Complex Numbers and Quadratic Equations
Complex Numbers and Quadratic Equations PDF Notes, Important Questions and Synopsis
SYNOPSIS
- A number of the form x + iy, where x, y Î ℝ 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) -
Complex conjugate
Argument
Magnitude
If z = x + iy, then the conjugate of z is
= x - iyamp(z) = arg(z) = q =
General argument: 2nπ + θ, n ϵ ℕ
Principal argument: -π < θ ≤ π
Least positive argument: 0 < θ ≤ 2πz = x + iy
|z|= - 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
- Square roots of a complex number
Let z = x + iy, then square root of z is
,for y>0
, for y<0
- Properties of the argument of a Complex Number:
- arg(any real positive number) = 0
- arg(any real negative number) = π
- 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] - If ABC is an equilateral triangle having vertices z1, z2, z3, then or
- If z1, z2, z3, z4 are vertices of a parallelogram, then z1 + z3 = z2 + z4.
- If z1, z2, z3 are affixes of the points A, B and C in the Argand plane, then
i. ÐBAC =ii. , where α = ÐBAC - The equation of a circle whose centre is at a point having affix z0 and radius R = |z - z0|.
- If a, b are positive real numbers, then.
- Integral powers of iota
Hence,
Quadratic Equations
- An equation of the form is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.
- Values of the variable which satisfies the quadratic equation are called its roots.
-
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.
-
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 .
-
Quadratic inequalities
Let y = be the quadratic polynomial. There are two inequalities:
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