Complex Numbers
JEE Concepts
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Complex numbers: a + bi, where a and b are real, i² = -1.
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Modulus or Magnitude: √(a² + b²).
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Conjugate: z* = a - bi
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Argand Diagram: Complex numbers represented on a plane, real =horizontal, imagnary = vertical.
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De Moivre’s theorem: r^(1/n)(cos(θ + 2kπ)/n + isin(θ + 2kπ)/n), k = 0, 1, 2, …, n-1.
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Polar form: z = r(cosθ + isinθ).
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Euler’s formula: e^(iθ) = cosθ + isinθ
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Applications: Quadratic equations, trigonometry, electrical engineering.
CBSE Board Concepts
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Basics: a + bi, where a and b are real.
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Modulus & Magnitude: |z| = √(a² + b²).
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Conjugate: a - bi
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Argand Diagram: Complex numbers represented on a plane.
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Operations: Add, subtract, multiply like a*(b+ci)+d(b-ci) = (ab+cd) +i(ac-db).
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Division: (a + bi)/(c + di) = [(ac + bd)/(c² + d²)] + [(bc - ad)/(c² + d²)]i
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Applications: Solve equations, trigonometric.
