Solutions of chapter 5. Complex numbers and quadratic equations

1

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

2

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

i⁹ + i¹⁹

3

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

i^–39

4

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

3(7 + i7) + i(7 + i7)

5

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

(1-i) – (-1 + i6)

6

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

.

7

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

8

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

(1 – i)⁴

9

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

10

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

11

Find the multiplicative inverse of the complex number (4-3i)

12

Find the multiplicative inverse of √5 + 3i

13

Find the multiplicative inverse of –i

14

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

1

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

z = –1 – i√ 3

2

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

z = – √3 + i

3

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

1 – i

4

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

– 1 + i

5

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

– 1 – i

6

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

– 3

7

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

√2 + i

8

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

i

1

Evaluate:

2

For any two complex numbers z₁ and z₂, prove that

Re (z₁ z₂) = Re z₁ Re z₂ – Imz₁ IMz₂

3

Reduce to the standard form.

4

If prove that

5

Convert the following in the polar form:

5

Convert the following in the polar form:

6

Solve each of the equation in Exercises 6 to 9:

7

Solve each of the equation in Exercises 6 to 9:

8

Solve each of the equation in Exercises 6 to 9:

27x² – 10 x + 1 = 0

9

Solve each of the equation in Exercises 6 to 9:

21x² − 28x + 10 = 0

10

If z₁ = 2 – i, z₂ = 1 + i, find

11

If prove that

12

Let z₁ = 2 – i, z₂ = –2 + i. Find:

(i)

(ii)

13

Find the modulus and argument of the complex number

14

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

15

Find the modulus of

16

If (x + iy)³ – u + iv, then show that:

17

If α and β are different complex numbers with |β| = 1 , then find

18

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

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

20

If , then find the least positive integral value of m.