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Complex Numbers

question 1

If (ω ≠ 1) be a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers

  1. 0, 0
  2. 0, 1
  3. 1, 1
  4. 1, 0
  5. - 1, 1

question 2

If ω (≠ 1) is a cube root of unity, then Properties of cube roots of unity equals

  1. 0
  2. 1
  3. i
  4. ω
  5. none of these

question 3

Let z and w be two non-zero complex numbers such that | z | = | w | and arg (z) + arg (w) = π. Then z equals

  1. w
  2. - w
  3. Conjugate of a complex number
  4. - Conjugate of a complex number
  5. none of these

question 4

The smallest positive integer n for which (1 + i)2n = (1 - i)2n is

  1. 4
  2. 8
  3. 2
  4. 12
  5. none of these

question 5

The complex number [(1 + 2i) ⁄ (1 - i)] lies in

  1. I st quadrant
  2. II nd quadrant
  3. III rd quadrant
  4. IV th quadrant
  5. cannot be determined

question 6

If α and β are different complex numbers with | β | = 1, then Modulus and conjugate of a complex number is

  1. 0
  2. ½
  3. 1
  4. 2
  5. none of these

question 7

If z1, z2, z3 are vertices of an equilateral triangle inscribed in a circle | z | = 2 and if z1 = 1 + i√3, then

  1. z2 = -2, z3 = 1 - i√3
  2. z2 = 2, z3 = 1 - i√3
  3. z2 = -2, z3 = - 1 - i√3
  4. z2 = 1 - i√3, z3 = - 1 - i√3
  5. none of these

question 8

The value of the expression 1.(2 - ω)(2 - ω²) + 2.(3 - ω)(3 - ω²) + ... + (n - 1)(n - ω)(n - ω²), where ω is an imaginary cube root of unity is

  1. Properties of omega
  2. Properties of omega
  3. Properties of omega
  4. 1
  5. 0

question 9

(1 + i)5 + (1 - i)5 =

  1. - 8
  2. 8i
  3. 8
  4. 32
  5. none of these

question 10

(1 + i)5 + (1 - i)5 =

  1. 28
  2. 25
  3. 24
  4. 23
  5. none of these

syllabus

Complex Numbers – Introduction; Iota (i), Powers of Iota (i), Imaginary Quantities; Equality of Complex Numbers; Addition, Subtraction, Multiplication and Division of Complex Numbers; Properties of Addition, Subtraction, Multiplication and Division of Complex Numbers; Conjugate, Modulus, Reciprocal (Inverse) and Square Root of a Complex Number; Properties of Conjugate, Properties of Modulus; Cube Roots of Unity, Properties of Cube Roots of Unity, Sum of Cube Roots of Unity, Product of Cube Roots of Unity; Representations of Complex Number – Geometrical Form, Vectorial Form and Trigonometrical Form or Polar Form; Geometrical Representation of Complex Number; Argument or Amplitude of a Complex Number; Modulus of a Complex Number; Geometrical Representations of Fundamental Operations – Geometrical Representation of Addition, Subtraction, Multiplication, Division and Conjugate of Complex Numbers; Multiplication of a Complex Number by Iota(i); Results on Modulus and Argument; Distance between Points, Affix of Point, Equation of Circle; De-Moivre's Theorem; Extraction of Roots of a Complex Number, nth roots of Complex Number, nth roots of Unity, Properties of nth roots of Unity.