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Complex Numbers and Quadratic Equations

NCERT Class 11 Mathematics Chapter 4

Complex Numbers and Quadratic Equations for Class 11 Maths and JEE: algebra, modulus, conjugate, the Argand plane, argument and quadratic equations.

6 topics Updated 8 Nov 2026

For: Class 11 and JEE, unless a topic says otherwise

In the syllabus: JEE Class 11 syllabus

With your textbook

NCERT Mathematics (Class 11), Chapter 4 “Complex Numbers and Quadratic Equations”, reprint 2026-27.

Open the official NCERT chapter (PDF, 1.6 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Complex numbers

    a + ib, iota and i = −1, real and imaginary parts, purely imaginary numbers and equality, with solved examples.

    NCERT Sections 4.1–4.2 10 min read 3 solved examples 8 practice questions

  2. Algebra of complex numbers

    Adding, subtracting, multiplying and dividing a + ib, powers of iota and square roots of negative numbers.

    NCERT Section 4.3 14 min read 4 solved examples 10 practice questions

  3. Modulus and conjugate of a complex number

    |z| = √(a + b), the conjugate a − ib, z times its conjugate, the inverse and properties.

    NCERT Section 4.4 10 min read 4 solved examples 9 practice questions

  4. The Argand plane

    x + iy as the point (x, y), |z| as a distance, the conjugate as a reflection, with solved examples.

    NCERT Section 4.5 10 min read 3 solved examples 8 practice questions

  5. Argument and polar form of a complex number

    Argument of a complex number for JEE Main: the principal argument in each quadrant, polar form r(cos θ + i sin θ), products and quotients, with examples.

    JEE only 12 min read 4 solved examples 9 practice questions

  6. Quadratic equations

    Quadratic equations for JEE Main (Class 11): complex roots, discriminant and nature of roots, sum and product of roots, forming equations, with examples.

    JEE only 13 min read 6 solved examples 9 practice questions

Key formulas in this chapter

Complex numbers

  • i = −1
  • z = a + ib, Re z = a, Im z = b
  • a + ib = c + id ⇔ a = c and b = d

Algebra of complex numbers

  • i = −1
  • (a + ib) + (c + id) = (a + c) + i(b + d)
  • (a + ib)(c + id) = (ac − bd) + i(ad + bc)
  • 1/(a + ib) = (a − ib)/(a + b), a + ib ≠ 0
  • i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i

Modulus and conjugate

  • |z| = √(a + b), z = a − ib
  • z z = |z|
  • 1/z = z /|z|, z ≠ 0
  • |z1z2| = |z1| |z2|

The Argand plane

  • x + iy ↔ the point (x, y)
  • |z| = distance of z from 0
  • |z1 − z2| = distance between z1 and z2

Argument and polar form

  • z = r(cos θ + i sin θ), r = |z|
  • cos θ = x/r, sin θ = y/r
  • −π < arg z ≤ π
  • |z1z2| = r1r2, arg adds (up to a multiple of 2π)

Quadratic equations

  • x = (−b ± √D)/(2a), D = b − 4ac
  • α + β = −b/a, αβ = c/a
  • x − (α + β)x + αβ = 0

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