Core5 mapped NDA questions
Build the idea
Build exam-ready understanding by connecting the core idea, formulas or facts, worked example and previous NDA questions. Focus on why each rule works, then test it under timed conditions.
Concept
A complex number is z = a + ib where i = √(−1), a is the real part and b the imaginary part. The modulus |z| = √(a²+b²) gives its distance from the origin in the Argand plane, and the argument is the angle it makes with the positive real axis.
The three cube roots of unity are 1, ω, ω² with 1 + ω + ω² = 0 and ω³ = 1.
Key Points & Formulas
- i² = −1, i³ = −i, i⁴ = 1 (powers of i repeat with period 4).
- |z| = √(a²+b²); arg(z)=tan⁻¹(b/a), adjusted for the quadrant.
- Conjugate of a+ib is a−ib; z·z̄ = |z|².
- Cube roots of unity: 1+ω+ω²=0, ω³=1.
Worked Example
Q. Find the modulus and the conjugate of z = 5 − 12i.
Modulus |z| = √(5² + (−12)²) = √(25+144) = √169 = 13.
Conjugate z̄ = 5 + 12i.