Complex Numbers and Quadratic Equations — Class 11 Mathematics

Core Concepts

The imaginary unit i satisfies i²=−1. A complex number has the form a+ib, where a,b are real. Its conjugate is a−ib and modulus is √(a²+b²).

Worked Examples

(3+2i)+(4−5i)=7−3i.

(3+2i)(3−2i)=9−(2i)²=13. Multiplying conjugates gives a real number.

For x²−5x+6=0, factorisation gives (x−2)(x−3)=0, so roots are 2 and 3.

Explained MCQs

1. i⁴=1. Powers repeat every four: i,−1,−i,1.
2. |3+4i|=5 because √(9+16)=5.
3. A quadratic has equal roots when its discriminant is 0.

Practice

1. Simplify (2+3i)(1−2i).
2. Find the roots of 2x²−7x+3=0.
3. Find the modulus and conjugate of −5+12i.