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📐 Mathematics  ·  Complex Numbers and Quadratic Equations  ·  JEE

If ω is a cube root of unity, find 1 + ω² + ω⁴.

Answer: 0.

  • A 0
  • B 1
  • C 3
  • D -1

Correct answer: A. 0

Explanation: Key property: ω³=1 (cube root of unity). So ω⁴=ω³·ω=1·ω=ω. The sum becomes 1+ω²+ω. Property of cube roots of unity: 1+ω+ω²=0. So 1+ω+ω²=0. Answer: 1+ω²+ω⁴ = 0

Argand Plane: z = a + ibReImz = a+iba (real part)b (imaginary part)θ = arg(z)|z| = length of the vector OZ = √(a²+b²); θ = angle OZ makes with the positive real axis

A complex number z = a+ib is plotted as a point (or vector from the origin) on the Argand plane, with the real part along the horizontal axis and the imaginary part along the vertical; its modulus |z| is the vector's length, and its argument θ is the angle from the positive real axis.

Concept context

Complex numbers in a+ib form, modulus, argument, polar form, cube roots of unity, and quadratic equations with complex roots

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