Exercise 5.1

Question 1:

Express the given complex number in the form a + ib: (5i)(-3i/5)

Answer:

(5i)(-3i/5) = (-5 * 3/5) * i * i

                   = -3 * i²

                   = -3 * (-1)                  [Since i² = -1]

                   = 3

Question 2:

Express the given complex number in the form a + ib: i⁹ + i¹⁹

Answer:

i⁹ + i¹⁹ = i^{4*2 + 1} + i^{4*4 + 3}

            = (i⁴)² * i + (i⁴)⁴ * i³

            = (1)² * i + (1)⁴ * i * i²                 [Since i⁴ = 1]

            = i + i * (-1)                                  [Since i² = -1]

            = i – i

            = 0

Question 3:

Express the given complex number in the form a + ib: i^-39

Answer:

i^-39 = i^{-4 * 9 - 3}

     = (i^-4)⁹ * i^-3

     = 1⁹ * i^-3                     [Since i⁴ = 1]

    = i^-3

    = 1/ i³

    = i⁴/ i³                        [Since i⁴ = 1]

    = i

Question 4:

Express the given complex number in the form a + ib:  3(7 + i7) + i(7 + i7)

Answer:

Given, 3(7 + i7) + i(7 + i7) = 21 + 21i + 7i + 7i²

                                              = 21 + 21i + 7i - 7            [Since i² = -1]

                                              = 14 + 28i

Question 5:

Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)

Answer:

Given, (1 – i) – (–1 + i6) = 1 – i + 1 - 6i

                                          = 2 - 7i   

Question 6:

Express the given complex number in the form a + ib: (1/5 + 2i/5) – (4 + 5i/2)

Answer:

Given, (1/5 + 2i/5) – (4 + 5i/2) = 1/5 + 2i/5 – 4 - 5i/2

                                                      = (1/5 - 4) + (2i/5 - 5i/2)

                                                      = -19/5 + (4i - 25i)/10    

                                                      = -19/5 - 21i/10     

 

Question 7:

Express the given complex number in the form a + ib: [(1/3 + 7i/3) + (4 + i/3)] – (-4/3 + i)

Answer:

Given, [(1/3 + 7i/3) + (4 + i/3)] – (-4/3 + i) = 1/3 + 7i/3 + 4 + i/3 + 4/3 – i

                                                                          = (1/3 + 4 + 4/3) + i(7/3 + 1/3 - 1)

                                                                          = 17/3 + 5i/3

Question 8:

Express the given complex number in the form a + ib: (1 – i)⁴

Answer:

Given, (1 – i)⁴ = [(1 – i)²]²   

             = [1 + i² – 2i]²

            

             = [1 – 1 – 2i]²                  [Since i² = -1]

            

             = (– 2i)²

                         = 4i²

                         = 4 * (-1)                         [Since i² = -1]

                         = -4

Question 9:

Express the given complex number in the form a + ib: (1/3 + 3i)³

Answer:

Given, (1/3 + 3i)³ = (1/3)³ + (3i)³ + 3(1/3)(3i) (1/3 + 3i)

                    = 1/27 + 27i³ + (3i) (1/3 + 3i)  

 

                    = 1/27 + 27i * i² + i + 9i²   

 

                    = 1/27 + 27i * (-1) + i + 9(-1)                  [Since i² = -1]

                    = 1/27 - 27i + i – 9

 

                    = (1/27 - 9) – 26i

 

                    = -242/27 – 26i

Question 10:

Express the given complex number in the form a + ib: (-2 – i/3)³

Answer:

Given, (-2 – i/3)³ = (-1)³ (2 + i/3)³

                              = -[2³ + (i/3)³ + 3(2)(i/3)(2 + i/3)]

                   = -[8 + i³/27+ (2i) (2 + i/3)]  

 

                    = -[8 – i/27 + 4i + 2i²/3]  

 

                    = -[8 – i/27 + 4i - 2/3]                  [Since i² = -1]

                   

                    = -[22/3 + 107i/27]

 

                    = -22/3 – 107i/27

Question 11:

Find the multiplicative inverse of the complex number 4 – 3i.

Answer:

Let z = 4 – 3i

Then,

z = 4 + 3i

|z|² = 4² + (-3)² = 16 + 9 = 25

Therefore, the multiplicative inverse of 4 – 3i is given by              

z^-1 = z /|z|² = (4 + 3i)/25 = 4/25 + 3i/25

 

 

Question 12:

Find the multiplicative inverse of the complex number √5 + 3i.

Answer:

Let z = √5 + 3i

Then,

z = √5 - 3i

|z|² = (√5)² + (-3)² = 5 + 9 = 14

Therefore, the multiplicative inverse of √5 - 3i is given by             

z^-1 = z/|z|² = (√5 - 3i)/14 = √5/14 - 3i/14

Question 13:

Find the multiplicative inverse of the complex number -i.

