JEE Maths Complex Numbers and Quadratic Equations
Complex Numbers and Quadratic Equations PDF Notes, Important Questions and 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)
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
- Representation of Complex Number
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
- Properties of the argument of a Complex Number:
- arg(any real positive number) = 0
- arg(any real negative number) = π
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
- 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
Let α, β be the roots of the quadratic equation then
i. Roots are given by the quadratic 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:
JEE Class Revise
- (1+w+w^2)= (complex number)
- The very first question : If z1 and z2...
- Please solve all 3 parts. Explain in detail. ThanQ!
- The question is on the picture😊
- two numbers are such that three times the first added to four times the second is 96 and the excess of four times the first over three times the second is three find the number
- Sir please provide solution
- Sir plz solve it step by step,thanks
- Z1 and Z2 are two complex numbers
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