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📐 Mathematics  ·  Complex Numbers and Quadratic Equations  ·  JEE

x² + 2(k + 1)x + (9k − 5) = 0 has both roots negative when:

Answer: 5/9 < k ≤ 1 or k ≥ 6.

  • A k > 6 only
  • B 5/9 < k ≤ 1 or k ≥ 6
  • C k < 5/9
  • D −1 < k < 5/9

Correct answer: B. 5/9 < k ≤ 1 or k ≥ 6

Explanation: Sum < 0 gives k > −1, product > 0 gives k > 5/9, and D ≥ 0 gives (k−1)(k−6) ≥ 0. Combining: 5/9 < k ≤ 1 or k ≥ 6.

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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