Solutions of chapter 5. Complex Numbers and Quadratic Equations

1

Evaluate:

(i) i¹⁹

(ii) i⁶²

(ii) i³⁷³.

1

Evaluate:

(i) i¹⁹

(ii) i⁶²

(ii) i³⁷³.

2

Evaluate:

(i)

(ii)

(iii) .

2

Evaluate:

(i)

(ii)

(iii) .

3

Evaluate:

(i) i^–50

(ii) i^–9

(ii) i^–131.

3

Evaluate:

(i) i^–50

(ii) i^–9

(ii) i^–131.

4

Evaluate:

(i)

(ii)

4

Evaluate:

(i)

(ii)

5

Prove that 1 + i² + i⁴ + i⁶ = 0

5

Prove that 1 + i² + i⁴ + i⁶ = 0

6

Prove that 6i⁵⁰ + 5i³³ – 2i¹⁵ + 6i⁴⁸ = 7i.

6

Prove that 6i⁵⁰ + 5i³³ – 2i¹⁵ + 6i⁴⁸ = 7i.

7

Prove that = 0.

7

Prove that = 0.

8

Prove that (1 + i¹⁰ + i²⁰ + i³⁰) is a real number.

8

Prove that (1 + i¹⁰ + i²⁰ + i³⁰) is a real number.

9

Prove that = 2i.

9

Prove that = 2i.

10

= 2(1 – i).

10

= 2(1 – i).

11

Prove that (1 – i)ⁿ= 2ⁿ for all values of n N

11

Prove that (1 – i)ⁿ= 2ⁿ for all values of n N

12

Prove tha= 0.

12

Prove tha= 0.

13

Prove that (1 + i² + i⁴ + i⁶ + i⁸ + …. + i²⁰) = 1.

13

Prove that (1 + i² + i⁴ + i⁶ + i⁸ + …. + i²⁰) = 1.

14

Prove that i⁵³ + i⁷² + i⁹³ + i¹⁰² = 2i.

14

Prove that i⁵³ + i⁷² + i⁹³ + i¹⁰² = 2i.

15

Prove that n N.

15

Prove that n N.

1

Simplify each of the following and express it in the form a + ib :

2(3 + 4i) + i(5 – 6i)

1

Simplify each of the following and express it in the form a + ib :

1

Simplify each of the following and express it in the form a + ib :

(–5 + 6i) – (–2 + i)

1

Simplify each of the following and express it in the form a + ib :

(8 – 4i) – (- 3 + 5i)

1

Simplify each of the following and express it in the form a + ib :

(1 – i)² (1 + i) – (3 – 4i)²

1

Simplify each of the following and express it in the form a + ib :

1

Simplify each of the following and express it in the form a + ib :

(3 + 4i) (2 – 3i)

1

Simplify each of the following and express it in the form a + ib :

2

Simplify each of the following and express it in the form (a + ib) :

2

Simplify each of the following and express it in the form (a + ib) :

(5 – 2i)²

2

Simplify each of the following and express it in the form (a + ib) :

(–3 + 5i)³

2

Simplify each of the following and express it in the form (a + ib) :

2

Simplify each of the following and express it in the form (a + ib) :

(4 – 3i)^–1

2

Simplify each of the following and express it in the form (a + ib) :

2

Simplify each of the following and express it in the form (a + ib) :

(2 + i)^–2

2

Simplify each of the following and express it in the form (a + ib) :

(1 + 2i)^–3

2

Simplify each of the following and express it in the form (a + ib) :

(1 + i)³ – (1 – i)³

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

3

Express each of the following in the form (a + ib):

4

Simplify each of the following and express it in the form (a + ib):

(i)

(ii)

5

Show that

(i) is purely real,

(ii) is purely real.

6

Find the real values of θ for which is purely real.

7

If |z + i| = |z – i|, prove that z is real.

8

Give an example of two complex numbers z₁ and z₂ such that z₁≠ z₂ and |z₁| = |z₂|.

