Complex Numbers Advanced Revision (JEE Advanced)

Algebraic Form

• $z = x + iy$, conjugate $\bar z = x - iy$, $|z|^2 = z\bar z$.
• Modulus: $|z| = \sqrt{x^2+y^2}$; argument $\theta \in (-\pi, \pi]$.

Polar / Exponential

• $z = r(\cos\theta + i\sin\theta) = r e^{i\theta}$.
• $z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$, $z_1/z_2 = (r_1/r_2)e^{i(\theta_1-\theta_2)}$.
• De Moivre: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$.

nth Roots of Unity

• Roots of $z^n = 1$: $\omega_k = e^{2\pi i k/n}$, $k = 0,\ldots,n-1$.
• Sum of all roots = 0; product = $(-1)^{n+1}$.
• Cube roots: $1, \omega, \omega^2$ with $1+\omega+\omega^2=0$, $\omega^3=1$.

Geometry with Complex Numbers

• Distance: $|z_1 - z_2|$.
• Circle: $|z - z_0| = r$; general form $z\bar z + \bar\alpha z + \alpha \bar z + k = 0$.
• Perpendicular bisector: $|z - z_1| = |z - z_2|$.
• Locus arg$((z-z_1)/(z-z_2)) = \alpha$ → arc of a circle.

Rotation

• Rotating $z$ about origin by θ: multiply by $e^{i\theta}$.
• Rotation about $z_0$ by θ: $z' = z_0 + (z - z_0)e^{i\theta}$.

Common Identities

• $|z_1+z_2|^2 + |z_1-z_2|^2 = 2(|z_1|^2 + |z_2|^2)$ (parallelogram law).
• $|z_1+z_2|\le |z_1|+|z_2|$ (triangle inequality); equality when $z_1, z_2$ collinear with origin, same direction.
• Reflection of $z$ about real axis: $\bar z$; about imaginary axis: $-\bar z$.

Advanced Tricks

• If $|z|=1$, then $\bar z = 1/z$.
• Sum $\sum \cos(k\theta) = \text{Re}\sum e^{ik\theta}$; use GP.
• Sum of $\omega^r$ over $n$-th roots of unity where $r$ divisible by n = n; otherwise 0.
• Equation of straight line: $\bar a z + a \bar z + b = 0$ with $b\in\mathbb R$.

Practice MCQs

Q1. If $|z|=1$ and $z \ne \pm 1$, then $z/(1-z^2)$ is A) Purely real B) Purely imaginary C) Zero D) Real if |z|=1

Q2. If $1, \omega, \omega^2$ are cube roots of unity, $(1+\omega)^6$ equals A) 1 B) −1 C) 64 D) 64ω

Q3. Locus of $z$ with $|z-1|+|z+1|=4$ is A) Circle B) Ellipse C) Parabola D) Hyperbola

Q4. If $z = e^{i\theta}$, $z^n + 1/z^n$ equals A) 2sin nθ B) 2cos nθ C) cos 2nθ D) 2i sin nθ

Q5. Number of solutions of $z² + |z| = 0$ is A) 1 B) 3 C) 4 D) Infinite

Q6. If arg(z − 1) − arg(z + 1) = π/2, locus of z is A) Line B) Circle x²+y²=1 (y>0) C) Parabola D) Ellipse

Q7. If $z_1 + z_2 + z_3 = 0$ and $|z_i|=1$, triangle formed is A) Right B) Equilateral C) Isoceles D) Scalene

Q8. Value of $(1+i)^{100}$ equals A) 2⁵⁰ B) −2⁵⁰ C) 2⁵⁰ i D) −2⁵⁰ i

Q9. Number of n-th roots of unity that are also n-th roots of −1 is A) 0 B) 1 C) 2 D) n

Q10. If $|z - i| = |z + i|$, z lies on A) Real axis B) Imaginary axis C) Circle |z|=1 D) Line y=x

Answers: 1-B (Multiply and simplify; result becomes 1/(z̄−z) which is purely imaginary.), 2-A (1+ω = −ω²; (−ω²)⁶ = ω¹² = 1.), 3-B (Sum of distances constant → ellipse.), 4-B (zⁿ + z⁻ⁿ = 2 cos nθ.), 5-B (Real: z=0 or z=−1 (via z²=−|z|); complex: two imaginary; total 3.), 6-B (Set of points where segment subtends right angle → semicircle.), 7-B (Centroid at origin equals circumcentre → equilateral.), 8-A ((1+i)² = 2i; (2i)⁵⁰ = 2⁵⁰ · i⁵⁰ = 2⁵⁰ · (−1)²⁵ = −2⁵⁰. Take B.), 9-A (No overlap: roots of z^n=1 and z^n=−1 are distinct.), 10-A (Equidistant from i and −i → real axis.)

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