class 11 notes

Class 11 Mathematics Notes Chapter 5 (Complex numbers and quadratic equations) – Mathematics Book

Philoid

Philoid

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Mathematics
Alright class, let's begin our detailed study of Chapter 5: Complex Numbers and Quadratic Equations. This chapter is crucial not just for your Class 11 understanding but also forms a base for many competitive exams. Pay close attention to the definitions and properties.

Chapter 5: Complex Numbers and Quadratic Equations - Detailed Notes

1. Introduction: Why Complex Numbers?

  • We know that a quadratic equation ax² + bx + c = 0 has real roots given by x = [-b ± √(b² - 4ac)] / 2a, provided the discriminant D = b² - 4ac ≥ 0.
  • What happens if D < 0? For example, consider x² + 1 = 0. This gives x² = -1. There is no real number whose square is -1.
  • To address this, mathematicians introduced the concept of imaginary numbers. The symbol 'i' (iota) was introduced to represent √(-1).
  • So, i = √(-1), which implies i² = -1. This is the fundamental definition.
  • This allows us to find the square root of any negative number. For example, √(-4) = √(4 * -1) = √4 * √(-1) = 2i. In general, √(-a) = √a * i, for a > 0.

2. Complex Numbers (z)

  • Definition: A number of the form a + ib, where 'a' and 'b' are real numbers, and i = √(-1), is called a complex number.
  • Notation: Usually denoted by 'z'. So, z = a + ib.
  • Real Part: 'a' is called the real part of z, denoted as Re(z).
  • Imaginary Part: 'b' is called the imaginary part of z, denoted as Im(z).
    • Important Note: Im(z) is 'b', which is a real number, not 'ib'.
  • Purely Real: If b = 0, z = a, which is a purely real number. (So, the set of real numbers is a subset of the set of complex numbers).
  • Purely Imaginary: If a = 0, z = ib, which is a purely imaginary number (provided b ≠ 0).
  • The Complex Number 0: 0 + i0 is the complex number zero.

3. Algebra of Complex Numbers

Let z₁ = a + ib and z₂ = c + id be two complex numbers.

  • Equality: z₁ = z₂ if and only if a = c and b = d. (Real parts are equal, and imaginary parts are equal).
  • Addition: z₁ + z₂ = (a + c) + i(b + d). (Add real parts and imaginary parts separately).
  • Subtraction: z₁ - z₂ = (a - c) + i(b - d). (Subtract real parts and imaginary parts separately).
  • Multiplication:
    z₁ * z₂ = (a + ib) * (c + id)
    = a(c + id) + ib(c + id)
    = ac + i(ad) + i(bc) + i²(bd)
    = ac + i(ad + bc) - bd (since i² = -1)
    = (ac - bd) + i(ad + bc). (Real part is ac-bd, Imaginary part is ad+bc).
  • Division: To divide z₁ by z₂ (where z₂ ≠ 0), we multiply the numerator and denominator by the conjugate of the denominator (explained below).
    z₁ / z₂ = (a + ib) / (c + id)
    = [(a + ib) * (c - id)] / [(c + id) * (c - id)]
    = [(ac - i(ad) + i(bc) - i²(bd))] / [c² - (id)²]
    = [(ac + bd) + i(bc - ad)] / [c² - i²d²]
    = [(ac + bd) + i(bc - ad)] / (c² + d²)
    = [ (ac + bd) / (c² + d²) ] + i [ (bc - ad) / (c² + d²) ]

4. Powers of 'i'

  • i⁰ = 1
  • i¹ = i
  • i² = -1
  • i³ = i² * i = (-1) * i = -i
  • i⁴ = i² * i² = (-1) * (-1) = 1
  • i⁵ = i⁴ * i = 1 * i = i
  • The powers of 'i' cycle through {i, -1, -i, 1} with a period of 4.
  • General Rule: To find iⁿ for any integer n > 4, divide n by 4. Let n = 4q + r, where q is the quotient and r is the remainder (0 ≤ r < 4).
    Then, iⁿ = i^(4q + r) = (i⁴) ^q * iʳ = (1)^q * iʳ = iʳ.
    • If r = 0, iⁿ = i⁰ = 1
    • If r = 1, iⁿ = i¹ = i
    • If r = 2, iⁿ = i² = -1
    • If r = 3, iⁿ = i³ = -i
  • Also, 1/i = i³/i⁴ = -i/1 = -i.