9

Find the conjugate of each of the following:

(–5 – 2i)

9

Find the conjugate of each of the following:

9

Find the conjugate of each of the following:

9

Find the conjugate of each of the following:

9

Find the conjugate of each of the following:

9

Find the conjugate of each of the following:

9

Find the conjugate of each of the following:

9

Find the conjugate of each of the following:

(2 – 5i)²

10

Find the modulus of each of the following:

10

Find the modulus of each of the following:

(–3 – 4i)

10

Find the modulus of each of the following:

(7 + 24i)

10

Find the modulus of each of the following:

3i

10

Find the modulus of each of the following:

10

Find the modulus of each of the following:

10

Find the modulus of each of the following:

5

10

Find the modulus of each of the following:

(1 + 2i) (i – 1)

11

Find the multiplicative inverse of each of the following:

11

Find the multiplicative inverse of each of the following:

(2 + 5i)

11

Find the multiplicative inverse of each of the following:

11

Find the multiplicative inverse of each of the following:

12

If = (a + ib), find the values of a and b.

13

If = x + iy, find x and y.

14

If, prove that x² + y² = 1.

15

If , where c is real, prove that a² + b² = 1 and .

16

Show that for all n N.

17

Find the smallest positive integer n for which (1 + i)²ⁿ = (1 – i)²ⁿ.

18

Prove that (x + 1 + i) (x + 1 – i) (x – 1 – i) (x – 1 – i) = (x⁴ + 4).

19

If a = (cosθ + i sinθ), prove that .

20

If z₁ = (2 – i) and z₂ = (1 + i), find .

21

Find the real values of x and y for which:

(1 – i) x + (1 + i) y = 1 – 3i

21

Find the real values of x and y for which:

(x + iy) (3 – 2i) = (12 + 5i)

21

Find the real values of x and y for which:

x + 4yi = ix + y + 3

21

Find the real values of x and y for which:

(1 + i) y² + (6 + i) = (2 + i)x

21

Find the real values of x and y for which:

= (1 – i)

21

Find the real values of x and y for which:

22

Find the real values of x and y for which (x – iy) (3 + 5i) is the conjugate of (-6 – 24i).

23

Find the real values of x and y for which the complex number (-3 + iyx²) and (x² + y + 4i) are conjugates of each other.

24

If z = (2 – 3i), prove that z² – 4z + 13 = 0 and hence deduce that 4z³ – 3z² + 169 = 0.

25

If (1 + i)z = (1 – i) then prove that z = -

26

If is purely an imaginary number and z ≠ -1 then find the value of |z|.

27

Solve the system of equations, Re(z²) = 0, |z| = 2.

28

Find the complex number z for which |z| = z + 1 + 2i.

1

Express each of the following in the form (a + ib) and find its conjugate.

(i)

(ii) (2 + 3i)²

(iii)

(iv)

(v)

(vi)

2

Express each of the following in the form (a + ib) and find its multiplicative inverse:

(i)

(ii)

(iii)

3

If (x + iy)³ = (u + iv) then prove that = 4 (x² – y²).

4

If (x + iy)^1/3 = (a + ib) then prove that = 4 (a² – b²).

5

Express (1 – 2i)^–3 in the form (a + ib).

6

Find real values of x and y for which

(x⁴ + 2xi) – (3x² + iy) = (3 – 5i) + (1 + 2iy).

7

If z² + |z|² = 0, show that z is purely imaginary.

8

If is purely imaginary and z = –1, show that |z| = 1.

9

If z₁ is a complex number other than –1 such that |z₁| = 1 and z₂ = then show that z2 is purely imaginary.

10

For all z C, prove that

(i)

(ii) .

(iii) = |z|²

(iv) is real

(v) is 0 or imaginary.

11

If z₁ = (1 + i) and z₂ = (–2 + 4i), prove that Im

12

If a and b are real numbers such that a² + b² = 1 then show that a real value of x satisfies the equation,

1

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 4

2

Find the modulus of each of the following complex numbers and hence express each of them in polar form: –2

3

Find the modulus of each of the following complex numbers and hence express each of them in polar form: –i

4

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 2i

5

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 1 – i

6

Find the modulus of each of the following complex numbers and hence express each of them in polar form: –1 + i

7

Find the modulus of each of the following complex numbers and hence express each of them in polar form: