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Cheat sheet · Class 11 · CBSE · Commerce Mathematics

Complex Numbers and Quadratic Equations cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 11 · Mathematics

Cheat sheet

Chapter 4Complex Numbers and Quadratic Equations

i² = −1, and the powers of i repeat every 4: i, −1, −i, 1.

1. Algebra

z=a+ibz = a + ibzˉ=a−ib\bar z = a - ib
∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}zzˉ=∣z∣2z\bar z = |z|^2

1z=zˉ∣z∣2\dfrac1z = \dfrac{\bar z}{|z|^2}; equal complex numbers have equal real and imaginary parts.

2. Polar Form

z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta)

r=∣z∣r = |z|; argument θ\theta with −π<θ≤π-\pi < \theta \le \pi is the principal value.

3. Quadratic Equations

x=−b±b2−4ac2ax = \dfrac{-b\pm\sqrt{b^2 - 4ac}}{2a}

If D=b2−4ac<0D = b^2 - 4ac < 0, roots are −b±i−D2a\dfrac{-b\pm i\sqrt{-D}}{2a} — complex conjugates.

4. Key Points

  • |z₁z₂| = |z₁||z₂|.
  • √a·√b = √(ab) fails when both are negative.
  • Argand plane: x real, y imaginary.
  • i⁴ⁿ = 1.

Remember: i powers | conjugate & modulus | polar form | complex roots

Complex Numbers and Quadratic Equations: questions and notes

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