Answer:

Let z = -i

Then,

z = i

|z|² = 1² = 1

Therefore, the multiplicative inverse of -i is given by             

z^-1 = z/|z|² = i/1 = i

Question 14:

Express the following expression in the form of a + ib.

{(3 + i√5)(3 - i√5)}/{(√3 + i√2) - (√3 - i√2)}

Answer:

Given, {(3 + i√5)(3 - i√5)}/{(√3 + i√2) - (√3 - i√2)}

= {3² - (i√5)²}/(√3 + i√2 - √3 + i√2)                               [(a - b)(a + b) = a² – b²]

= (9 - 5i²)/i2√2

= {9 - 5 * (-1)}/i2√2            [Since i² = -1]

= (9 + 5)/i2√2

= 14/i2√2

= 7/i√2

= (7/i√2) * (i/i)

= 7i/(√2 * i²)

= 7i/(-√2)                             [Since i² = -1]

= -7i/√2

= (-7i/√2) * (√2/√2)

= -7√2i/2

 

                                                                     Exercise 5.2

Question 1:

Find the modulus and the argument of the complex number z = -1 - √3

Answer:

Given, z = -1 - √3

Let r cos θ = -1 and r sin θ = -√3

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = (-1)² + (-√3)²

=> (r² cos² θ + r² sin² θ) = 1 + 3

=> r² (cos² θ + sin² θ) = 4  

=> r² = 4                                   [Since cos² θ + sin² θ = 1]

=> r = 2                                    [Since r > 0]

So, Modulus = 2

Now, 2cos θ = -1 and 2sin θ = -√3

=> cos θ = -1/2 and sin θ = -√3/2

Since both the values of sin θ and cos θ are negative and sin θ and cos θ are negative in III

quadrant,

So, argument θ = -(π – π/3) = -2π/3

Thus, the modulus and argument of the complex number -1 - √3 are 2 and -2π/3 respectively.

Question 2:

Find the modulus and the argument of the complex number z = -√3 + i

Answer:

Given, z = -√3 + i

Let r cos θ = -√3 and r sin θ = 1

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = (-√3)² + 1²

=> (r² cos² θ + r² sin² θ) = 3 + 1

=> r² (cos² θ + sin² θ) = 4  

=> r² = 4                                   [Since cos² θ + sin² θ = 1]

=> r = 2                                    [Since r > 0]

So, Modulus = 2

Now, 2cos θ = -√3 and 2sin θ = 1

=> cos θ = -√3/2 and sin θ = 1/2

Since sin θ is positive and cos θ is negative.

So, sin θ and cos θ are in II quadrant.

Now, argument θ = π – π/6 = 5π/6

Thus, the modulus and argument of the complex number -√3 + i are 2 and 5π/6 respectively.

Question 3:

Convert the given complex number in polar form: 1 – i

Given, z = 1 – i

Let r cos θ = 1 and r sin θ = -1

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = 1² + (-1)²

=> (r² cos² θ + r² sin² θ) = 1 + 1

=> r² (cos² θ + sin² θ) = 2  

=> r² = 2                                   [Since cos² θ + sin² θ = 1]

=> r = √2                                    [Since r > 0]

So, Modulus = √2

Now, √2cos θ = 1 and √2sin θ = -1

=> cos θ = 1/√2 and sin θ = -1/√2

Since sin θ is negative and cos θ is positive.

So, sin θ and cos θ are in IV quadrant.

Now, argument θ = -π/4

Now, 1 – i = r cos θ + i * r sin θ

                   = √2{cos (-π/4) + i sin (-π/4)}

                   = √2(cos π/4 - i sin π/4)

This is the required polar form.

Question 4:

Convert the given complex number in polar form: -1 + i

Given, z = -1 + i

Let r cos θ = -1 and r sin θ = 1

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = (-1)² + 1²

=> (r² cos² θ + r² sin² θ) = 1 + 1

=> r² (cos² θ + sin² θ) = 2  

=> r² = 2                                     [Since cos² θ + sin² θ = 1]

=> r = √2                                    [Since r > 0]

So, Modulus = √2

Now, √2cos θ = -1 and √2sin θ = 1

=> cos θ = -1/√2 and sin θ = 1/√2

Since sin θ is positive and cos θ is negative.

So, sin θ and cos θ are in II quadrant.

Now, argument θ = π - π/4 = 3π/4

Now, -1 + i = r cos θ + i * r sin θ

                   = √2{cos (3π/4) + i sin (3π/4)}

                   = √2(cos 3π/4 + i sin 3π/4)

This is the required polar form.

