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# Complex Numbers and Quadratic Equations

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

## 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:        Download complete content for FREE ## JEE Main Video Lectures By Experts

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