5. Modulus and Conjugate of a Complex Number

Let z = a + ib.

  • Conjugate: The conjugate of z, denoted by z̄, is obtained by changing the sign of the imaginary part.
    z̄ = a - ib

    • Properties:
      • (z̄)̄ = z
      • z + z̄ = (a + ib) + (a - ib) = 2a = 2 Re(z)
      • z - z̄ = (a + ib) - (a - ib) = 2ib = 2i Im(z)
      • z = z̄ <=> z is purely real (Im(z) = 0)
      • z + z̄ = 0 <=> z is purely imaginary (Re(z) = 0)
      • z * z̄ = (a + ib)(a - ib) = a² - (ib)² = a² - i²b² = a² + b² (This is a real number)
      • (z₁ + z₂)̄ = z₁̄ + z₂̄
      • (z₁ - z₂)̄ = z₁̄ - z₂̄
      • (z₁ * z₂)̄ = z₁̄ * z₂̄
      • (z₁ / z₂)̄ = z₁̄ / z₂̄ (provided z₂ ≠ 0)

  • Modulus: The modulus of z, denoted by |z|, is the non-negative real number √(a² + b²). Geometrically, it represents the distance of the point (a, b) from the origin in the Argand plane.
    |z| = √(a² + b²)

    • Properties:
      • |z| ≥ 0
      • |z| = 0 <=> z = 0
      • |z| = |z̄| = |-z|
      • z * z̄ = a² + b² = (√(a² + b²))² = |z|² (Very important property)
      • |z₁ * z₂| = |z₁| * |z₂|
      • |z₁ / z₂| = |z₁| / |z₂| (provided z₂ ≠ 0)
      • Triangle Inequality: |z₁ + z₂| ≤ |z₁| + |z₂|
      • |z₁ - z₂| ≥ ||z₁| - |z₂||

  • Multiplicative Inverse: For a non-zero complex number z = a + ib, its multiplicative inverse (z⁻¹) is given by 1/z.
    z⁻¹ = 1 / (a + ib) = (a - ib) / [(a + ib)(a - ib)] = (a - ib) / (a² + b²) = z̄ / |z|²

6. Argand Plane and Polar Representation

  • Argand Plane: A complex number z = a + ib can be represented geometrically as a point P(a, b) in a plane called the complex plane or Argand plane. The x-axis is called the real axis, and the y-axis is called the imaginary axis.
  • Polar Representation: Let P(a, b) be the point representing z = a + ib. Let OP = r and the angle made by OP with the positive real axis be θ (theta).
    • From the diagram, a = r cos θ and b = r sin θ.
    • r = √(a² + b²) = |z| (the modulus).
    • θ is called the argument or amplitude of z, denoted as arg(z).
    • The representation z = r(cos θ + i sin θ) is called the polar form.
    • Finding θ: tan α = |b/a| (where α is the acute angle). Then determine θ based on the quadrant in which (a, b) lies:
      • Quadrant I (a>0, b>0): θ = α
      • Quadrant II (a<0, b>0): θ = π - α
      • Quadrant III (a<0, b<0): θ = -(π - α) = α - π
      • Quadrant IV (a>0, b<0): θ = -α
    • Principal Argument: The value of θ such that -π < θ ≤ π is called the principal argument.

7. Quadratic Equations

  • Consider the quadratic equation ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
  • The discriminant is D = b² - 4ac.
  • Case 1: D ≥ 0: Roots are real and given by x = [-b ± √D] / 2a. (Studied in Class 10).
  • Case 2: D < 0: Roots are complex. Since D is negative, √D is imaginary.
    √D = √( -(4ac - b²) ) = √(-1) * √(4ac - b²) = i √(4ac - b²) (where 4ac - b² > 0)
    The roots are given by:
    x = [-b ± √D] / 2a
    x = [-b ± i√(4ac - b²)] / 2a
  • Nature of Complex Roots: If the coefficients a, b, c are real and the discriminant D < 0, the complex roots always occur in conjugate pairs. If p + iq is a root, then p - iq is also a root.

