PROPERTIES OF MODULUS OF COMPLEX NUMBERS

Practice Questions

Questions 1-4 : Find the modulus of each of the following complex numbers

Question 1 :

2/(3 + 4i)

Answer :

|(2/(3 + 4i)| = |2|/|(3 + 4i)|

= 2/√(3² + 4²)

= 2/√(9 + 16)

= 2/√25

= 2/5

Question 2 : 

(2 - i)/(1 + i) + (1 - 2i)/(1 - i)

Answer :

= |(2 - i)/(1 + i) + (1 - 2i)/(1 - i)|

= |(2 - i)|/|(1 + i)| + |(1 - 2i)|/|(1 - i)| ----(1)

|(2 - i)| = √(2² + 1²) = √5

|1 + i| = √(1² + 1²) = √2

|1 - 2i| = √(1² + 2²) = √5

|1 - i| = √(1² + 1²) = √2

Substitute the values in (1). 

 = (√5/√2) + (√5/√2)

= 2√5/√2

= √2√5

= √10

Question 3 : 

(1 - i)¹⁰

Answer :

|zⁿ|  =  |z|ⁿ

(1 - i)¹⁰ =  {(1 - i)²}⁵

= (1² + i² - 2i)⁵

= (1 - 1 - 2i)⁵

= (- 2i)⁵

= -32i⁵

= |-32i|

= √(-32)²

= 32

Question 4 :

2i(3− 4i)(4 − 3i) 

Answer :

|2i(3− 4i)(4 − 3i)| = |2i||3 - 4i||4 - 3i|

= √2² √3² + (-4)²√4² + (-3)² 

= √4√25√25

= 2(5)(5)

= 50 

Question 5 :

For any two complex numbers z₁ and z₂ , such that |z₁| = |z₂|  =  1 and z₁ z₂ ≠ -1, then show that z₁ + z₂/(1 + z₁ z₂) is a real number.

Answer :

Let z₁  =  1 and z₂  =  i

|z₁|  =  √1² + 0²  =  1

|z₂|  =  √0 + 1²  =  1

z₁ + z₂  =  1 + i

z₁  z₁  =   i

By applying the  values of z₁ + z₂ and z₁  z₂  in the given statement, we get 

z₁ + z₂/(1 + z₁ z₂)    =  (1 + i)/(1 + i)  =  1

1 is real. Hence it is proved.

Question 6 :

Which one of the points 10 − 8i , 11 + 6i is closest to 1 + i

Answer :

Let the given points as A(10 - 8i), B (11 + 6i) and C (1 + i).

To find which point is more closer, we have to find the distance between the points AC and BC.

√162 > √125

Hence the point B is closer to C.

Question 7 :

Simplify (2 + 6i) + (9 - 2i)

Answer :

= (2 + 6i) + (9 - 2i)

= 2 + 9 + 6i - 2i

= 11 + 4i

Question 8 :

Simplify (8 + 3i) - (-1 + 5i)

Answer :

= (8 + 3i) - (-1 + 5i)

= 8 + 3i + 1 - 5i

= 8 + 1 + 3i - 5i

= 9 - 2i

Question 9 :

Simplify (1 + 4i)/(3 + 2i)

Answer :

= (1 + 4i)/(3 + 2i)

Multiplying the numerator and denominator by the conjugate of the denominator.

Conjugate of 3 + 2i is 3 - 2i

= [(1 + 4i)/(3 + 2i)] · [(3 - 2i) / (3 - 2i)]

= [(1 + 4i)(3 - 2i)/(3 + 2i)(3 - 2i)]

(1 + 4i)(3 - 2i) = 3 - 2i + 12i - 8i²

= 3 + 10i - 8(-1)

= 3 + 10i + 8

= 11 + 10i

(3 + 2i)(3 - 2i) = 3² - (2i)²

= 9 - 4i²

= 9 - 4(-1)

= 9 + 4

= 13

So, the required complex number is (11 + 10i)/13.

Question 10 :

i³

Answer :

i³ = i² · i

= -1 · i

= -·i

Question 11 :

Solve the following equations for real x and y.

3 + 5i + x - iy = 6 - 2i

Answer :

3 + 5i + x - iy = 6 - 2i

Combining the real and imaginary parts, we get

3 + x + 5i - iy = 6 - 2i

(3 + x) + i(5 - y) = 6 - 2i

So, the values of x and y are 3 and 7 respectively.

Question 12 :

Solve the following equations for real x and y.

x + iy = (1 - i)(2 + 8i)

Answer :

x + iy = (1 - i)(2 + 8i)

x + iy = 2 + 8i - 2i - 8i²

= 2 + 6i - 8(-1)

= 2 + 6i + 8

x + iy = 10 + 6i

By comparing the corresponding terms, we get

x = 10 and y = 6

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