Lesson Worksheet: Operations on Complex Numbers in Polar Form Mathematics • 12th Grade

In this worksheet, we will practice performing calculations with complex numbers in polar form.

Q1:

Given that 𝑧=2ο€»πœ‹6+π‘–πœ‹6ο‡οŠ§cossin and 𝑧=1√3ο€»πœ‹3+π‘–πœ‹3ο‡οŠ¨cossin, find π‘§π‘§οŠ§οŠ¨.

  • A2√33ο€»πœ‹2+π‘–πœ‹2cossin
  • B2√33ο€Ό11πœ‹6+𝑖11πœ‹6cossin
  • Cο€Ώ2+1√3ο‹ο€»πœ‹2+π‘–πœ‹2cossin
  • Dο€Ώ2+1√311πœ‹6+𝑖11πœ‹6cossin
  • E2√33ο€»πœ‹2+π‘–πœ‹2cossin

Q2:

Given that 𝑧=5ο€Ό5πœ‹6+𝑖5πœ‹6cossin and 𝑧=4(180+𝑖180)∘∘cossin, determine π‘§π‘§οŠ§οŠ¨.

  • A9(330+𝑖330)cossin∘∘
  • B20(330+𝑖330)cossin∘∘
  • C20ο€Ί330+𝑖330cossin∘∘
  • D9(330βˆ’π‘–330)cossin∘∘
  • E20(30+𝑖30)cossin∘∘

Q3:

If 𝑍=7(πœƒ+π‘–πœƒ)cossin, 𝑍=16(πœƒ+π‘–πœƒ)cossin, and πœƒ+πœƒ=πœ‹οŠ§οŠ¨, then what is π‘π‘οŠ§οŠ¨?

  • Aβˆ’112𝑖
  • Bβˆ’112
  • C112𝑖
  • D112

Q4:

Given that 𝑧=2((5π‘Žβˆ’2𝑏)+𝑖(5π‘Žβˆ’2𝑏))cossin and 𝑧=4((4π‘Žβˆ’3𝑏)+𝑖(4π‘Žβˆ’3𝑏))cossin, find π‘§π‘§οŠ§οŠ¨.

  • A6((20π‘Ž+6𝑏)+𝑖(20π‘Ž+6𝑏))cossin
  • B12((π‘Ž+𝑏)+𝑖(π‘Ž+𝑏))cossin
  • C8((20π‘Ž+6𝑏)+𝑖(20π‘Ž+6𝑏))cossin
  • D8((9π‘Žβˆ’5𝑏)+𝑖(9π‘Žβˆ’5𝑏))cossin
  • E6((9π‘Žβˆ’5𝑏)+𝑖(9π‘Žβˆ’5𝑏))cossin

Q5:

Given that 𝑧=20ο€»πœ‹2+π‘–πœ‹2ο‡οŠ§cossin and 𝑧=4ο€»πœ‹6+π‘–πœ‹6ο‡οŠ¨cossin, find π‘§π‘§οŠ§οŠ¨ in polar form.

  • A5ο€Ό2πœ‹3+𝑖2πœ‹3cossin
  • B16ο€»πœ‹3+π‘–πœ‹3cossin
  • C5ο€»πœ‹2+π‘–πœ‹2cossin
  • D5ο€»πœ‹3+π‘–πœ‹3cossin
  • E80ο€»πœ‹3+π‘–πœ‹3cossin

Q6:

Given that 𝑍=5(5πœƒ+𝑖5πœƒ)cossin, 𝑍=4πœƒ+𝑖4πœƒοŠ¨cossin, tanπœƒ=43, and πœƒβˆˆο€»0,πœ‹2, find π‘π‘οŠ§οŠ¨.

  • A3+4𝑖
  • B4+3𝑖
  • C35+45𝑖
  • D45+35𝑖

Q7:

Given that 𝑧=ο€Ό7πœ‹6+𝑖7πœ‹6cossin, find 1𝑧.

  • Acossinο€»πœ‹6+π‘–ο€»πœ‹6
  • Bcossinο€Ό5πœ‹6+𝑖5πœ‹6
  • Ccossinο€Ό7πœ‹6+𝑖7πœ‹6
  • Dsincosο€Ό5πœ‹6+𝑖5πœ‹6

Q8:

Consider the complex number 𝑧=1+√3𝑖.

Find the modulus of 𝑧.

Find the argument of 𝑧.

  • A2
  • Bπœ‹6
  • Cπœ‹3
  • D√10
  • E2πœ‹3

Hence, use the properties of multiplication of complex numbers in polar form to find the modulus and argument of π‘§οŠ©.

  • Amodulus = 8, argument = πœ‹
  • Bmodulus = √10, argument = πœ‹
  • Cmodulus = 8, argument = πœ‹2
  • Dmodulus = √3, argument= πœ‹
  • Emodulus = √10, argument = πœ‹2

Hence, find the value of π‘§οŠ©.

Q9:

Given that |𝑍|=2 where principal argument (𝑍)=6π‘Ž+5π‘οŠ§, and |𝑍|=6 where principal argument (𝑍)=6π‘Ž+4π‘οŠ¨, find π‘π‘οŠ§οŠ¨.

  • A12((36π‘Ž+20𝑏)+𝑖(36π‘Ž+20𝑏))cossin
  • B8((12π‘Ž+9𝑏)+𝑖(12π‘Ž+9𝑏))coscos
  • C12((12π‘Ž+9𝑏)+𝑖(12π‘Ž+9𝑏))cossin
  • D8((12π‘Ž+9𝑏)+𝑖(12π‘Ž+9𝑏))cossin
  • E8((36π‘Ž+20𝑏)+𝑖(36π‘Ž+20𝑏))cossin

Q10:

Given that 𝑧=6(4πœƒ+𝑖4πœƒ)cossin and 𝑧=13(2πœƒ+𝑖2πœƒ)sincos, where 0<πœƒ<90∘, determine the trigonometric form of π‘§π‘§οŠ§οŠ¨.

  • A2((90βˆ’2πœƒ)+𝑖(90βˆ’2πœƒ))cossin∘∘
  • B2(2πœƒ+𝑖2πœƒ)cossin
  • C193(2πœƒ+𝑖2πœƒ)cossin
  • D2((90+2πœƒ)+𝑖(90+2πœƒ))cossin∘∘
  • E193((90+2πœƒ)+𝑖(90+2πœƒ))cossin∘∘

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