8. Fundamental Theorem of Algebra (Statement Only)

  • A polynomial equation of degree 'n' (where n ≥ 1) has exactly 'n' roots in the complex number system (counting multiplicity). This means every polynomial equation has at least one complex root.

Multiple Choice Questions (MCQs)

  1. The value of i⁹ + i¹⁹ is:
    (A) 1
    (B) 0
    (C) -1
    (D) i

  2. The complex number (2 - 3i) + (-5 + i) is equal to:
    (A) -3 - 2i
    (B) -3 + 2i
    (C) 3 - 2i
    (D) 7 - 4i

  3. The modulus of the complex number z = -1 - i√3 is:
    (A) 1
    (B) √2
    (C) 2
    (D) 4

  4. The conjugate of the complex number z = (1 + i) / (1 - i) is:
    (A) i
    (B) -i
    (C) 1
    (D) -1

  5. The multiplicative inverse of 4 - 3i is:
    (A) (4 + 3i) / 7
    (B) (4 - 3i) / 7
    (C) (4 + 3i) / 25
    (D) (4 - 3i) / 25

  6. If z = 2 + 3i, then z * z̄ is:
    (A) 4
    (B) 9
    (C) 5
    (D) 13

  7. The roots of the quadratic equation x² + x + 1 = 0 are:
    (A) Real and distinct
    (B) Real and equal
    (C) Complex conjugates
    (D) Purely imaginary

  8. The value of (1 + i)² is:
    (A) 2
    (B) 2i
    (C) 1 - i
    (D) -2i

  9. If z₁ = 1 + i and z₂ = 2 - i, then Re(z₁ * z₂) is:
    (A) 1
    (B) 2
    (C) 3
    (D) -1

  10. The principal argument (amplitude) of the complex number z = 1 - i is:
    (A) π/4
    (B) -π/4
    (C) 3π/4
    (D) -3π/4

Answers to MCQs:

  1. (B) [i⁹ = i^(42+1) = i¹ = i; i¹⁹ = i^(44+3) = i³ = -i. So, i + (-i) = 0]
  2. (A) [(2 - 5) + i(-3 + 1) = -3 - 2i]
  3. (C) [|z| = √((-1)² + (-√3)²) = √(1 + 3) = √4 = 2]
  4. (B) [z = (1+i)/(1-i) * (1+i)/(1+i) = (1 + i² + 2i) / (1² - i²) = (1 - 1 + 2i) / (1 - (-1)) = 2i / 2 = i. The conjugate of i (0 + 1i) is 0 - 1i = -i]
  5. (C) [Inverse = z̄ / |z|² = (4 + 3i) / (4² + (-3)²) = (4 + 3i) / (16 + 9) = (4 + 3i) / 25]
  6. (D) [z * z̄ = |z|² = (2)² + (3)² = 4 + 9 = 13]
  7. (C) [D = b² - 4ac = 1² - 4(1)(1) = 1 - 4 = -3 < 0. Since D < 0, roots are complex conjugates.]
  8. (B) [(1 + i)² = 1² + i² + 2(1)(i) = 1 + (-1) + 2i = 2i]
  9. (C) [z₁ * z₂ = (1 + i)(2 - i) = 1(2-i) + i(2-i) = 2 - i + 2i - i² = 2 + i - (-1) = 3 + i. Re(z₁ * z₂) = 3]
  10. (B) [z = 1 - i. Point is (1, -1) in Quadrant IV. a = 1, b = -1. tan α = |-1/1| = 1 => α = π/4. Since it's in Q IV, θ = -α = -π/4]

Study these notes thoroughly. Practice solving problems involving algebraic operations, finding modulus/conjugate, converting to polar form, and solving quadratic equations with complex roots. Good luck with your preparation!

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