# Complex Numbers And Quadratic Equations class 11 Notes Mathematics

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## Complex Numbers And Quadratic Equations class 11 Notes Mathematics

CBSE Class 11 Mathematics
Revision Notes
Chapter-5

1. Algebra, Modulus and Conjugate of Complex Numbers
2. Argand Plane and Polar Representation
• $\Rightarrow$     ${i^2} = - 1$
• Imaginary Number: Square root of a negative number is called an Imaginary number. For example, $\sqrt { - 5} ,\sqrt { - 16} ,$ etc. are imaginary numbers.
• Integral power of Iota ($i$) : ${i^p}\left( {p > 4} \right) = {i^{4q + r}}$ = ${\left( {{i^4}} \right)^q}.{i^r} = {i^r},$ where $\sqrt { - 1} = i$ and ${i^4} = 1$
• Complex Number: A number of the form $\text{a }+~\text{ib},$ where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number. It is denoted by $z.$
• Real part of $z=a+ib$ is $a$ and is denoted by $Re(z) = a$.
• Imaginary part of $z=a+ib$ is $b$ and is written as $Im(z) = b$.
• Equality of complex numbers: Two complex numbers ${z_1} = a + ib$ and ${z_2} = c + id$ are said to be equal, if $a=c$ and $b=d$.
• Conjugate of a complex number: Two complex numbers are said to be conjugate of each other, if their sum is real and their product is also real. Conjugate of a complex number $z=a+ib$ is $\overline z = a - ib$ i.e., conjugate of a complex number is obtained by changing the sign of imaginary part of z.
• Modulus of a complex number: Modulus of a complex number $z=x+iy$ is denoted by $\left| z \right| = \sqrt {{x^2} + {y^2}}$.
• Argument of a complex number $x+iy$ : Arg$(x+iy)=$ ${\tan ^{ - 1}}{y \over x}$.
• Representation of complex number as ordered pair: Any complex number $a+ib$ can be written in ordered pair as $\left( {a,b} \right)$, where a is the real past and b is the imaginary part of a complex number.
• $\text{Let }{{\text{z}}_{1}}\text{ }=\text{ a }+\text{ ib and }{{\text{z}}_{2}}\text{ }=\text{ c }+\text{ id}.\text{ Then}$

(i) ${{\text{z}}_{1}}\text{+ }{{\text{z}}_{2}}\text{= (a + c) + i (b + d)}$

(ii) ${{\text{z}}_{1}}{{\text{z}}_{2}}\text{= (ac -bd) + i (ad +bc)}$

• Division of a complex number: If ${z_1} = a + ib$ and ${z_2} = c + id$, then,

${{{z_1}} \over {{z_2}}} = {{a + ib} \over {c + id}} = {{\left( {a + ib} \right)\left( {c - id} \right)} \over {\left( {c + id} \right)\left( {c - id} \right)}}$= ${{ac + bd} \over {{c^2} + {d^2}}} + i{{bc - ad} \over {{c^2} + {d^2}}}$

•   For any non-zero complex number $\text{z }=\text{ a }+\text{ ib }\left( \text{a }\ne \text{ }0,\text{ b }\ne \text{ }0 \right),$ there exists the complex number  $\frac{a}{{{a}^{2}}+{{b}^{2}}}+i\frac{-b}{{{a}^{2}}+{{b}^{2}}}$denoted by $\frac{1}{z}$or  ${{z}^{-1}}$, called the multiplicative inverse of z such that $\text{(a + ib) }\left( \frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}+i\frac{-b}{{{a}^{2}}+{{b}^{2}}} \right)\text{=1+io=1}$
• Polar form of a complex number: The polar form of the complex number $\text{z }=\text{ x }+\text{ iy is r }\left( \text{cos}\theta \text{ }+\text{ i sin}\theta \right)$, where $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ (the modulus of z) and $~\text{cos}\theta ~=\frac{x}{r},$ $~\sin \theta ~=\frac{y}{r},$. (θ is known as the argument of z. The value of θ, such that is called the principal argument of  z.
• Important properties:      (i) $\left| {{z_1}} \right| + \left| {{z_2}} \right| \ge \left| {{z_1} + {z_2}} \right|$,      (ii) $\left| {{z_1}} \right| - \left| {{z_2}} \right| \le \left| {{z_1} + {z_2}} \right|$
• Fundamental Theorem of algebra: A polynomial equation of n degree has n roots.

• Quadratic Equation: Any equation containing a variable of highest degree 2 is known as quadratic equation. e.g., $a{x^2} + bx + c = 0$.
• Roots of an equation: The values of variable satisfying a given equation are called its roots. Thus, $x = \alpha$ is a root of the equation $p\left( x \right) = 0$ if $p\left( \alpha \right) = 0.$
• Solution of quadratic equation: The solutions of the quadratic equation $a{x^2} + bx + c = 0$, where $a,b,c \in {\rm{R,}}$ $a \ne 0,$  ${b^2} - 4ac < 0,$  are given by $x = {{ - b \pm i\sqrt {4ac - {b^2}} } \over {2a}}.$