Complex Numbers and Quadratic Equations
Board: CBSE |Class:Class11
ComprehensivestudynotesforComplexNumbersandQuadraticEquationsbyAjayYadav(MathKingofKatargam).Mastereveryconceptwithclearexplanations,solvedexamples,andpracticeproblems.
√(-1)=i,theimaginaryunit.i²=-1,i³=-i,i⁴=1.Powersofirepeatevery4.
Acomplexnumberz=a+ibwherea=Re(z)(realpart)andb=Im(z)(imaginarypart).Operations:addition,subtraction,multiplication,division(usingconjugate).
Conjugate:z̅=a-ib.Modulus:|z|=√(a²+b²).Properties:zz̅=|z|²,|z1z2| = |z1||z₂|.
Argand Plane
Complex numbers are represented as points (a,b) in the plane. x-axis = real axis, y-axis = imaginary axis. The distance from origin = |z|. The angle θ = tan⁻¹(b/a) is the argument.
Polar Form
z = r(cosθ + isinθ), where r = |z| and θ = arg(z). Euler’s formula: e¹θ = cosθ + isinθ.
De Moivre's Theorem
(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ). Used to find powers and roots of complex numbers.
Important Formulas
| iⁿ | i⁴k⁾ = ik (k = 0,1,2,3) |
| Modulus | |a+ib| = √a²+b² |
| Conjugate | a+ib = a-ib |
| Polar Form | z = r(cosθ+isinθ) |
| De Moivre | (cosθ+isinθ)ⁿ = cos(nθ)+isin(nθ) |
Solved Examples
Example 1: Simplify i&sup9; + i¹¹.
Solution: i&sup9; = i⁴ × i⁴ × i = i. i¹¹ = i⁴ × i⁴ × i³ = -i. Sum = i + (-i) = 0.
Example 2: Express (1+i)/(1-i) in a+ib form.
Solution: (1+i)/(1-i) × (1+i)/(1+i) = (1+2i+i²)/(1-i²) = (1+2i-1)/(1+1) = 2i/2 = i.
Example 3: Find modulus and argument of 1 + i.
Solution: r = √1+1 = √2. θ = tan⁻¹(1) = 45° = π/4. z = √2(cosπ/4 + isinπ/4).
Practice Questions
- Simplify: i⁸ + i⁶ + i⁺ + i⁻.
- Find the conjugate of (3+2i)/(2-3i).
- Express -1 + i√3 in polar form.
- Find the square root of 3 + 4i.
- If z = 2 + 3i, find |z| and arg(z).
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