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📐 Mathematics  ·  Complex Numbers and Quadratic Equations  ·  JEE

Both roots of x² + ax + b = 0 are real and exceed 2. Which must hold?

Answer: a 4.

  • A a > -4 and b > 4
  • B a < -4 and b > 4
  • C a > 4 and b > 4
  • D a < 4 and b < -4

Correct answer: B. a < -4 and b > 4

Explanation: Both roots > 2: D = a²-4b ≥ 0; f(2) = 4+2a+b > 0; axis of symmetry -a/2 > 2 (so a < -4). With both roots > 2 and f(2) > 0: b > -2a - 4 > 4. So a < -4 and b > 4.

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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