Question 5:

Convert the given complex number in polar form: -1 - i

Given, z = -1 - i

Let r cos θ = -1 and r sin θ = -1

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = (-1)² + (-1)²

=> (r² cos² θ + r² sin² θ) = 1 + 1

=> r² (cos² θ + sin² θ) = 2  

=> r² = 2                                   [Since cos² θ + sin² θ = 1]

=> r = √2                                    [Since r > 0]

So, Modulus = √2

Now, √2cos θ = -1 and √2sin θ = -1

=> cos θ = -1/√2 and sin θ = -1/√2

Since both sin θ and cos θ are negative.

So, sin θ and cos θ are in III quadrant.

Now, argument θ = -(π - π/4) = -3π/4

Now, -1 - i = r cos θ + i * r sin θ

                   = √2{cos (-3π/4) + i sin (-3π/4)}

                   = √2(cos 3π/4 - i sin 3π/4)

This is the required polar form.

Question 6:

Convert the given complex number in polar form: -3

Given, z = -3

Let r cos θ = -3 and r sin θ = 0

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = 3² + 0²

=> (r² cos² θ + r² sin² θ) = 9

=> r² (cos² θ + sin² θ) = 9  

=> r² = 9                                   [Since cos² θ + sin² θ = 1]

=> r = 3                                    [Since r > 0]

So, Modulus = 3

Now, 3cos θ = -3 and 3sin θ = 0

=> cos θ = -1 and sin θ = 0

Now, argument θ = π

Now, -3 = r cos θ + i * r sin θ

               = 3(cos π + i sin π)

This is the required polar form.

 

Question 7:

Convert the given complex number in polar form: √3 + i

Given, z = √3 + i

Let r cos θ = √3 and r sin θ = 1

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = (√3)² + (1)²

=> (r² cos² θ + r² sin² θ) = 3 + 1

=> r² (cos² θ + sin² θ) = 4  

=> r² = 4                                   [Since cos² θ + sin² θ = 1]

=> r = 2                                    [Since r > 0]

So, Modulus = 2

Now, 2cos θ = √3 and 2sin θ = 1

=> cos θ = √3/2 and sin θ = 1/2

Since both sin θ and cos θ are positive.

So, sin θ and cos θ are in I quadrant.

Now, argument θ = π/6

Now, √3 + i = r cos θ + i * r sin θ

                   = 2(cos π/6 + i sin π/6)

This is the required polar form.

Question 8:

Convert the given complex number in polar form: i

Given, z = i

Let r cos θ = 0 and r sin θ = 1

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = 0² + 1²

=> (r² cos² θ + r² sin² θ) = 1

=> r² (cos² θ + sin² θ) = 1  

=> r² = 1                                   [Since cos² θ + sin² θ = 1]

=> r = 1                                    [Since r > 0]

So, Modulus = 1

Now, cos θ = 0 and sin θ = 1

Now, argument θ = π/2

Now, i = r cos θ + i * r sin θ

             = cos π/2 + i sin π/2

This is the required polar form.                           

 

 

                                                                     Exercise 5.3

Question 1:

Solve the equation x² + 3 = 0

Answer:

The given quadratic equation is x² + 3 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = 1, b = 0, and c = 3

Therefore, the discriminant of the given equation is

D = b² – 4ac = 0² – 4 * 1 * 3 = –12

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {0 ± √(-12)}/2

                                                              = {±√(-1) * √12}/2

                                                              = ±2√3i/2                    [Since i = √(-1)]

                                                              = ±i√3

Question 2:

Solve the equation 2x² + x + 1 = 0

Answer:

The given quadratic equation is 2x² + x + 1 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = 2, b = 1, and c = 1

Therefore, the discriminant of the given equation is

D = b² – 4ac = 1² – 4 * 2 * 1 = 1 – 8 = –7

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {-1 ± √(-7)}/(2 * 2)

                                                             = {-1 ± √(-1) * √7}/4

                                                             = (-1 ± i√7)/4                          [Since i = √(-1)]

Question 3:

Solve the equation x² + 3x + 9 = 0

Answer:

The given quadratic equation is x² + 3x + 9 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = 1, b = 3, and c = 9

Therefore, the discriminant of the given equation is

D = b² – 4ac = 3² – 4 * 1 * 9 = 9 – 36 = –27

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {-3 ± √(-27)}/(2 * 1)

                                                             = {-3 ± √(-1) * √27}/2

                                                             = (-3 ± i3√3)/2                          [Since i = √(-1)]

Question 4:

Solve the equation –x² + x – 2 = 0

Answer:

The given quadratic equation is –x² + x – 2 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = –1, b = 1, and c = –2

