Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 6 Complex Numbers and Quadratic Equations will enable students to study smartly.
CHSE Odisha 11th Class Math Notes Chapter 6 Complex Numbers and Quadratic Equations
Unit imaginary number ‘i’.
The unit imaginary number i = √-1
i2 = -1
i3 = -i
i4 = 1
In general (i)4n = 1, (i)4n+1 = i, (i)4n+2 = -1, and (i)4n+3 = -i.
⇒ If a and b are positive real numbers then
√-a × √-b = -√ab
√a × √b = √ab
Complex Number
General form: = z = a +ib
- a = Real part of (z) = Re (z)
- b = Imaginary part of (z) = Im(z)
- a + i0 is purely real and 0 + ib is purely imaginary .
- a + ib = c + id iff a = c and b = d
Complex Algebra
(a) Addition of complex numbers
If z1 = a + ib and z2 = c + id then z1 + z2 = (a + c) + i(b + d)
Properties:
- Addition is commutative: z1 + z2 = z2 + z1
- Addition is associative: (z1 + z2) + z3 = z1 + (z2 + z3)
- 0 + i0 is the additive identity.
- -z is the additive inverse of z.
(b) Subtraction of complex numbers:
z1 = a + ib and z2 = c + id then z1 – z2 = (a – c) + i(b – d)
(c) Multiplication of complex numbers:
z1 = a + ib and z2 = c + id then z1z2 = (ac – bd) + i(bc + ad)
Properties:
- Multiplication is commutative: z1z2 = z2z1
- Multiplication is associative: z1(z2z3) = z1z2(z3)
- 1 = 1 + i0 is the multiplicative identity.
- If z = a + ib then the inverse of z.
z-1 = \(\frac{1}{a+i b}=\frac{a-i b}{(a+i b)(a-i b)}\)
= \(\frac{a-i b}{a^2+b^2}=\frac{a}{a^2+b^2}-\frac{i b}{a^2+b^2}\) - Multiplication is distributive over addition. z1(z2 + z3) = z1z2 + z1z3
Conjugate and modulus of a complex number:
If z = a + ib the conjugate of z is \(\bar{Z}\) = a – ib.
⇒ We get conjugate by replacing i by (-i) Modulus of z = a + ib is denoted by |z| and |z| = \(\sqrt{a^2+b^2}\)
Properties Of Conjugate:
(i) \((\overline{\bar{z}})\) = z
(ii) z + \(\bar{z}\) = 2 Re (z)
(iii) z – \(\bar{z}\) = 2i m̂ (z)
(iv) z – \(\bar{z}\) ⇔ z is purely real
(v) Conjugate of real number is itself.
(vi) z + \(\bar{z}\) = 0 ⇒ z is purely imaginary.
(vii) z. \(\bar{z}\) = [Re(z)]2 + [m̂(z)]2
= a2 + b2
= |z|2
(viii) \(\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\)
(ix) \(\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}\)
(x) \(\overline{z_1z_2}=\overline{z_1}\overline{z_2}\)
(xi) \(\left(\overline{\frac{z_1}{z_2}}\right)=\frac{\overline{z_1}}{\overline{z_2}}\)
Properties of modulus:
(1) Order relations are not defined for complex numbers. i,e,. z1 > z2 or z1 < z2 has no meaning but |z1| < |z2| or |z1| > |z2| is meaningful because |z1| and |z2| are real numbers.
(2) |z| = 0 ⇔ z = 0
(3) |z| = |\(\bar{z}\)| = |-z|
(4) |z| ≤ Re (z) ≤ |z| and -|z| ≤ m̂ (z) ≤ |z|
(5) |z1z2| = |z1| |z2|
(6) \(\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}\)
(7) |z1 ± z2|2 = |z1|2 + |z2|2 ± 2 Re (z1\(\bar{z}_2\))
(8) |z1 + z2|2 = |z1 – z2|2 = 2(|z1|2 + |z2|2)
(9) |z1 + z2|2 ≤ |z1| + |z2|
Square Root Of Complex Number:
Let z = a + ib
Let √z = x + iy
If b > 0 then x and y are taken as same sign.
