Solution of Chapter 5. Complex numbers and quadratic equations (Mathematics Book)

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

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

i⁹ + i¹⁹

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

i^–39

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

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

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

(1-i) – (-1 + i6)

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

.

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

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

(1 – i)⁴

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

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

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

Find the multiplicative inverse of √5 + 3i

Find the multiplicative inverse of –i

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

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

z = –1 – i√ 3

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

z = – √3 + i

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

1 – i

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

– 1 + i

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

– 1 – i

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

– 3

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

√2 + i

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

i

Evaluate:

For any two complex numbers z₁ and z₂, prove that

Re (z₁ z₂) = Re z₁ Re z₂ – Imz₁ IMz₂

Reduce to the standard form.

If prove that

Convert the following in the polar form:

Convert the following in the polar form:

Solve each of the equation in Exercises 6 to 9:

Solve each of the equation in Exercises 6 to 9:

Solve each of the equation in Exercises 6 to 9:

27x² – 10 x + 1 = 0

Solve each of the equation in Exercises 6 to 9:

21x² − 28x + 10 = 0

If z₁ = 2 – i, z₂ = 1 + i, find

If prove that

Let z₁ = 2 – i, z₂ = –2 + i. Find:

(i)

(ii)

Find the modulus and argument of the complex number

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

Find the modulus of

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

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

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

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

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

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

Solve each of the following equations:

x² + 3 = 0

Solve each of the following equations:

2x² + x + 1 = 0

Solve each of the following equations:

x² + 3x + 9 = 0

Solve each of the following equations:

– x² + x – 2 = 0

Solve each of the following equations:

x² + 3x + 5 = 0

Solve each of the following equations:

x² – x + 2 = 0

Solve each of the following equations:

√2x² + x + √2 = 0

Solve each of the following equations:

√3x² – √2x + 3√3 = 0

Solve each of the following equations:

x² + x + 1/√2 = 0

Solve each of the following equations:

x² + x/√2 + 1 = 0