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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
- Imaginary Number: Square root of a negative number is called an Imaginary number. For example, etc. are imaginary numbers.
- Integral power of Iota ( ) : = where and
- Complex Number: A number of the form 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
- Real part of is and is denoted by .
- Imaginary part of is and is written as .
- Equality of complex numbers: Two complex numbers and are said to be equal, if and .
- 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 is 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 is denoted by .
- Argument of a complex number : Arg .
- Representation of complex number as ordered pair: Any complex number can be written in ordered pair as , where a is the real past and b is the imaginary part of a complex number.
(i)
(ii)
- Division of a complex number: If and , then,
=
- For any non-zero complex number there exists the complex number denoted by or , called the multiplicative inverse of z such that
- Polar form of a complex number: The polar form of the complex number , where (the modulus of z) and . (θ is known as the argument of z. The value of θ, such that is called the principal argument of z.
- Important properties: (i) , (ii)
- 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., .
- Roots of an equation: The values of variable satisfying a given equation are called its roots. Thus, is a root of the equation if
- Solution of quadratic equation: The solutions of the quadratic equation , where are given by
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