Therefore, the discriminant of the given equation is

D = b² – 4ac = 1² – 4 * (–1) * (–2) = 1 – 8 = –7

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {-1 ± √(-7)}/(2 * -1)

                                                              = {-1 ± √(-1) * √7}/(-2)

                                                               = (-1 ± i√7)/ (-2)                       [Since i = √(-1)]

Question 5:

Solve the equation x² + 3x + 5 = 0

Answer:

The given quadratic equation is x² + 3x + 5 = 0

On comparing the given equation with ax2 + bx + c = 0, we get

a = 1, b = 3, and c = 5

Therefore, the discriminant of the given equation is

D = b² – 4ac = 3² – 4 * 1 * 5 =9 – 20 = –11

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {-3 ± √(-11)}/(2 * 1)

                                                             = {-3 ± √(-1) * √11}/2

                                                             = (-3 ± i√11)/2                       [Since i = √(-1)]

Question 6:

Solve the equation x² – x + 2 = 0

Answer:

The given quadratic equation is x² – x + 2 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = 1, b = –1, and c = 2

Therefore, the discriminant of the given equation is

D = b² – 4ac = (–1)² – 4 * 1 * 2 = 1 – 8 = –7

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {1 ± √(-7)}/(2 * 1)

                                                             = {1 ± √(-1) * √7}/2

                                                             = (1 ± i√7)/2                       [Since i = √(-1)]

Question 7:

Solve the equation √2x² + x + √2 = 0

Answer:

The given quadratic equation is √2x² + x + √2 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = √2, b = 1, and c = √2

Therefore, the discriminant of the given equation is

D = b² – 4ac = 1² – 4 * √2 * √2 = 1 – 8 = –7

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {-1 ± √(-7)}/(2 * √2)

                                                             = {-1 ± √(-1) * √7}/2√2

                                                             = (-1 ± i√7)/2√2                       [Since i = √(-1)]

Question 8:

Solve the equation √3x² - √2x + 3√3 = 0

Answer:

The given quadratic equation is √3x² - √2x + 3√3 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = √3, b = -√2, and c = 3√3

Therefore, the discriminant of the given equation is

D = b² – 4ac = (-√2)² – 4 * √3 * 3√3 = 2 – 36 = –34

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {√2 ± √(-34)}/(2 * √3)

                                                             = {√2 ± √(-1) * √34}/2√3

                                                             = (√2 ± i√34)/2√3                       [Since i = √(-1)]

Question 9:

Solve the equation: x² + x + 1/√2 = 0

Answer:

The given quadratic equation is x² + x + 1/√2 = 0

=> √2x² + √2x + 1 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = √2, b = √2, and c = 1

Therefore, the discriminant of the given equation is

D = b² – 4ac = (√2)² – 4 * √2 * 1 = 2 – 4√2

Therefore, the required solutions = (-b ± √D)/2a

                                                             = [-√2 ± √(2 - 4√2)/(2 * √2)

                                                             = [-√2 ± √{2(1 - 2√2)}/2√2

                                                             = [-√2 ± √2√{2√2 - 1)} * √(-1)]/2√2

                                                             = [-1 ± √(2√2 - 1)i]/2                             [Since i = √(-1)]

Question 10:

Solve the equation x² + x/√2 + 1 = 0

 

Answer:

The given quadratic equation is x² + x/√2 + 1 = 0

=> √2x² + x + √2 = 0

On comparing the given equation with ax² + bx + c = 0, we get

a = √2, b = 1, and c = √2

Therefore, the discriminant of the given equation is

D = b² – 4ac = 1² – 4 * √2 * √2 = 1 – 8 = –7

Therefore, the required solutions = (-b ± √D)/2a

                                                             = {-1 ± √(-7)}/(2 * √2)

                                                             = {-1 ± √(-1) * √7}/2√2

                                                             = (-1 ± i√7)/2√2                       [Since i = √(-1)]

 

  

                                                       Miscellaneous Exercise on chapter 5

Question 1:

Evaluate: [i¹⁸ + (1/i)²⁵]³

Answer:

Given, [i¹⁸ + (1/i)²⁵]³

= [i^4*4+2 + 1/i^4*6+1]³

= [(i⁴)² * i² + 1/{(i⁴)⁶ * i}]³

= [-1 + 1/i]³                                  [Since i⁴ = 1 and i² = -1]

= [-1 + i/i²]³

= [-1 - i]³

= (-1)³ * [1 + i]³

= - {1 + i³ + 3 * 1 * i(1 + i)}

= - {1 - i + 3i + 3i²}                       [Since i² = -1]

= - {1 - i + 3i - 3}

= -(-2 + 2i)

= 2 – 2i

Question 2:

For any two complex numbers z₁ and z₂, prove that

Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

Answer:

Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂

Now, z₁z₂ = (x₁ + iy₁)(x₂ + iy₂)

                  = x₁(x₂ + iy₂) + iy₁(x₂ + iy₂)  

                  = x₁x₂ + i x₁y₂ + iy₁x₂ + i² y₂y₂

                  = x₁x₂ + i x₁y₂ + iy₁x₂ - y₂y₂              [Since i² = -1]

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

So, Re (z₁z₂) = x₁x₂ – y₁y₂

=> Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

Hence, proved.

