ 𝑖 ο€Ό 1 9 πœ‹ 1 2   1 2 c o s s i n and 𝑍 = 3 𝑒 2 𝑖 1 1 πœ‹ 6 , where 𝑖 = βˆ’ 1 2 , express 𝑍 = 𝑍 𝑍 1 2 2 in trigonometric form.

  • A 𝑍 = 7 2 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n
  • B 𝑍 = 7 2 ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n
  • C 𝑍 = ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n
  • D 𝑍 = ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n

Q7:

Given that and , find in trigonometric form.

  • A
  • B
  • C
  • D

Q8:

Express 6 𝑒 Γ· 2 𝑒 3 πœ‹ 2 4 πœ‹ 3 𝑖 𝑖 in exponential and trigonometric forms.

  • A 3 𝑒 πœ‹ 6 𝑖 , 3 ο€» ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   c o s s i n
  • B 3 𝑒 βˆ’ 𝑖 πœ‹ 6 , 3 ο€» ο€» βˆ’ πœ‹ 6  + 𝑖 ο€» βˆ’ πœ‹ 6   c o s s i n
  • C 𝑒 βˆ’ 𝑖 πœ‹ 6 , ο€» ο€» βˆ’ πœ‹ 6  + 𝑖 ο€» βˆ’ πœ‹ 6   c o s s i n
  • D 𝑒 πœ‹ 6 𝑖 , ο€» ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   c o s s i n

Q9:

Put 8 + 8 √ 3 𝑖 1 6 ο€» ο€» βˆ’  + 𝑖 ο€» βˆ’   c o s s i n πœ‹ 2 πœ‹ 2 in the form π‘₯ + 𝑦 𝑖 , where π‘₯ , 𝑦 ∈ ℝ , and then express it in trigonometric form.

  • A βˆ’ √ 3 2 + 1 2 𝑖 , c o s s i n ο€Ό 5 πœ‹ 6  + 𝑖 ο€Ό 5 πœ‹ 6 
  • B βˆ’ √ 3 2 βˆ’ 1 2 𝑖 , c o s s i n ο€Ό βˆ’ 5 πœ‹ 6  + 𝑖 ο€Ό βˆ’ 5 πœ‹ 6 
  • C √ 3 2 + 1 2 𝑖 , c o s s i n ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6 
  • D √ 3 2 βˆ’ 1 2 𝑖 , c o s s i n ο€» βˆ’ πœ‹ 6  + 𝑖 ο€» βˆ’ πœ‹ 6 

Q10:

Simplify 𝑍 = 5 ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4   Γ— 1 2 𝑒 c o s s i n 1 3 πœ‹ 1 2 𝑖 , giving your answer in algebraic form.

  • A 3 0 √ 3 βˆ’ 3 0 𝑖
  • B βˆ’ 3 0 √ 3 βˆ’ 3 0 𝑖
  • C βˆ’ 3 0 √ 3 + 3 0 𝑖
  • D 3 0 √ 3 + 3 0 𝑖

Q11:

Simplify 𝑍 = 1 0 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   Γ— 7 𝑒 c o s s i n 2 πœ‹ 3 𝑖 , giving your answer in algebraic form.

  • A βˆ’ 3 5 √ 3 βˆ’ 3 5 𝑖
  • B 3 5 √ 3 βˆ’ 3 5 𝑖
  • C 3 5 √ 3 + 3 5 𝑖
  • D βˆ’ 3 5 √ 3 + 3 5 𝑖

Q12:

Given that 𝑍 = 1 8 βˆ’ 1 8 𝑖 1 and 𝑍 = ο€Ό 7 πœ‹ 6  βˆ’ 𝑖 ο€Ό 7 πœ‹ 6  2 c o s s i n , find 𝑍 𝑍 1 2 , giving your answer in exponential form.

  • A 1 8 √ 2 𝑒 1 1 πœ‹ 1 2 𝑖
  • B βˆ’ 1 1 8 √ 2 𝑒 1 1 πœ‹ 1 2 𝑖
  • C 1 1 8 √ 2 𝑒 1 1 πœ‹ 1 2 𝑖
  • D 1 8 √ 2 𝑒 βˆ’ 𝑖 1 1 πœ‹ 1 2

Q13:

Express 4 √ 7 𝑒 Γ— √ 7 𝑒 2 𝑒 Γ— 7 𝑒 3 πœ‹ 4 7 πœ‹ 6 1 1 πœ‹ 1 2 πœ‹ 3 𝑖 𝑖 𝑖 𝑖 in exponential and trigonometric forms.

  • A 2 𝑒 2 πœ‹ 3 𝑖 , 2 ο€Ό ο€Ό 2 πœ‹ 3  + 𝑖 ο€Ό 2 πœ‹ 3   c o s s i n
  • B 2 𝑒 βˆ’ 𝑖 2 πœ‹ 3 , 2 ο€Ό ο€Ό βˆ’ 2 πœ‹ 3  + 𝑖 ο€Ό βˆ’ 2 πœ‹ 3   c o s s i n
  • C 𝑒 βˆ’ 𝑖 2 πœ‹ 3 , ο€Ό ο€Ό βˆ’ 2 πœ‹ 3  + 𝑖 ο€Ό βˆ’ 2 πœ‹ 3   c o s s i n
  • D 𝑒 2 πœ‹ 3 𝑖 , ο€Ό ο€Ό 2 πœ‹ 3  + 𝑖 ο€Ό 2 πœ‹ 3   c o s s i n

Q14:

Put ο€» + 𝑖  ο€» + 𝑖  ο€» + 𝑖  c o s s i n c o s s i n c o s s i n πœ‹ 3 πœ‹ 3 3 3 πœ‹ 2 3 πœ‹ 2 2 πœ‹ 3 πœ‹ 3 2 in the form π‘Ž + 𝑏 𝑖 .

  • A βˆ’ 1 2 βˆ’ √ 3 2 𝑖
  • B βˆ’ 𝑖
  • C 𝑖
  • D βˆ’ 1 2 + √ 3 2 𝑖
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