Polar form of Complex Numbers MCQ Quiz - Objective Question with Answer for Polar form of Complex Numbers - Download Free PDF

Last updated on Apr 23, 2025

Latest Polar form of Complex Numbers MCQ Objective Questions

Polar form of Complex Numbers Question 1:

The polar form of the complex number (i21)3 is:

  1. cosπ2isinπ2
  2. cosπ2+isinπ2
  3. 2cosπ23isinπ2
  4. cos3π2isin3π2

Answer (Detailed Solution Below)

Option 1 : cosπ2isinπ2

Polar form of Complex Numbers Question 1 Detailed Solution

Concept:

Power of i,

For any integer k , i4k + 1  = i

  • i1 = i
  • i2 = -1
  • i3 = -i
  • i4 = 1

Polar form of z = r(cos θ + i sinθ)

where r = a2+b2 ,rcosθ = a and rsinθ = b

cosπ2  = 0

sin 3π2 = -1

Calculation:

Let  z = (i21)3 = (i)63 = i4 × 15 + 3 = (i4)15 i3 = i3 = -i

From the above concept  r = 1 and cosθ = 0 and sinθ = -1 

⇒ z  = cosπ2 + i sin 3π2 = cosπ2 - i sin π2

Option 1 is correct

Polar form of Complex Numbers Question 2:

F2 Savita Engineering 28-6-22 D6

Which below option is true for fourth quadrant as per above figure?

  1. (3π2<x<2π) a is positive and b is negative
  2. (0<x<π2) a is positive and b is negative
  3. (π2<x<π) a is positive and b is negative
  4. (π<x<3π2) a is positive and b is negative

Answer (Detailed Solution Below)

Option 1 : (3π2<x<2π) a is positive and b is negative

Polar form of Complex Numbers Question 2 Detailed Solution

Explanation:

For Ist quadrant : x > 0, y > 0 and 0 < θ < π/2

For IInd quadrant : x < 0, y > 0 and π/2 < θ < π

For IIIrd quadrant : x < 0, y < 0 and π < θ < 3π/2

For IVth quadrant : x > 0, y < 0 and 3π/2 < θ < 2π

Polar form of Complex Numbers Question 3:

What is the polar form of the complex number (i25)3 ?

  1. cosπ2+isinπ2
  2. 2cosπ23isinπ2
  3. cosπ2isinπ2
  4. 3cosπ25isinπ2

Answer (Detailed Solution Below)

Option 3 : cosπ2isinπ2

Polar form of Complex Numbers Question 3 Detailed Solution

Concept:

Complex Number: 

  • It is a combination of a real number and an imaginary number. The complex number is in the form of a + ib, where a and b are real numbers.
  • It is represented by 'z'.

Modulus of z: It is represented by |z|.

|z| = a2 + b2

Polar form: 

  • The ordered pair (r, θ) is called the polar coordinates of point A, as the point. The origin is called the pole and the positive X-axis is called the initial line.

x = r cosθ and y = r sinθ 

z = x + iy as z = r cosθ + ir sinθ = r(cosθ + isinθ), which is called the polar form of the complex number.

Here, r = |z| = x2 + y2 is the modulus of z and θ is known as the argument or amplitude of z.

Formula Used:

(ax)y = axy

a+  y = ax. ay

Calculation: 

We have,

⇒ z = (i25)3        -------(1)

⇒ z = i25 × 3

⇒ z = i75

⇒ z = i(72 + 3)     -------(2)

⇒ z = (i72) × (i3)

⇒ z = (i4 × 18) × (i3)   -----(3)

⇒ z = (i4)18 × (i3)     ------(4)

⇒ z = (1)18 × (i2) × i

⇒ z = 1 × (-1) × i

⇒ z = -i

We can write in the form, z = x + iy

⇒ z = 0 - i

⇒ |z| = 02+(1)2

⇒ |z| = 1 = 1

⇒ r = |z| = 1

Similarly, to get the value of θ,

tanα = |Im(z)Re(z)| = |10| = ∞

⇒ α = π2

As x = 0 and y = - 1 < 0. the coordinate (x, y) lies in the IV quadrant.