Question 3:

Reduce {1/(1 – 4i) – 2/(1 + i)}{(3 – 4i)/(5 + i)} to the standard form.

Answer:

Given, {1/(1 – 4i) – 2/(1 + i)}{(3 – 4i)/(5 + i)}

= [{(1 + i) – 2(1 – 4i)}/{(1 – 4i)(1 + i)}]{(3 – 4i)/(5 + i)}

= [{1 + i – 2 + 8i}/{1 + i – 4i – 4i²}]{(3 – 4i)/(5 + i)}

= [{-1 + 9i}/{1 + i – 4i + 4}]{(3 – 4i)/(5 + i)}

= [{-1 + 9i}/{5 – 3i}]{(3 – 4i)/(5 + i)}

= [{-3 + 4i + 27i - 36 i²}/{25 + 5i – 15i – 3i²}]

= [{-3 + 4i + 27i + 36}/{25 + 5i – 15i + 3}]

= (33 + 31i)/(28 - 10i)

= (33 + 31i)/{2 * (14 - 5i)}

= [(33 + 31i)/{2 * (14 - 5i)}] *{(14 + 5i)/(14 + 5i)}

= [{462 + 165i + 434i + 155 i²}/[2 * (14² - (5i)²]

= [{462 + 165i + 434i - 155}/{2 * (196 + 25)}

= (307 + 599i)/(2 * 221)

= (307 + 599i)/442

= 307/442 + 599i/442

This is the required standard form.

Question 4:

If x – iy = √{(a - ib)/(c - id)}, prove that (x² + y²)² = (a² + b²)/(c² + d²)

Answer:

Given, x – iy = √{(a - ib)/(c - id)}

                       = √[{(a - ib)/(c - id)} * {(c + id)/(c + id)}]

                       = √[{(ac + bd) + i(ad - bc)}/(c² + d²)]

Now, (x – iy)² = {(ac + bd) + i(ad - bc)}/(c² + d²)

=> x² - y² - 2ixy = {(ac + bd) + i(ad - bc)}/(c² + d²)

=> (x² - y²) - 2ixy = (ac + bd)/(c² + d²) + i(ad - bc)/(c² + d²)

On comparing real and imaginary part, we get

x² - y² = (ac + bd)/(c² + d²) and -2xy = (ad - bc)/(c² + d²)  ………………1

Now, (x² + y²)² = (x² - y²)² + 4 x²y²

                          = {(ac + bd)/(c² + d²)}² + {(ad - bc)/(c² + d²)}²

                          = (a²c² + b²d² + 2abcd + a²d² + b²c² – 2abcd)/(c² + d²)²

                          = (a²c² + b²d² + a²d² + b²c²)/(c² + d²)²   

                          = {a²(c² + d²) + b²(c² + d²)}/(c² + d²)² 

                          = {(a² + b²)(c² + d²)}/(c² + d²)²

                          = (a² + b²)/(c² + d²)  

Hence, proved.

 

Question 5:

Convert the following in the polar form:

(i) (1 + 7i)/(2 - i)²                                         (ii) (1 + 3i)/(1 – 2i)

Answer:

(i) Let z = (1 + 7i)/(2 - i)²             

              = (1 + 7i)/(4 + i² – 4i)              

              = (1 + 7i)/(4 - 1 – 4i)

              = (1 + 7i)/(3 – 4i)

              = {(1 + 7i)/(3 – 4i)} * {(3 + 4i)/ (3 + 4i)}       

              = (3 + 4i + 21i + 28i²)/{3² – (4i)²}

              = (3 + 4i + 21i - 28)/(9 + 16)

              = (-25 + 25i)/25          

              = -1 + i

Let r cos θ = –1 and r sin θ = 1

On squaring and adding, we obtain

      r²(cos² θ + sin² θ) = 1 + 1

=> r² = 2                                              [Since cos² θ + sin² θ = 1]

=> r = √2                 [Since r > 0]

Now, √2 cos θ = -1 and √2 sin θ = 1

=> cos θ = -1/√2 and sin θ = 1/√2

=> θ = π – π/4 = 3π/4

Now, z = r cos θ + i r sin θ

             = √2(cos 3π/4 + i sin 3π/4)     

This is the required polar form.

