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

i^{2} = -1

i^{3} = -i

i^{4} = 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) = I
_{m}(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 z_{1} = a + ib and z_{2} = c + id then z_{1} + z_{2} = (a + c) + i(b + d)

Properties:

- Addition is commutative: z
_{1}+ z_{2}= z_{2}+ z_{1} - Addition is associative: (z
_{1}+ z_{2}) + z_{3}= z_{1}+ (z_{2}+ z_{3}) - 0 + i0 is the additive identity.
- -z is the additive inverse of z.

(b) Subtraction of complex numbers:

z_{1} = a + ib and z_{2} = c + id then z_{1} – z_{2} = (a – c) + i(b – d)

(c) Multiplication of complex numbers:

z_{1} = a + ib and z_{2} = c + id then z_{1}z_{2} = (ac – bd) + i(bc + ad)

Properties:

- Multiplication is commutative: z
_{1}z_{2}= z_{2}z_{1} - Multiplication is associative: z
_{1}(z_{2}z_{3}) = z_{1}z_{2}(z_{3}) - 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. z
_{1}(z_{2}+ z_{3}) = z_{1}z_{2}+ z_{1}z_{3}

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}

= a^{2} + b^{2}

= |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,. z_{1} > z_{2} or z_{1} < z_{2} has no meaning but |z_{1}| < |z_{2}| or |z_{1}| > |z_{2}| is meaningful because |z_{1}| and |z_{2}| are real numbers.

(2) |z| = 0 ⇔ z = 0

(3) |z| = |\(\bar{z}\)| = |-z|

(4) |z| ≤ Re (z) ≤ |z| and -|z| ≤ m̂ (z) ≤ |z|

(5) |z_{1}z_{2}| = |z_{1}| |z_{2}|

(6) \(\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}\)

(7) |z_{1} ± z_{2}|^{2} = |z_{1}|^{2} + |z_{2}|^{2} ± 2 Re (z_{1}\(\bar{z}_2\))

(8) |z_{1} + z_{2}|^{2} = |z_{1} – z_{2}|^{2} = 2(|z_{1}|^{2} + |z_{2}|^{2})

(9) |z_{1} + z_{2}|^{2} ≤ |z_{1}| + |z_{2}|

**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 e^{iθ}, because e^{iθ} = cos θ + i sin θ where θ is the argument and r is the modulus if z.

Note:

(1) |z_{1} z_{2} z_{3} ….. z_{n}| = |z_{1}||z_{2}| …. |z_{n}|

(2) arg (z_{1}z_{2} …. Z_{n}) = arg (z_{1}) + arg (z_{2}) + ….. + arg (z_{n})

(3) arg \(\left(\frac{z_1}{z_2}\right)\) = arg (z_{1}) – arg (z_{2})

(4) arg \((\bar{z})\) = -arg (z)

**Cube Roots Of Unity:**

Cube roots of unity are 1, ω, ω^{2} 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 + ω^{2} = 0

(iii) Cube roots of -1 are -1, -ω, -ω^{2}

(iv) 1 + ω^{n} + ω^{2n }\(=\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) z^{3} + 1 = (z + 1) (z + ω) (z + ω^{2})

(vi) -ω and -ω^{2} are roots of z^{2} – 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, α^{2}, α^{3} …..α^{n-1}. where α = e^{i\(\frac{2 \pi}{n}\)} = cos \(\frac{2 \pi}{n}\) + i sin \(\frac{2 \pi}{n}\)

Properties:

- 1 + α + α
^{2}….. + α^{n-1}= 0 - 1 + α
^{p}+ α^{2p}+ ….. + 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. α
^{2}….. α^{n-1}= (-1)^{n-1} - z
^{n}– 1 = (z – 1) (z – α) (z – α^{2}) …..(z – α^{n-1})

**Quadratic Equations:**

The general form: ax^{2} + bx + c = 0 …(i)

Solutions of quadratic equation(1) are

x = \(\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\)

D = b^{2} – 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.