If b < 0 then x and y are of opposite sign.
Representation of a complex number:
We represent a complex number in different forms like
(i) Geometrical form
(ii) Vector form
(iii) Polar form
(iv) Eulerian form or Exponential form
(i) Geometrical form:
Geometrically z = x + iy = (x, y) represents a point in a coordinate plane known as Argand plane or Gaussian plane.
(ii) Vector form:
In vector form a complex number z = x + iy is the vector \(\overrightarrow{\mathrm{OP}}\) where p(x, y) is the point in the cartesian plane.
(iii) Polar form:
A complex number z = x + iy in polar form can be written as z = r(cos θ + i sin θ) where r = \(\sqrt{x^2+y^2}\) = |z| and θ is called the argument and -π < θ ≤ π. Technique to write z = x + iy in polar form.
Step – 1: Find r = |z| = \(\sqrt{x^2+y^2}\)
Step – 2: Find α = tan-1 \(\left|\frac{y}{x}\right|\)
Step – 3:
θ = α for x > 0, y > 0
θ = π – α for x > 0, y > 0
θ = -π + α for x > 0, y > 0
θ = -α for x > 0, y > 0
Step – 4: Write z = r(cos θ + i sin θ)
(iv) Eulerian form or Exponential form z = r eiθ, because eiθ = cos θ + i sin θ where θ is the argument and r is the modulus if z.
Note:
(1) |z1 z2 z3 ….. zn| = |z1||z2| …. |zn|
(2) arg (z1z2 …. Zn) = arg (z1) + arg (z2) + ….. + arg (zn)
(3) arg \(\left(\frac{z_1}{z_2}\right)\) = arg (z1) – arg (z2)
(4) arg \((\bar{z})\) = -arg (z)
Cube Roots Of Unity:
Cube roots of unity are 1, ω, ω2 where ω = \(\frac{-1 \pm i \sqrt{3}}{2}\)
Properties of Cube roots of unity:
(i) Cube roots of unity lie on unit circle |z| = 1
(ii) 1 + ω + ω2 = 0
(iii) Cube roots of -1 are -1, -ω, -ω2
(iv) 1 + ωn + ω2n \(=\left\{\begin{array}{l}
0 \text { if } n \text { is not a multiple of } 3 \\
3 \text { if } n \text { is a multiple of } 3
\end{array}\right.\)
(v) z3 + 1 = (z + 1) (z + ω) (z + ω2)
(vi) -ω and -ω2 are roots of z2 – z + 1 = 0.
De-moivre’s theorem:
(a) (De-moivre’s theorem for integral index)
(cos θ + i sin θ)n = cos (nθ) + i sin (nθ)
(b) (De-moivre’s theorem for rational index)
cos (nθ) + i sin (nθ) is one of the values of (cos θ + i sin θ)n
(c) nth roots of unity
nth roots of unity are 1, α, α2, α3 …..αn-1. where α = ei\(\frac{2 \pi}{n}\) = cos \(\frac{2 \pi}{n}\) + i sin \(\frac{2 \pi}{n}\)
Properties:
- 1 + α + α2 ….. + αn-1 = 0
- 1 + αp + α2p + ….. + α(n-1)p \(= \begin{cases}0 & \text { if } p \text { is not a multiple of } n \\ n & \text { if } p \text { is a multiple of } n\end{cases}\)
- 1. α. α2 ….. αn-1 = (-1)n-1
- zn – 1 = (z – 1) (z – α) (z – α2) …..(z – αn-1)
Quadratic Equations:
The general form: ax2 + bx + c = 0 …(i)
Solutions of quadratic equation(1) are
x = \(\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\)
D = b2 – 4ac is called the discrimination of a quadratic equation.
D > 0 ⇒ The equation has real and distinct roots.
D = 0 ⇒ The equation has real and equal roots.
D < 0 ⇒ The equation has complex roots.
Note:
In a quadratic equation with real coefficients, the complex roots occur in conjugate pairs.