Complex Numbers & Quadratic Equations Speed Test for Exam Prep - Interactive Mini Game Updated on Wednesday, August 6, 2025

Sharpen your Complex Numbers & Quadratic Equations skills with this interactive mini-game designed to improve your speed and accuracy for competitive exams.

Updated: just now

Categories: Mini Game, Math, Class 11
Tags: Mini Game, Math, Class 11, Complex Numbers, Quadratic Equations, JEE, NEET
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Complex Numbers & Quadratic Equations Cheatsheet & Quiz

Complex Numbers & Quadratic Equations Cheatsheet

Cheat Codes & Shortcuts

  • Complex Number: \( z = a + bi \) where \( i^2 = -1 \)
  • Modulus: \( |z| = \sqrt{a^2 + b^2} \)
  • Conjugate: \( \overline{z} = a - bi \)
  • Polar Form: \( z = r(\cos \theta + i \sin \theta) \), where \( r = |z| \) and \( \theta = \arg(z) \)
  • Euler's Formula: \( z = r e^{i \theta} \)
  • Quadratic Equation Standard Form: \( ax^2 + bx + c = 0 \)
  • Discriminant: \( \Delta = b^2 - 4ac \)
  • Roots of Quadratic: \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)
  • Nature of Roots: Real and distinct if \( \Delta > 0 \), real and equal if \( \Delta = 0 \), complex if \( \Delta < 0 \)
  • Sum and Product of Roots: \( \alpha + \beta = -\frac{b}{a} \), \( \alpha \beta = \frac{c}{a} \)

Quick Reference Table

Type Form Solution / Property
Complex Number \( z = 3 + 4i \) \( |z| = 5 \), \( \overline{z} = 3 - 4i \)
Polar Form \( z = 1 + i \) \( r = \sqrt{2} \), \( \theta = \frac{\pi}{4} \), so \( z = \sqrt{2} e^{i\pi/4} \)
Quadratic Equation \( 2x^2 - 4x + 2 = 0 \) \( \Delta = 0 \Rightarrow \) One real root \( x = 1 \)
Quadratic Roots (Complex) \( x^2 + 4x + 5 = 0 \) \( \Delta = -4 \Rightarrow x = -2 \pm i \)
Sum & Product of Roots \( ax^2 + bx + c = 0 \) \( \sum = -\frac{b}{a}, \quad \prod = \frac{c}{a} \)

Advice

Discriminant Rules: Quickly determine the nature of roots using \( \Delta \).

Complex Roots: Occur in conjugate pairs for real-coefficient quadratics.

Use Polar Form: For multiplication/division of complex numbers, polar form and Euler's formula simplify calculations.

Sum/Product Check: Use to verify roots without solving.

Practice: Familiarize with imaginary unit powers: \( i^2 = -1, i^3 = -i, i^4 = 1 \).

Complex Numbers & Quadratic Equations Quick Tips

  • Modulus & Argument: Find magnitude and angle to express complex numbers in polar form.
  • Euler's Formula: Useful for powers and roots of complex numbers.
  • Quadratic Roots: Use quadratic formula, consider discriminant carefully.
  • Nature of Roots: Distinguish real vs complex from discriminant.
  • Sum & Product: Use relationships without solving quadratic.

Complex Numbers & Quadratic Equations Speed Quiz

Test your speed with 5 questions! You have 30 seconds per question.