The value of i³⁵ is:

−i. 35 = 4(8) + 3, so i³⁵ = i³ = −i.

(2 + 3i) + (4 − 5i) equals:

6 − 2i. Add real parts and imaginary parts separately.

The conjugate of 3 − 4i is:

3 + 4i. Change the sign of the imaginary part.

For any positive integer k, i⁴ᵏ equals:

1. i⁴ = 1, so any multiple of 4 as the power gives 1.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 3: Complex Numbers

Short notes, key concepts, quick revision and practice MCQs with answers

Equations such as x² + 1 = 0 have no real solution, so the number i = √−1 is introduced. The chapter covers imaginary and complex numbers, their algebra, conjugates, powers of i, solving quadratic equations with complex roots and the cube roots of unity.

Key concepts: Complex Numbers

Imaginary number: A number of the form ki, where k is real, k ≠ 0 and i = √−1. For example √−25 = 5i. i² = −1.
Complex number: z = a + ib where a, b are real. a is the real part Re(z) and b is the imaginary part Im(z). If b = 0, z is purely real; if a = 0 (b ≠ 0), z is purely imaginary.
Equality: a + ib = c + id if and only if a = c and b = d.
Algebra of complex numbers: (a + ib) ± (c + id) = (a ± c) + i(b ± d); (a + ib)(c + id) = (ac − bd) + i(ad + bc). To divide, multiply numerator and denominator by the conjugate of the denominator.
Conjugate: The conjugate of z = a + ib is z̄ = a − ib. z + z̄ = 2a (real) and z·z̄ = a² + b² (real and non-negative).
Powers of i: i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = −1, i⁴ᵏ⁺³ = −i. Divide the power by 4 and use the remainder.
Quadratic equations: For ax² + bx + c = 0 with b² − 4ac < 0, the roots are (−b ± i√(4ac − b²)) / 2a, a pair of complex conjugates.
Cube roots of unity: The roots of x³ = 1 are 1, ω and ω², where ω = (−1 + i√3)/2. 1 + ω + ω² = 0 and ω³ = 1.

Quick revision: important points

i² = −1, i³ = −i, i⁴ = 1; i + i² + i³ + i⁴ = 0.
1/i = −i.
(1 + i)² = 2i and (1 − i)² = −2i.
Multiplicative inverse of a + ib is (a − ib)/(a² + b²).
ω² + ω = −1 and ω³ = 1.
Complex roots of a real quadratic equation always occur in conjugate pairs.

Practice MCQs with answers

The value of i³⁵ is:
1. 1
2. i
3. −1
4. −i
Show answer
Answer: D. −i
35 = 4(8) + 3, so i³⁵ = i³ = −i.
(2 + 3i) + (4 − 5i) equals:
1. 6 + 8i
2. 6 − 2i
3. −2 + 8i
4. 8 − 15i
Show answer
Answer: B. 6 − 2i
Add real parts and imaginary parts separately.
(1 + i)(1 − i) equals:
1. 0
2. 1
3. 2
4. 2i
Show answer
Answer: C. 2
1 − i² = 1 + 1 = 2.
The conjugate of 3 − 4i is:
1. −3 + 4i
2. 3 + 4i
3. −3 − 4i
4. 4 − 3i
Show answer
Answer: B. 3 + 4i
Change the sign of the imaginary part.
For any positive integer k, i⁴ᵏ equals:
1. 1
2. i
3. −1
4. −i
Show answer
Answer: A. 1
i⁴ = 1, so any multiple of 4 as the power gives 1.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters