Complex Numbers and Quadratic Equations:Class 11 Maths NCERT Chapter 5
Key Features of NCERT Material for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations
In the previous Chapter 4:Principle of Mathematical Induction we will study about different properties of mathematical inductions.In this Chapter 5:Complex Numbers and Quadratic Equations we will study about complex numbers and its properties.
Quick revision notes
Imaginary Numbers
The square base of a negative real number is called an imaginary number, for example √-2, √-5 and so on.
The quantity √-1 is an imaginary unit and it is indicated by ‘i’ called Iota.
Integral Power of IOTA (i)
i = √-1, i2 = – 1; i3 = – i and i4 = 1
Thus, i4n+1 = i, i4n+2 = – 1, i4n+3 = – i, i4n = 1
Note:
- For any two real numbers an and b, the outcome √a × √b : √ab is genuine just, when atleast one of the given numbers i.e. either zero or positive. √-a × √-b ≠ √ab
- In this way, i2 = √-1 × √-1 ≠ 1 ‘i’ is neither positive, zero nor negative.
- in + in+1 + in+2 + in+3 = 0
Complex Number
Some of the structure x + iy, where x and y are real numbers, is known as a complex number, x is called real part and y is called imaginary aspect of the complex number i.e. Re(Z) = x and Im(Z) = y.
Purely Real and Purely Imaginary Complex Number
A complex number Z = x + iy is purely real if it’s imaginary part is 0, i.e. Im(z) = 0 and purely imaginary if it’s real part is 0 i.e. Re (z) = 0.
Equality of Complex Number
Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equivalent, iff x1 = x2 and y1 = y2 i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2)
Note: The Order relation “greater than” and “less than” are not defined for complex number.
Variable based math of Complex Numbers
Addition of complex numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, at that point their sum defined as
z1 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2)
Properties of Addition
- Commutative Property: z1 + z2 = z2 + z1
- Associative Property: z1 + (z2 + z3) = (z1 + z2) + z3
- Property of Additive Identity: z + 0 = z = 0 + z Here, 0 is additive identity.
Subtraction of complex numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, at that point their difference is defined as
z1 – z2 = (x1 + iy1) – (x2 + iy2) = (x1 – x2) + i(y1 – y2)
Multiplication of complex numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, at that point their multiplication is defined as
z1z2 = (x1 + iy1) (x2 + iy2) = (x1x2 – y1y2) + i (x1y2 + x2y1)
Properties of Multiplication
- Commutative: z1z2 = z2z1
- Additive: z1(z2z3) = (z1z2)z3
- Multiplicative character: z . 1 = z = 1 . z Here, 1 is multiplicative character of a component z.
- Multiplicative inverse: For each non-zero complex number z, there exists a complex number z1 with the end goal that z . z1 = 1 = z1 . z
- Distributive law: z1(z2 + z3) = z1z2 + z1z3
Division of Complex Numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, at that point their division is characterized as
Conjugate of Complex Number
Let z = x + iy, in the event that ‘I’ is supplanted by (- I), at that point said to be conjugate of the complex number z and it is meant by , for example = x – iy
Properties of Conjugate
Modulus of a Complex Number
Take z = x + iy to be a complex number. At that point, the positive square base of the entirety of square of genuine part and square of nonexistent part is called modulus (total estimations) of z and it is indicated by |z| for example |z| =
It speaks to a separation of z from inception in the arrangement of complex number c, the request connection isn’t characterized
for example z1 > z2 or z1 < z2 has no importance except for |z1| > |z2| or |z1|<|z2| has got its significance, since |z1| and |z2| are genuine numbers.
Properties of Modulus of a Complex number
Argand Plane
Any complex number z = x + iy can be spoken to mathematically by a (x, y) in a plane, called argand plane or gaussian plane. A simply number x, for example (x + 0i) is spoken to by the point (x, 0) on X-pivot. Hence, X-hub is called genuine hub. A simply nonexistent number iy for example (0 + iy) is spoken to by the point (0, y) on the y-pivot. Along these lines, the y-hub is known as the fanciful pivot.
Argument of a complex Number
The angle made by line joining guide z toward the source, with the positive course of X-pivot in an enemy of clockwise sense is called contention or sufficiency of complex number. It is meant by the image arg(z) or amp(z).
arg(z) = θ = tan-1(y/x)
Argument of z isn’t novel, general estimation of the argument of z is 2nπ + θ, however arg(0) isn’t characterized. The special estimation of θ with the end goal that – π < θ ≤ π is known as the principal estimation of the sufficiency or principal argument.
Principal Value of Argument
- if x > 0 and y > 0, at that point arg(z) = θ
- if x < 0 and y > 0, at that point arg(z) = π – θ
- if x < 0 and y < 0, at that point arg(z) = – (π – θ)
- if x > 0 and y < 0, at that point arg(z) = – θ
Polar Form of a Complex Number
If z = x + iy is a complex number, at that point z can be composed as z = |z| (cosθ + isinθ), where θ = arg(z). This is called polar structure. If the overall estimation of the argument is θ, at that point the polar type of z will be z = |z| [cos (2nπ + θ) + isin(2nπ + θ)], where n is a whole number.
Square Root of a Complex Number
Solution of a Quadratic Equation
The equation ax2 + bx + c = 0, where a, b and c are numbers (genuine or complex, a ≠ 0) is known as the overall quadratic equation in factor x. The estimations of the variable fulfilling the given equation are called roots of the equation.
The quadratic equation ax2 + bx + c = 0 with genuine coefficients has two roots given by and , where D = b2 – 4ac, called the discriminant of the equation.
Note:
(I) When D = 0, roots mineral genuine and equivalent. At the point when D > 0 roots are genuine and inconsistent. Further If a,b, c ∈ Q and D is impeccable square, at that point the roots of quadratic equation are genuine and inconsistent and if a, b, c ∈ Q and D isn’t immaculate square, at that point the roots are irrational and happen in pair. At the point when D < 0, roots of the equation are non genuine (or complex).
(ii) Let α, β be the roots of quadratic equation ax2 + bx + c = 0, at that point sum of roots α + β = and the product of roots αβ = .