Complex Numbers and Quadratic Equations – Class 11 Mathematics notes, important questions and MCQs

Chapter overview

Complex numbers a + ib extend real numbers so that every quadratic equation has roots. They can be added, multiplied, conjugated and shown on the Argand plane.

Algebra

• i = √−1, i² = −1, i³ = −i, i⁴ = 1.
• Conjugate z̄ = a − ib; modulus |z| = √(a² + b²); z·z̄ = |z|².
• 1/z = z̄/|z|².

Polar form and quadratics

• z = r(cos θ + i sin θ), r = |z|, θ = arg z.
• Roots of ax² + bx + c = 0: x = [−b ± √(b² − 4ac)]/2a; if D < 0, roots are complex conjugates.
• Sum of roots = −b/a, product = c/a.

Key terms

Modulus

distance of z from the origin, √(a² + b²)

Conjugate

a − ib for z = a + ib

Argument

angle θ the line joining z to the origin makes with the positive real axis; principal value in (−π, π]

Formulas

• |z| = √(a² + b²)
• x = [−b ± √(b² − 4ac)]/2a

Important questions with answers

Quick revision

• i² = −1.
• Complex roots occur in conjugate pairs (real coefficients).
• z·z̄ = |z|².

Exam tips

• iⁿ repeats every 4: find n mod 4.
• To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

Practice MCQs

1. i² equals

   1. −1
   2. 1
   3. i
   4. −i
   Show answer

   Answer: (a) −1. Definition.

2. Modulus of 3 + 4i is

   1. 5
   2. 7
   3. 25
   4. 1
   Show answer

   Answer: (a) 5. √(9 + 16).

3. i¹⁰⁵ equals

   1. i
   2. 1
   3. −1
   4. −i
   Show answer

   Answer: (a) i. 105 mod 4 = 1.

4. Conjugate of 2 − 5i is

   1. 2 + 5i
   2. −2 + 5i
   3. −2 − 5i
   4. 5 − 2i
   Show answer

   Answer: (a) 2 + 5i. Change sign of imaginary part.

5. (2 + 3i)(1 − i) equals

   1. 5 + i
   2. −1 + 5i
   3. 5 − i
   4. 2 − 3i
   Show answer

   Answer: (a) 5 + i. 2 − 2i + 3i + 3.

6. Roots of x² + x + 1 = 0 are

   1. real and equal
   2. real and distinct
   3. complex conjugates
   4. zero
   Show answer

   Answer: (c) complex conjugates. D = −3.

7. (1 + i)² equals

   1. 2i
   2. 2
   3. 0
   4. 1 + 2i
   Show answer

   Answer: (a) 2i. 1 + 2i − 1.

8. The multiplicative inverse of 3 + 4i is

   1. (3 − 4i)/25
   2. (3 + 4i)/25
   3. 3 − 4i
   4. (3 − 4i)/5
   Show answer

   Answer: (a) (3 − 4i)/25. z̄/|z|².