(ii) Let z = (1 + 3i)/(1 – 2i)    

              = {(1 + 3i)/(1 – 2i)} * {(1 + 2i)/(1 + 2i)}       

              = (1 + 2i + 3i + 6i²)/{1² – (2i)²}

              = (1 + 2i + 3i - 6)/(1 + 4)

              = (-5 + 5i)/5

              = -1 + i                

Let r cos θ = –1 and r sin θ = 1

On squaring and adding, we obtain

      r²(cos² θ + sin² θ) = 1 + 1

=> r² = 2                                              [Since cos² θ + sin² θ = 1]

=> r = √2                 [Since r > 0]

Now, √2 cos θ = -1 and √2 sin θ = 1

=> cos θ = -1/√2 and sin θ = 1/√2

=> θ = π – π/4 = 3π/4

Now, z = r cos θ + i r sin θ

             = √2(cos 3π/4 + i sin 3π/4)

This is the required polar form.

Question 6:

Solve the equation 3x² – 4x + 20/3 = 0

Answer:

The given quadratic equation is 3x² – 4x + 20/3 = 0

This equation can also be written as 9x² – 12x + 20 = 0

On comparing this equation with ax² + bx + c = 0, we get

a = 9, b = –12, and c = 20

Therefore, the discriminant of the given equation is

D = b² – 4ac = (–12)² – 4 * 9 * 20 = 144 – 720 = –576

Therefore, the required solutions are = (-b ± √D)/2a

                                                                   = {12 ± √(-576)}/(2 * 9)

                                                                   = {12 ± √(-1) * √576}/18

                                                                   = (12 ± i24)/18                       [Since i = √(-1)]

                                                                   = 6(2 ± i4)/18

                                                                   = (2 ± i4)/3          

Question 7:

Solve the equation x² – 2x + 3/2 = 0

Answer:

The given quadratic equation is x² – 2x + 3/2 = 0

This equation can also be written as 2x² – 4x + 3 = 0

On comparing this equation with ax² + bx + c = 0, we get

a = 2, b = –4, and c = 3

Therefore, the discriminant of the given equation is

D = b² – 4ac = (–4)² – 4 * 2 * 3 = 16 – 24 = –8

Therefore, the required solutions are = (-b ± √D)/2a

                                                                    = {4 ± √(-8)}/(2 * 2)

                                                                    = {4 ± √(-1) * √8}/4

                                                                    = (4 ± i2√2)/4                       [Since i = √(-1)]

                                                                   = 2(2 ± i√2)/4

                                                                   = (2 ± i√2)/2

Question 8:

Solve the equation 27x² – 10x + 1 = 0

Answer:

The given quadratic equation is 27x² – 10x + 1 = 0

On comparing this equation with ax² + bx + c = 0, we get

a = 27, b = –10, and c = 1

Therefore, the discriminant of the given equation is

D = b² – 4ac = (–10)² – 4 * 27 * 1 = 100 – 108 = –8

Therefore, the required solutions are = (-b ± √D)/2a

                                                                    = {10 ± √(-8)}/(2 * 27)

                                                                    = {10 ± √(-1) * √8}/54

                                                                    = (10 ± i2√2)/54                       [Since i = √(-1)]

                                                                   = 2(5 ± i√2)/54

                                                                   = (5 ± i√2)/27

Question 9:

Solve the equation 21x² – 28x + 10 = 0

Answer:

The given quadratic equation is 28x² – 28x + 10 = 0

On comparing this equation with ax² + bx + c = 0, we get

a = 28, b = –28, and c = 10

Therefore, the discriminant of the given equation is

D = b² – 4ac = (–28)² – 4 * 21 * 10 = 784 – 840 = –56

Therefore, the required solutions are = (-b ± √D)/2a

                                                                    = {28 ± √(-56)}/(2 * 21)

                                                                    = {28 ± √(-1) * √56}/42

                                                                    = (28 ± i2√14)/42                       [Since i = √(-1)]

                                                                   = 2(14 ± i√14)/42

                                                                   = (14 ± i√14)/21

                                                                   = 14/21 ± i√14/21

                                                                   = 2/3 ± i√14/21

Question 10:             

If z₁ = 2 – i, z₂ = 1 + i, find |z₁ + z₂ + 1|/|z₁ – z₂ + 1|

Answer:

Given, z₁ = 2 – i, z₂ = 1 + i                                                        

Now, |z₁ + z₂ + 1|/|z₁ – z₂ + 1| =|(2 – i) + (1 + i) + 1|/|(2 – i) – (1 + i) + 1|

