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📐 Mathematics  ·  Complex Numbers and Quadratic Equations  ·  JEE

The set of real k for which x² − 2kx + (k² + k − 5) = 0 has both roots less than 5 is:

Answer: k < 4.

  • A k > 4
  • B k < 4
  • C k = 4
  • D all real k

Correct answer: B. k < 4

Explanation: Need D ≥ 0 (k ≤ 5), vertex k < 5, and f(5) = k² − 9k + 20 = (k−4)(k−5) > 0. The intersection is k < 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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