Test: Representation of Complex Numbers (May 13) - JEE MCQ

Concept:

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

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

Therefore, required polar form is z = 

For I^st quadrant : x > 0, y > 0 and 0 < θ < π/2
For II^nd quadrant : x < 0, y > 0 and π/2 < θ < π
For III^rd quadrant : x < 0, y < 0 and π < θ < 3π/2
For IV^th quadrant : x > 0, y < 0 and 3π/2 < θ < 2π

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|.

 

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| =  is the modulus of z and θ is known as the argument or amplitude of z.
Formula Used:
(a^x)^y = a^xy
a^{x +  y} = a^x. a^y
Calculation: 
We have,
⇒ z = (i²⁵)³        -------(1)
⇒ z = i^{25 × 3}
⇒ z = i⁷⁵
⇒ z = i^{(72 + 3)}     -------(2)
⇒ z = (i⁷²) × (i³)
⇒ z = (i^{4 × 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

⇒ r = |z| = 1
Similarly, to get the value of θ,


As x = 0 and y = - 1 < 0. the coordinate (x, y) lies in the IV quadrant.
In the IV quadrant tangent function is negative.

The polar form of a complex number,
⇒ z = r(cosθ + isinθ)

∴ The polar form of the complex number 

Concept:
Power of i,
for any integer, i^{4k + 1}  = i

Calculation:
Let  z = (i¹⁵)³ = (i)⁴⁵ = i^{4 × 11 + 1} = (i⁴)¹¹ i = i = 0 + i
polar form of z = r(cos θ + i sinθ)

Concept:
when z = x + iy then, Principal amplitude of a complex number, 

x > 0, y < 0, The point lies in IVth quadrant.
Calculation:
Let θ be the principal value of amplitude of  
Since, tan θ =  lies in IV^th quadrant.
tan θ = tan 

CONCEPT:
Let the point P represent the nonzero complex number z = x + iy.
Here  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  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 –

CALCULATION:
Given complex number is 

By squaring and adding, we get:


Since it is in third quadrant

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

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

CONCEPT:
If P represent the nonzero complex number z = x + iy.
Here  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, 
For the second quadrant 
For the third quadrant 
For the fourth quadrant 
Note: The polar form z = r (cosθ + i sinθ) is abbreviated as r.cisθ.
CALCULATION:

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

Concept:
Power of i,
for any integer, i^{4k + 1}  = i
cos (π/2)  = 0
sin (π/2) = 1
Calculation:
Let  z = (i¹⁵)³ = (i)⁴⁵ = i^{4 × 11 + 1} = (i⁴)¹¹ i = i = 0 + i
polar form of z = r(cos θ + i sinθ)

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

Here  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 –

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.

∴ on comparing with z = r (cosθ + i sinθ), we can write as