In the IV quadrant tangent function is negative.

⇒ α = π2

⇒ arg(z) = θ = α = π2

The polar form of a complex number,

⇒ z = r(cosθ + isinθ)

⇒ z = 1[cos(π2) + isin(π2)]

⇒ z = cos(π2)  isin(π2)

(i25)3=cos(π2)  isin(π2)

∴ The polar form of the complex number (i25)3 is cos(π2)  isin(π2)

Polar form of Complex Numbers Question 4:

What is the polar form of the complex number (i15)3?

  1. cosπ3 + i sin π3
  2. cosπ2 + i sin π2
  3. cosπ2 + i sin π2
  4. -cosπ2 + i sin π2

Answer (Detailed Solution Below)

Option 3 : cosπ2 + i sin π2

Polar form of Complex Numbers Question 4 Detailed Solution

Concept:

Power of i,

for any integer, i4k + 1  = i

cosπ2  = 0

sin π2 = 1

Calculation:

 Let  z = (i15)3 = (i)45 = i4 × 11 + 1 = (i4)11 i = i = 0 + i

polar form of z = r(cos θ + i sinθ)

z = 1{ cos(π2) + i sin (π2) } = cosπ2 + i sin π2

Polar form of Complex Numbers Question 5:

What is the principal value of amplitude of 3 - i ?

  1. -π/3
  2. π/6
  3. π/3
  4. -π/6

Answer (Detailed Solution Below)

Option 4 : -π/6

Polar form of Complex Numbers Question 5 Detailed Solution

Concept:

when z = x + iy then, Principal amplitude of a complex number, θ = tan-1(yx)

tan π6 = 13

x > 0, y < 0, The point lies in IVth quadrant.

Calculation:

Let θ be the principal value of amplitude of 3  - i

Since, tan θ = 13 and 3 - i lies in IVth quadrant.

 tan θ = tan (-π6), θ = -π6

Top Polar form of Complex Numbers MCQ Objective Questions

Represent the complex number Z = - 2 - i 2√3 in the polar form.

  1. 4(cos(π3)+isin(π3))
  2. 4(cos(2π3)isin(2π3))
  3. 4(cos(2π3)+isin(2π3))
  4. 4(cos(π3)isin(2π3))

Answer (Detailed Solution Below)

Option 2 : 4(cos(2π3)isin(2π3))

Polar form of Complex Numbers Question 6 Detailed Solution

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CONCEPT:

Let the point P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called modulus of given complex number.

The argument of Z is measured from positive x-axis only.

Let the point P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called modulus of given complex number.

The argument of Z is measured from positive x-axis only.

Let z = r (cosθ + i sinθ) is polar form of any complex number then following ways are used while writing θ for different quadrants –

For first quadrant, θ=tan1yx

For second quadrant θ=πtan1yx

For third quadrant θ=π+tan1yx

For fourth quadrant, θ=tan1yx

CALCULATION:

Given complex number is Z=2i23

rcosθ = - 2, rsinθ=23

By squaring and adding, we get:

r2(cos2θ+sin2θ)=4+12

∴ r = 4

cosθ=2r=24=12andsinθ=23r=234=32

Since it is in third quadrant

θ=π+π3=2π3

So, on comparing with z = r (cosθ + i sinθ),

we can write as

→  4(cos(2π3)+isin(2π3))i.e.4(cos(2π3)isin(2π3)).

If the area of the triangle on the complex plane formed by the points z, z + iz and iz is 50, then |z| is

  1. 1
  2. 5
  3. 10
  4. 100

Answer (Detailed Solution Below)

Option 3 : 10

Polar form of Complex Numbers Question 7 Detailed Solution

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Formula:

Area of the triangle = (1/2) × |z|2

Calculation:

Three points z, z + iz, and iz on the complex plane.

