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Complex Numbers And Quadratic Equations class 11 Notes Mathematics

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CBSE Class 11 Mathematics
Revision Notes
Chapter-5
COMPLEX NUMBERS AND QUADRATIC EQUATIONS

Algebra, Modulus and Conjugate of Complex Numbers
Argand Plane and Polar Representation
Quadratic Equations

$\Rightarrow$ ${i^2} = - 1$
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:

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 $x = {{ - b \pm i\sqrt {4ac - {b^2}} } \over {2a}}.$

The Complex Numbers And Quadratic Equations class 11 Notes

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CBSE Class-11 Revision Notes and Key Points

Complex Numbers And Quadratic Equations class 11 Notes Mathematics. CBSE quick revision note for class-11 Mathematics, Physics, Chemistry, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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