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# Complex Numbers and Quadratic Equations

## Complex Numbers and Quadratic Equations PDF Notes, Important Questions and Synopsis

SYNOPSIS

1. A number of the form x + iy, where x, y Î  and  (i is iota), is called a complex number.
It is denoted by z, and a set of complex numbers is denoted by ℂ.
x = real part or Re(z), y = imaginary part or Im(z)

2.  Complex conjugate Argument Magnitude If z = x + iy, then the conjugate of z is = x - iy amp(z) = arg(z) = q = General argument: 2nπ + θ, n ϵ ℕ Principal argument: -π < θ ≤ π Least positive argument: 0 < θ ≤ 2π z = x + iy |z|=
3. Representation of Complex Number

 Polar Representation Exponential Form Vector Representation x = r cos  θ, y = r sin  θ z = r eiθ (where = cos eiθ + I sin θ) z = x + iy is considered a position vector of point p

4. Square roots of a complex number
Let z = x + iy, then square root of z is
 ,for y>0 , for y<0
5.  Properties of the argument of a Complex Number:
 arg(any real positive number) = 0 arg(any real negative number) = π

6. Inequalities
 I. Triangle inequalities       1. |z1 ± z2| £ | z1| ± | z2|       2. |z1 ± z2| ³ | z1| - | z2| II. Parallelogram inequalities       | z1 + z2|2+ | z1 - z2|2 = 2 [|z1|2+| z2|2]
7. If ABC is an equilateral triangle having vertices z1, z2, z3, then  or

8. If z1, z2, z3, z4 are vertices of a parallelogram, then z1 + z3 = z2 + z4.

9. If z1, z2, z3 are affixes of the points A, B and C in the Argand plane, then

 i. ÐBAC = ii.   , where α = ÐBAC
10. The equation of a circle whose centre is at a point having affix z0 and radius R = |z - z0|.
11. If a, b are positive real numbers, then.
12. Integral powers of iota

Hence,

1. An equation of the form  is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.
2. Values of the variable which satisfies the quadratic equation are called its roots.
3. Nature of Roots
Let f(x) =  be the quadratic equation, the discriminant D = .

 If a > 0 If a < 0 1. 1. 2. 2. 3. 3.
4. Let α, β be the roots of the quadratic equation then

 i. Roots are given by the quadratic formula:      formula:      a, b = ii.  Relation between roots and coefficients:1. Sum of the roots =a+b = -2.  Product of the roots = a×b =

Note: Quadratic equation can be rewritten as .

Let y =  be the quadratic polynomial. There are two inequalities:

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