Area of the triangle formed on the complex plane = 50

⇒ 50 = (1/2) × |z|2

⇒ 100 = |z|2

⇒ |z| = 10

The polar form of -√3 + i will be –

  1. 2eπ6
  2. e5π6
  3. 2ei5π6
  4. 2e7π6

Answer (Detailed Solution Below)

Option 3 : 2ei5π6

Polar form of Complex Numbers Question 8 Detailed Solution

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CONCEPT:

If P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called the modulus of the given complex number.

The argument of Z is measured from the positive x-axis only.

Let z = r (cos θ + i sin θ) is a polar form of any complex number then following ways are used while writing θ for different quadrants –

For the first quadrant, θ=tan1yx

For the second quadrant θ=πtan1yx

For the third quadrant θ=π+tan1yx

For the fourth quadrant θ=tan1yx

Note: The polar form z = r (cosθ + i sinθ) is abbreviated as r.cisθ.

CALCULATION:

Given Z=3+i

x=3,y=1

r=(3)2+(1)2=2

tanθ=13

θ=tan1(13)tan1(13) =  π6

Here the reference angle and for θ is 30 °. Since the complex number is in the second quadrant –

θ=ππ6=5π6

Z=3+i

2ei5π6

What is the polar form of the complex number (i15)3?

  1. cosπ3 + i sin π3
  2. cosπ2 + i sin π2
  3. cosπ2 + i sin π2
  4. -cosπ2 + i sin π2

Answer (Detailed Solution Below)

Option 3 : cosπ2 + i sin π2

Polar form of Complex Numbers Question 9 Detailed Solution

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Concept:

Power of i,

for any integer, i4k + 1  = i

cosπ2  = 0

sin π2 = 1

Calculation:

 Let  z = (i15)3 = (i)45 = i4 × 11 + 1 = (i4)11 i = i = 0 + i

polar form of z = r(cos θ + i sinθ)

z = 1{ cos(π2) + i sin (π2) } = cosπ2 + i sin π2

What is the principal value of amplitude of 3 - i ?

  1. -π/3
  2. π/6
  3. π/3
  4. -π/6

Answer (Detailed Solution Below)

Option 4 : -π/6

Polar form of Complex Numbers Question 10 Detailed Solution

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Concept:

when z = x + iy then, Principal amplitude of a complex number, θ = tan-1(yx)

tan π6 = 13

x > 0, y < 0, The point lies in IVth quadrant.

Calculation:

Let θ be the principal value of amplitude of 3  - i

Since, tan θ = 13 and 3 - i lies in IVth quadrant.

 tan θ = tan (-π6), θ = -π6

If a complex number having absolute value of √2 is making 45° angle with x-axis in third quadrant, then it can be written as –

  1. [r, θ] = [2,3π4]
  2. [r, θ] = [2,π4]
  3. [r, θ] = [2,3π4]
  4. [r, θ] = [2,π4]

Answer (Detailed Solution Below)

Option 3 : [r, θ] = [2,3π4]

Polar form of Complex Numbers Question 11 Detailed Solution

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CONCEPT:

Point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P.

If P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called modulus of the given complex number.

The argument of Z is measured from the positive x-axis only.

Let z = r (cos θ + i sin θ) is a polar form of any complex number then following ways are used while writing θ for different quadrants –

For the first quadrant, θ=tan1yx

For the second quadrant θ=πtan1yx

For the third quadrant θ=π+tan1yx

For the fourth quadrant θ=tan1yx

CALCULATION:

Given that |z|=2 and angle with x-axis is 45°.

Since its in the third quadrant -

θ=π+π4=3π4

⇒ [r, θ] = [2,3π4]

Express the complex number 2i using polar coordinates.

  1. 2(cos(π2)+isin(π2))
  2. 4(cos(π2)isin(π2))
  3. 2(isin(π2))
  4. 4(cos(π4)+isin(π4))

Answer (Detailed Solution Below)

Option 1 : 2(cos(π2)+isin(π2))

Polar form of Complex Numbers Question 12 Detailed Solution

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CONCEPT:

Let the point P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called modulus of given complex number.

The argument of Z is measured from positive x-axis only.

