Properties of Complex Numbers MCQ Quiz in मल्याळम - Objective Question with Answer for Properties of Complex Numbers - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

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:

Let z = 

⇒ z =   = 

⇒ z =   =  

⇒ z =  ×  =  = i 

⇒ z = i  

We know that, Conjugate of z = z̅ = x - iy   

⇒  = - i  . 

The correct option is 1

Concept:

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

Calculation:

Let two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂

Where x₁ = Re (z₁), x₂ = Re (z₂) and y₁ = Im (z₁), y₂ = Im (z₂).

Now,

z₁z₂ = (x₁ + iy₁) (x₂ + iy₂)

z₁z₂ = (x₁x₂ – y₁y₂)  + i (x₁y₂ + y₁x₂)

∴ Im (z₁z₂) = (x₁y₂ + y₁x₂)

= Re (z₁) Im (z₂) + Im (z₁) Re (z₂)

Concept:

Conjugate of a complex number:

For any complex number z = x + iy  the conjugate z̅ is given by  z̅ = x - iy

Calculation:

z = 

z = 

z = 

z = 

z = 

z = 1 - i

Conjugate of z = z̅  = 1 + i

Concept:

Modulus of the Complex Number z = a + ib is given by:

Mod(z) = |z| = 

= 

Solution:

 =  =  =  =  =  = 2 - i

Modulus of the complex number z = 2 - i is 

 =  =  

Concept:

The argument of a complex number z = x + iy

arg(z) = tan^-1

The angle is according to the sign of the y and x

• Both positive then angle ∈ [0°, 90°]
• Negative x and positive y then angle ∈ [90°, 180°]

Calculation:

Given complex number z = 2 + i 2√3

arg(z) = tan^-1

arg(z) = tan^-1

arg(z) = 60° (∵ Both positive then angle ∈ [0°, 90°])

Concept:

Complex Numbers: For a complex number z = a + ib, the following are defined:

• Conjugate: z̅ = a - ib
• arg(z) = θ = .
• arg(z₁.z₂) = arg(z₁) + arg(z₂).

Calculation:

Given that z̅ + iw̅ = 0.

⇒ z̅ = -iw̅

Taking conjugates, we get:

⇒ z = iw

Multiplying by i, we get:

⇒ w = -iz

Now, arg(zw) = π.

⇒ arg(z.(-iz)) = π

⇒ arg(-i.z²) = π

⇒ arg(-i) + 2 arg(z) = π

⇒  + 2 arg(z) = π

⇒ 2 arg(z) = 

⇒ arg(z) = .

Concept:

Multiplicative Inverse of a complex number, z  = 

Application:

Multiplicative Inverse of z = 1 + i is z^-1 i.e.

Multiplicative Inverse of z = 

Puttng z = 1 + i, we get,

Multiplicative Inverse of 1 + i  = 

Rationalizing, 

=  

=  (Since i²=-1)

=  

Concept

The general form of a complex number is z = x + iy. The polar representation of z

is z = r(cos θ + i sin θ). Here, r is the modulus of z and θ is called the amplitude or argument of the complex number. The formula to find the amplitude of a complex

number is:  and ∣z∣ = 

Calculation

⇒  = 

⇒ 

⇒  = 0 + i

Now, Argument =  = 

Modulus is ∣z∣ =   = 

∴ Argument and modulus of  are respectively  and 1.

Concept:

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

Modulus of z = |z| = 

Calculations:

Let z = x + iy = (1 + i)² = 1² + 2i + i²         (∵ (a + b)² = a² + b² + 2ab)

z = 1 + 2i - 1                                (∵ i² = -1)

∴ z = x + iy = 2i

So, x = 0  and y = 2

As we know that if z = x + iy be any complex number, then its modulus is given by, |z| = 

∴ |z| =  

Concept:

Properties of complex numbers

z  = , if z is purely real

z  = - , if z is either 0 or purely imaginary

arg(zn​) = n arg(z)

Calculation:

Given that

 = 0

z1 = -i z22

arg(z1) - arg(z2) = 2π

arg(-i z22) - arg(z2) = 2π

arg(-i) + arg(z22) - arg(z2) = 2π

+ 2 arg(z2) - arg(z2) = 2π

arg(z2) = 2π +  =