                                                      =|4/(2 – 2i)|

                                                      =|4/{2(1 – i)|

                                                      = |2/(1 - i)|

                                                      = |{2/(1 - i)} * {(1 + i)/ (1 + i)}|

                                                      = |2(1 + i)/(1² – i²)|     

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

                                                      = |2(1 + i)/2|

                                                      = |(1 + i)|

                                                      = √(1² + 1²) 

                                                      = √2

Thus, the value of |z₁ + z₂ + 1|/|z₁ – z₂ + 1| = √2

Question 11:

If a + ib = (x + i)²/(2x² + 1), prove that a² + b² = (x² + 1)²/(2x + 1)²

Answer:

Given, a + ib = (x + i)²/(2x² + 1)

                       = (x² + i² + 2ix)/(2x² + 1) 

                       = (x² - 1 + 2ix)/(2x² + 1)  

                       = (x² – 1)/(2x² + 1) + 2ix/(2x² + 1)

On comparing real and imaginary parts, we get

a = (x² – 1)/(2x² + 1), b = 2x/(2x² + 1)

Now, a² + b² = [(x² – 1)/(2x² + 1)]² + [2x/(2x² + 1)]²

                       = (x⁴ + 1 – 2x² + 4x²)/(2x² + 1)²      

                       = (x⁴ + 1 + 2x²)/(2x² + 1)²   

                       = (x² + 1)²/(2x² + 1)²

So, a² + b² = (x² + 1)²/(2x² + 1)²

Hence, proved.

Question 12:

Let z₁ = 2 – i, z₂ = -2 + i. Find

(i) Re{(z₁z₂)/z₁}                                         (ii) Im(1/z₁z₁)

Answer:

Given, z₁ = 2 – i, z₂ = -2 + i

(i) z₁z₂ = (2 - i)(-2 + i)

            = -4 + 2i + 2i – i²

           = -4 + 4i – (-1)

           = -4 + 4i + 1

           = -3 + 4i   

z₁ = 2 + i  

So, (z₁z₂)/z₁ = (-3 + 4i)/(2 + i)

                      = {(-3 + 4i)/(2 + i)} * {(2 - i)/ (2 - i)}

                      = (-6 + 3i + 8i – 4i²)/(2² + 1²)

                      = (-6 + 3i + 8i + 4)/5

                      = (-2 + 11i)/5

                      = -2/5 + 11i/5

On comparing real part, we get

Re{(z₁z₂)/z₁} = -2/5      

(ii) (1/z₁z₁) = 1/{(2 - i)(2 + i)} = 1/(2² - i²) = 1/(4 + 1) = 1/5

On comparing imaginary part, we get

Im (1/z₁z₁) = 0

Question 13:

Find the modulus and argument of the complex number (1 + 2i)/(1 – 3i)

Answer:

Let z = (1 + 2i)/(1 – 3i)

         = {(1 + 2i)/(1 - 3i)} * {(1 + 3i)/(1 + 3i)}

         = (1 + 3i + 2i + 6i²)/(1² + 3²)

         = (1 + 3i + 2i - 6)/10

         = (-5 + 5i)/10

         = -1/2 + 1i/2

Given, z = r cos θ + i r sin θ

Let r cos θ = -1/2 and r sin θ = 1/2

On squaring and adding, we obtain

     (r cos θ)² + (r sin θ)² = (-1/2)² + (1/2)²

=> (r² cos² θ + r² sin² θ) = 1/4 + 1/4

=> r² (cos² θ + sin² θ) = 1/2  

=> r² = 1/2                                   [Since cos² θ + sin² θ = 1]

=> r = 1/√2                                  [Since r > 0]

So, Modulus = 1/√2

Now, cos θ/√2 = -1/2 and sin θ/√2 = 1/2

=> cos θ = -1/√2 and sin θ = 1/√2

Now, argument θ = π - π/4 = 3π/4

Therefore, the modulus and argument of the given complex number are 1/√2 and 3π/4

respectively.

Question 14:

Find the real numbers x and y if (x – iy)(3 + 5i) is the conjugate of –6 – 24i.

Answer:

Given, (x - iy)(3 + 5i) = 3x + 5xi – 3yi - 5yi²

                                     = 3x + 5xi – 3yi + 5y

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

Congugate of (x - iy)(3 + 5i) = (3x + 5y) - i(5x - 3y)

Now, (3x + 5y) - i(5x - 3y) = -6 - 24i

Compare real and imaginary term, we get

        3x + 5y = -6 ...........1

and 5x - 3y = 24 ...........2

After solving equation 1 and 2, we get

x =3 and y= -3

Thus, the values of x and y are 3 and –3 respectively.

