Complex Numbers & Quadratic Equations
Practice official JEE Main Complex Numbers PYQs online. Master cube roots of unity (omega), De Moivre’s theorem, locus & geometry with step-by-step solutions.
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🎯 High-Yield Syllabus Breakdown
These high-frequency concepts account for over 90% of all questions asked in Complex Numbers & Quadratic Equations:
- Algebra of Complex Numbers & Conjugates
- Modulus & Argument (Triangle Inequalities)
- Cube Roots of Unity (ω³ = 1, 1 + ω + ω² = 0)
- De Moivre’s Theorem & n-th Roots of Unity
- Geometry & Locus of Complex Numbers (|z - z₁| = |z - z₂|)
⚡ Essential Formulas & Speed Shortcuts
Formulas and time-saving shortcuts to solve Complex Numbers questions in under 45 seconds:
Cube roots of unity: 1 + ω + ω² = 0 and ω³ = 1; (1 - ω)(1 - ω²) = 3.
Triangle Inequality: ||z₁| - |z₂|| ≤ |z₁ ± z₂| ≤ |z₁| + |z₂|.
Locus |z - z₁| / |z - z₂| = k: Straight line if k = 1 (perpendicular bisector), Circle (Circle of Apollonius) if k ≠ 1.
📖 Sample Solved Exam Questions
Preview authentic questions from this chapter with full step-by-step solutions:
💡 View Step-by-Step Solution & Answer
Correct Answer: -1
- Step 1: Given |z - 1| = |z + i|. This geometrically represents the perpendicular bisector of the line segment joining (1, 0) and (0, -1).
- Step 2: Algebraically, |(x - 1) + iy|² = |x + i(y + 1)|².
- Step 3: Expand: (x - 1)² + y² = x² + (y + 1)² ⇒ x² - 2x + 1 + y² = x² + y² + 2y + 1.
- Step 4: Cancel common terms: -2x = 2y ⇒ y = -x. The line is y = -x, which has slope m = -1.
⚡ Speed Tip: Geometric Shortcut: The segment joining (1, 0) and (0, -1) has slope (-1 - 0)/(0 - 1) = 1. Its perpendicular bisector has slope -1/1 = -1.
💡 View Step-by-Step Solution & Answer
Correct Answer: 3 and 7
- Step 1: The condition |z - (3 + 4i)| ≤ 2 represents a circular disc with center z₀ = 3 + 4i and radius r = 2.
- Step 2: The distance of the center from the origin is |z₀| = √(3² + 4²) = 5.
- Step 3: By the triangle inequality, the minimum distance from the origin is |z₀| - r = 5 - 2 = 3.
- Step 4: The maximum distance from the origin is |z₀| + r = 5 + 2 = 7.
⚡ Speed Tip: Circle Distance Formula: For |z - z₀| ≤ r, |z|_min = |z₀| - r and |z|_max = |z₀| + r.
💡 View Step-by-Step Solution & Answer
Correct Answer: 32
- Step 1: Recall the fundamental cube roots identity: 1 + ω + ω² = 0.
- Step 2: Therefore, 1 + ω² = -ω, which implies (1 - ω + ω²) = (-ω - ω) = -2ω.
- Step 3: Also, 1 + ω = -ω², which implies (1 + ω - ω²) = (-ω² - ω²) = -2ω².
- Step 4: The expression becomes (-2ω)⁵ + (-2ω²)⁵ = -32(ω⁵ + ω¹⁰) = -32(ω² + ω) = -32(-1) = 32.
⚡ Speed Tip: Cube Root Substitution: Replace 1 + ω² with -ω and 1 + ω with -ω². The sum collapses into -32(ω² + ω) = 32.
❓ Frequently Asked Questions (FAQs)
How to solve cube roots of unity problems quickly?
Express quadratic expressions like x² + x + 1 = 0 roots as ω and ω², then use ω³ = 1 to reduce high powers.