Test: Modulus & Argument (May 15) - JEE MCQ

Concept:
Multiplicative Inverse of a complex number, z  = 1/z
Application:
Multiplicative Inverse of z = 1 + i is z^-1 i.e.
Multiplicative Inverse of z = 1/z
Puttng z = 1 + i, we get,
Multiplicative Inverse of 1 + i  = 1/1+i
Rationalizing, 

Concept:

• (a + b)² = a² + b² + 2ab and
• i² = -1
• The conjugate of the complex function is given by changing the sign of i.
• If z = x + iy , then   = x - iy

Explanation:
Given complex number, z = (6 + 5i)2
i.e.  (6 + 5i) ² = 6² + (5i)² +2(6)(5i)
⇒(6 + 5i) ² = 36 - 25 +60i
⇒(6 + 5i) ² = 11 + 60i
⇒ z = 11 + 60i
Thus,
Conjugate of z = = 11 - 60i
ADDITIONAL INFORMATION :
If z = x + iy , then

Concept Used:
The multiplicative inverse of a complex number a + bi is given by:

Explanation:
Multiplicative inverse of the complex number 2 - 3i, is 

So, the multiplicative inverse of 2 - 3i is 

Concept:
Quadratic Polynomial: A polynomial in which the highest degree monomial is of the second degree, A quadratic polynomial is also known as a second-order polynomial.
The general form of a quadratic polynomial is ax² + bx + c, where a ≠ 0.
Conjugate of Complex Number: A conjugate of a complex number is another complex number which has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign.
Formula Used:
(a - b)² = a² + b² - 2ab
(a² - b²) = (a - b)(a + b)
Calculation:
We have,
z = 1 + 3i is one of the root.
Other root is conjugate of z = 1 - 3i
So the polynomial is quadratic polynomial with two roots. 
we can write the expression for our quadratic polynomial as follows:
⇒ x = (z - z₁)(z - z₂)
⇒ x = (z - (1 + 3i)) (z - (1 - 3i))
⇒ x = (z - 1 - 3i)(z - 1 + 3i)
⇒ x = [(z - 1) - 3i] [(z - 1) + 3i]
⇒ x = (z - 1)² - (3i)²
⇒ x = z² - 2z + 1 - 9i²
⇒ x = z² - 2z + 1 - 9(-1)
⇒ x = z² - 2z + 1 + 9
⇒ x = z² - 2z + 10
∴ If z = 1 + 3i, then the polynomial generated by its roots is z² - 2z + 10.

Concept:
If a number ω = a + ib is purely imaginary, then

• a = 0 
• and ω =

Calculation:
Given, z is a complex number such that  is purely imaginary,


∴ The correct option is (3).

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: (1 + i) (x + iy) = 2 + 4i
⇒ x + iy + ix + i²y = 2 + 4i
⇒ (x – y) + i(x + y) = 2 + 4i
Equating real and imaginary part,
x - y = 2         …. (1)
x + y = 4        …. (2)
Adding equation 1 and 2, we get
x = 3
Now,
5x = 5 × 3 = 15

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 = 
Calculations:

As we know i² = -1 

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

1 Concept:
Let z = x + iy be a complex number.

• Modulus of z = 
• arg (z) = arg (x + iy) = 
• For calculating the conjugate, replace i with -i.
• Conjugate of z = x – iy

Calculation:
Let z = (i - i²)³
⇒ z = i³ (1 - i) 3  = - i (1 - i)³
For calculating the conjugate, replace i with -i.
⇒ z̅  =  -(- i) (1 - (- i))³
⇒ z̅  =  i(1 + i)³
Using (a + b)³ = a³ + b³ + 3a²b + 3ab²
⇒ z̅  =  i(1 + i³ +3 ×1² × i + 3 × i² × 1 ) 
⇒ z̅  =  i(1 - i + 3i - 3) 
⇒ z̅  =  i(-2 + 2i)
⇒ z̅  = -2i + 2i²
⇒ z̅  = -2 - 2 i
So, the conjugate of  (i - i²)³ is -2 - 2i

Concept:
i² = -1
i³ = - i
i⁴ = 1
i⁴ⁿ = 1
Calculation:
We have to find the value of (i² + i⁴ + i⁶ +... + i²ⁿ)
(i² + i⁴ + i⁶ +... + i²ⁿ) = (i² + i⁴) + (i⁶ + i⁸) + …. + (i^2n-2 + i²ⁿ)
= (-1 + 1) + (-1 + 1) + …. (-1 + 1)
= 0 + 0 + …. + 0
= 0

Concept: 
Let z = x + iy be a complex number.

• Modulus of z = 
• arg (z) = arg (x + iy) = 
• Conjugate of z = z̅ = x – iy

Calculation:
Given complex number is z = 

Conjugate of z =