Real and Imaginary parts MCQ Quiz - Objective Question with Answer for Real and Imaginary parts - Download Free PDF

Formula Used: 

(a + b) (a - b) = a² - b²

i² = -1

Calculation:

Let x =, where 

Squaring both sides,

x² = 

x² = 

x2 = 

x2 = 

x2 = 

x = 

Calculation

2z² - 32 - 2i = 0

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Similarly

⇒ 

⇒ 

Real = 

Im = 9

Hence option 4 is correct

Answer (1)

⇒ 

∴ 

= 

Calculation

Given:

A complex number  such that  is purely imaginary.

Let , where and are real numbers. Then

Since is purely imaginary, its real part must be zero:

⇒ 

⇒ 

⇒ 

This is the equation of a circle with center and radius .

The area of the region bounded by this circle and lying in the first quadrant is of the total area of the circle.

Area of the circle is .

The area in the first quadrant is .

Hence option 2 is correct

Calculation

Let z = r.e^iθ

So, , where a, b ∈ Z

So, 

For 

We get 

Hence option 1 is correct

Concept:

Equality of complex numbers.

Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if x1 = x2 and y1 = y2

"OR"

Re (z1) = Re (z2) and Im (z1) = Im (z2).

Calculation:

Given:

(2 - i) (x - iy) = 3 + 4i

⇒ 2x - 2iy - ix + i2y = 3 + 4i

⇒ 2x - 2iy - ix - y = 3 + 4i                 (∵ i² = -1)

⇒ (2x – y) + i(-x - 2y) = 3 + 4i

Equating real and imaginary parts,

2x - y = 3      ----(1)

-x - 2y = 4       ----(2)

Solving equation 1 and 2, we get

x =  and y = 

Now, the value of 5x can be calculated as:

5x = 5 × = 2

Concept:

Euler's Formula on Complex Numbers:

• e^ix = cos x + i sin x
• e^-ix = cos x - i sin x

Calculation:

(sin x + icos x)³

Take i common, we get

(sin x + icos x)³

= 

= -i  × (-i sin x + cos x)³                

(∵ i³ = -i and 1/i = -i)

= -i × (cos x - i sin x) ³

(∵e^-ix = cos x - i sin x)

= -i (cos 3x – i sin 3x)

= (-i cos 3x + i² sin 3x)

= -sin3x – i cos 3x

∴ Real part = -sin 3x

Concept:

Let z = x + iy be a complex number, Where x is called the real part of a complex number or Re (z) and y is called the imaginary part of the complex number or Im (z)

Condition for purely real: Imaginary part equals zero.

Condition for purely imaginary: Real part equals zero.

Calculation:

Multiplying the numerator and denominator by 2 + i sin θ

       (∵ i² = -1)

The imaginary part of z 

For z to be purely real Im(z) = 0

⇒ sin θ = 0

So, θ = nπ, where n belongs to an integer

Concept:

Let z = x + iy be a complex number.

• Modulus of z =  
• Arg (z) = Arg (x + iy) =   

Calculation: 

Let 

Multiplying numerator and denominator by 1 + i

= 5 + 5i

Hence, option 4 is correct.

Concept:

Complex Numbers:

• A complex number is a number of the form a + ib, where a and b are real numbers and i is the complex unit defined by i = .
• 'a' is called the real part Re{z} and b is called the imaginary part Im{z}.
• If z₁ = z₂, then Re{z₁} = Re{z₂} and Im{z1} = Im{z2}.
• The number 0 can be written as: 0 + i0.

Calculation:

Since, z = (2x - 3y) + i(x2 - y2) = 0, it means that both the real and the imaginary parts of the z are equal to 0.

i.e. Re{z} = 2x - 3y = 0.

Concept:

Equality of complex numbers:

Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if x1 = x2 and y1 = y2

Or Re (z1) = Re (z2) and Im (z1) = Im (z2)

Calculation:

Given that, 2x2 + (x2 - y)i = (8 - 3i)

Comparing the real and imaginary parts,

⇒ 2x² = 8 

⇒ x² = 4

⇒ x = -2, 2

And, x2 - y = -3

⇒ 4 - y = -3               [∵ x² = 4]

⇒ y = 7 

Hence, option (3) is correct.

Concept:

Equality of complex numbers.

Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if x1 = x2 and y1 = y2

Or Re (z1) = Re (z2) and Im (z1) = Im (z2).

Calculations:

Multiplying numerator and denominator by i

Re(z) = -1 

Im(z) = -1

Hence , option 4 is correct 

Concept:

Equality of complex numbers.

Two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ are equal if and only if x₁ = x₂ and y₁ = y₂

Or Re (z₁) = Re (z₂) and Im (z₁) = Im (z₂).

Calculations:

Given:  

As we know i2 = -1 

Comparing real and imaginary parts, we get.

Concept:

Let z = x + iy be a complex number,

Where x is called the real part of the complex number or Re (z) and y is called the Imaginary part of the complex number or Im (z)

Conjugate of z = z̅ = x - iy  

Calculation:

1. The difference of Z and its conjugate is an imaginary number.

Consider z = a + ib           ....(i)

conjugate of z = z̅ = a - ib      ....(ii)

eq(i) - eq (ii)

z - z̅ = a + ib - a + ib

⇒ 2ib

Thus it is clear that the difference of z and its conjugate is an imaginary number.

2. The sum of Z and its conjugate is a real number.

eq (i) + eq(ii)

z + z̅ = a + ib + a - ib

⇒ 2a

Thus it is clear that the sum of Z and its conjugate is a real number.

So, Both 1 and 2 are correct.

Concept:

Let A = x₁ + iy₁ and B = x₂ + iy₂ 

If A = B then x₁ = x₂ and y₁ = y₂

Calculations:

Given 

⇒

⇒     

We know i² = -1 

⇒

⇒

⇒A + iB = 2i

⇒ A + iB = 0 + 2i

Comparing real and imaginary parts, we get.

⇒A = 0