Chapter 4 · Class XI Mathematics · NCERT Exercises

Complex Numbers & Quadratic Equations — Exercises

From i²=−1 to the Argand Plane — All 52 NCERT Solutions Explained

📂 2 Exercises 📝 28 Questions 🎓 High

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Exercise Index

①

Exercise 4.1

Arithmetic on ℂ; a+bi form; modulus; conjugate

14 Q Easy-Moderate

★

Miscellaneous

Powers of i; polar proofs; mixed quadratics

14 Q Hard

2 exercise files · 28 total questions

Chapter at a Glance

JEE MainJEE AdvancedCBSE BoardsBITSATKVPY

12Concepts

20Formulas

HighDifficulty

7–9%Weightage

Before You Begin

Prerequisites

✓Real number algebra
✓Quadratic equations (Class X)
✓Coordinate plane plotting

Have Ready

🔧Graph paper for Argand plane
🔧Powers-of-i table
🔧Scientific calculator

Exercise Topic Map

Exercise 4.1Add/subtract/multiply/divide; rationalise using conjugate; |z|

Miscellaneousi^n via cycle-of-4; prove modulus identities; combined

Key Formulae

\(i^{4k}=1,\; i^{4k+1}=i,\; i^{4k+2}=-1,\; i^{4k+3}=-i\)

\(|z| = \sqrt{a^2 + b^2}\)

\(z^{-1} = \dfrac{\bar{z}}{|z|^2}\)

\(z = r(\cos\theta + i\sin\theta)\)

\(D < 0:\; \text{roots} = \dfrac{-b \pm i\sqrt{|D|}}{2a}\)

NCERT Solving Method

Step 1 — Division: multiply top and bottom by conjugate of denominator (a−bi). Step 2 — Argument: compute arctan(b/a) then adjust for quadrant using signs of a,b. Step 3 — Quadratic: compute D; if D<0 write √(−|D|)=i√|D| and proceed normally. Step 4 — Powers of i: divide exponent by 4; use the remainder.

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