Complex Numbers and Quadratic Equations Class 11 Maths Notes – Formulas & Quick Revision

Complex Numbers mein iota, modulus-argument aur quadratic ke complex roots aate hain. Yahan aapko is chapter ke sabse important formulas aur quick revision points free milte hain — NCERT/CBSE syllabus ke hisaab se, boards aur JEE/NEET dono ke liye.

Important Formulas & Concepts — Complex Numbers and Quadratic Equations

• i² = −1; i⁴ⁿ = 1 (powers cycle: i, −1, −i, 1)
• z = a + ib; |z| = √(a² + b²)
• z·z̄ = |z|²
• Polar form: z = r(cosθ + i·sinθ)
• arg(z₁z₂) = arg(z₁) + arg(z₂)
• Quadratic: x = [−b ± √(b²−4ac)]/2a
• D < 0 → complex conjugate roots
• Sum of roots = −b/a; product = c/a

Quick Revision Points

• Complex roots hamesha conjugate pairs mein (real coefficients)
• 1/z nikalne ke liye conjugate se multiply karo
• |z₁z₂| = |z₁||z₂|; |z₁/z₂| = |z₁|/|z₂|
• Argument quadrant ke hisaab se adjust karo
• √(-x) = i√x pehle convert karo, phir multiply

FAQ

i ki powers kaise simplify karun?

Power ko 4 se divide karo, remainder se answer — PDF mein cycle diagram hai.

Ye JEE mein kitna aata hai?

JEE mein complex numbers major topic hai — modulus-argument aur roots ke questions pakke.

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