Question 15:

Find the modulus of {(1 + i)/(1 - i)} - {(1 - i)/(1 + i)}

Answer:

Given expression is

    {(1 + i)/(1 - i)} - {(1 - i)/(1 + i)}

= [{(1 + i)*(1 + i)}/{(1 - i)*(1 + i)}] - [{(1 - i)*(1 - i)}/{(1 + i)*(1 - i)}]

= {(1 + i)²/(1 - i²)} - {(1 - i)²/(1 - i²)}

= (1 + i² + 2i)/(1 + 1) - (1 + i² - 2i)/(1 + 1)            (since i² = -1)

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

= 2i/2 + 2i/2

= i + i

= 2i

= 0 + 2i

Now modulus = √(0² + 2²) = √4 = 2

So, modulus of the expression = 2

 

Question 16:

If (x + iy)³ = u + iv, then show that: u/x + v/y = 4(x² – y²)

Answer:

Given, (x + iy)³ = u + iv

=> x³ + (iy)³ + 3 * x * iy(x + iy) = u + iv

=> x³ - iy³ + 3x²yi + 3xy²i² = u + iv

=> x³ - iy³ + 3x²yi - 3xy² = u + iv

=> (x³ - 3xy²) + i(3x²y - y³) = u + iv

On comparing real and imaginary part, we get

u = x³ - 3xy² and v = 3x²y - y³

Now, u/x + y/v = (x³ - 3xy²)/x + (3x²y - y³)/y

                            = x² - 3y² + 3x² – y²

                            = 4x² - 4y²     

                            = 4(x² – y²)

So, u/x + v/y = 4(x² – y²)

Question 17:

If α and β are different complex numbers with |β| = 1, then find |(β - α)/(1 - αβ)|

Answer:

Let α = a + ib and β = x + iy

It is given that |β| = 1

=> √(x² + y²) = 1

=> x² + y² = 1   ………..1

Now, |(β - α)/(1 - αβ)|= |{(x + iy) – (a + ib)}/{1 – (a - ib)(x + iy)}|

                                         = |{(x - a) + i(y - b)}/{1 – (ax + aiy - ibx + by)}|

                                         = |{(x - a) + i(y - b)}/{(1 – ax – by) + i(bx - ay)}|

                                         = |{(x - a) + i(y - b)}|/|{(1 – ax – by) + i(bx - ay)}|   

                                         = √{(x - a)² + (y - b)²}/√{(1 – ax – by)² + (bx - ay)²} 

                                         =                     √(x² + a² – 2ax + y² + b² – 2by)

                                             √(1 + a²x² + b²y² -2ax + 2abxy – 2by + b²x² + a²y² – 2abxy)

                                        =                     √{(x² + a²) – 2ax + y² + b² – 2by)}

                                                        √{1 + a²(x² + y²) + b² (y² + x²) – 2ax – 2by}

                                       = √{1 + a² + b² – 2ax - 2by)}/ √{1 + a² + b² – 2ax - 2by)}      [From eq. 1]

                                       = 1

So, |(β - α)/(1 - αβ)|= 1

Question 18:

Find the number of non-zero integral solutions of the equation |1 – i|^x = 2^x

Answer:

Given, |1 – i|^x = 2^x

=> √{1² + (-1)²} ^x = 2^x

=> √(1 + 1) ^x = 2^x

=> (√2)^x = 2^x

=> 2^x/2 = 2^x

=> x/2 = x

=> x = 2x

=> 2x – x = 0

=> x = 0

Thus, 0 is the only integral solution of the given equation. Therefore, the number of nonzero

integral solutions of the given equation is 0.

Question 19:

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that:

(a² + b²)(c² + d²)(e² + f²)(g² + h²) = A² + B²

Answer:

Given, (a + ib)(c + id)(e + if)(g + ih) = A + iB

=> |(a + ib)(c + id)(e + if)(g + ih)| = |A + iB|

=> |(a + ib)||(c + id)||(e + if)||(g + ih)| = |A + iB|

=> √(a² + b²) √(c² + d²) √(e² + f²) √(g² + h²) = √(A² + B²)

On squaring both sides, we get

(a² + b²)(c² + d²)(e² + f²)(g² + h²) = A² + B²

Hence proved.

Question 20:

If {(1 + i)/(1 - i)}^m then find the least positive integral value of m.

Answer:

Given, {(1 + i)/(1 - i)}^m = 1

=> [{(1 + i) * (1 + i)}/{(1 - i) * (1 + i)}]^m = 1

=> [{(1 + i)²}/{(1 - i²)}]^m = 1

=> [(1 + i² + 2i)/{1 - (-1)}]^m = 1                      [Since i² = -1]

=> [(1 - 1 + 2i)/{1 + 1}]^m = 1

=> [2i/2]^m = 1

=> i^m = 1

Now, i^m is 1 when n = 4                                [Since i⁴ = 1]

So, the least value of n is 4.