## Unit 1: Complex Numbers (Solutions)

This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.

After reading this unit the students will be able to

- Recall complex number $z$ represented by an expression of the form $z=a+ib$ or of the form $(a,b)$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$.
- Recognize $a$ as real part of $z$ and $b$ as imaginary part of $z$.
- Know condition for equality of complex numbers.
- Carry out basic operations on complex numbers.
- Define $\bar{z} = a —ib$ as the complex conjugate of $z=a+ib$.
- Define $|z| = \sqrt{a^2+b^2}$ as the absolute value or modulus of a complex number $z=a+ib$
- Describe algebraic properties of complex numbers (e.g. commutative, and distributive) with respect to $'+'$ and $'\times'$.
- Know additive identity and multiplicative identity for the set of complex numbers.
- Find additive inverse and multiplicative inverse of a complex $z$.
- Demonstrate the following properties $|z|=|-z|=|\bar{z}=|-\bar{z}|$
- Find real and imaginary parts of complex numbers.
- Solve simultaneous linear equations with complex coefficients.
- Write the polynomial as a product of linear factors.
- Solve quadratic equation of the form $pz^2+ qz+ r = 0$ by completing squares, where $p,q,r$ are real numbers and $z$ a complex number.