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📐 Mathematics  ·  Complex Numbers and Quadratic Equations  ·  JEE

If alpha and beta are roots of ax² + bx + c = 0, then alpha³ + beta³ =

Answer: -(b³ - 3abc)/a³.

  • A (3abc - b³)/a³
  • B -(b³ - 3abc)/a³
  • C (b³ - 3abc)/a³
  • D 3abc/a³

Correct answer: B. -(b³ - 3abc)/a³

Explanation: Vieta's formulas: α+β = -b/a, αβ = c/a. Use identity: α³+β³ = (α+β)³ - 3αβ(α+β) = (-b/a)³ - 3(c/a)(-b/a) = -b³/a³ + 3bc/a². Common denominator a³ gives (-b³ + 3abc)/a³ = -(b³ - 3abc)/a³. Answer: -(b³ - 3abc)/a³

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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