Let z = r (cosθ + i sinθ) is polar form of any complex number then following ways are used while writing θ for different quadrants –

For first quadrant, θ=tan1yx

For second quadrant θ=πtan1yx

For third quadrant θ=π+tan1yx

For fourth quadrant θ=tan1yx

CALCULATION:

On the complex plane, the number z = 2i is the same as z = 0 + 2i. Writing it in polar form, we have to calculate r first.

r=x2+y2=02+22=4=2

cosθ=xr=02=0andsinθ=2r=22=1θ=π2

∴ on comparing with z = r (cosθ + i sinθ), we can write as 2(cos(π2)+isin(π2)).

The locus represented by |z - 1| = |z + i| is:

  1. a circle of radius 1
  2. an ellipse with foci at (1, 0) and (0, -1)
  3. a straight line through the origin
  4. a circle on the line joining (1, 0), (0, 1) as diameter

Answer (Detailed Solution Below)

Option 3 : a straight line through the origin

Polar form of Complex Numbers Question 13 Detailed Solution

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Calculation:

Given that,

∣z − 1∣ = ∣z + i∣

Let x = x + iy

∣x + iy − 1∣ = ∣x + iy + i∣

∣(x − 1) + iy∣ = ∣x + i(y + 1)∣

√(x - 1)2 + y2 = √x2 + (y + 1)2

(x − 1)2 + y2  = x2 + (y + 1)

⇒ (x − 1)2 + y2  = x2 + y2 + 1 + 2y 

⇒ x2 + 1 − 2x + y2 = x2 +  y2 + 2y + 1

⇒ − 2x  =  2y

⇒ -2x = 2y

Hence, It can be written as

by + ax = 0    a = 2 b = 2 (as they are constant)

∴ It represent the straight line passing through origin.

Express z = 1 + i  in the polar form.

  1. z = 2 (cos π4 + i sin π4
  2. z = 2 (cos π3 + i sin π3
  3. z =  (cos π4 + i sin π4
  4. z = 2 (cos π6 + i sin π6

Answer (Detailed Solution Below)

Option 1 : z = 2 (cos π4 + i sin π4

Polar form of Complex Numbers Question 14 Detailed Solution

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Concept:

Polar form of complex number, z = r cos θ + i r sin θ 

cos2θ + sin2θ = 1

cos π4 = 12

sin π4 = 12

Calculation:

Given

z = 1 + i        ....(1)

Let Polar form of Given Equation be 

z = r cos θ + ir sin θ        ....(2)

Comparing (1) and (2)

we get,

1 = r cos θ, 1 = r sin θ

by squaring and adding, we get

r2(cos2θ + sin2θ) = 2

r2 = 2

r = 2

Therefore, cos θ = 12, sin θ  = 12, which gives θ = π4

Therefore, required polar form is z = 2 (cos π4 + i sin π4)

Represent the complex number Z = -4 + i4√3 in the polar form.

  1. 8(cos2π3+isin2π3)
  2. 8(cosπ3+isinπ3)
  3. 8(cos2π3+isin2π3)
  4. 8(cos2π3isin2π3)

Answer (Detailed Solution Below)

Option 1 : 8(cos2π3+isin2π3)

Polar form of Complex Numbers Question 15 Detailed Solution

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CONCEPT:

Let the point P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called modulus of given complex number.

The argument of Z is measured from positive x-axis only.

Let z = r (cosθ + i sinθ) is polar form of any complex number then following ways are used while writing θ for different quadrants –

For first quadrant, θ=tan1yx

For second quadrant θ=πtan1yx

For third quadrant θ=π+tan1yx

For fourth quadrant θ=tan1yx

CALCULATION:

Given complex number is Z = -4 + i 4√3

rcosθ = - 4, rsinθ = 4√3

By squaring and adding, we get –

r2(cos2θ+sin2θ)=16+48

∴ r = 8

cosθ=4r=48=12 and sinθ=43r=438=32

Since it is in second quadrant θ=ππ3=2π3

So, on comparing with z = r (cosθ + i sinθ), we can write as 8(cos2π3+isin2π3).

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