Quick Revision Notes

Class 11 Mathematics

Chapter: Complex Numbers and Quadratic Equations

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1. Introduction to Complex Numbers (सम्मिश्र संख्याएँ)

A number of the form $z = a + ib$, where $a$ and $b$ are real numbers, is defined as a complex number.

• Real part ($\text{Re}(z)$): $a$
• Imaginary part ($\text{Im}(z)$): $b$
• $i$ is called the iota, where $i = \sqrt{-1}$.

Integral Powers of $i$:

• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$
• In general, $i^{4n} = 1$, $i^{4n+1} = i$, $i^{4n+2} = -1$, $i^{4n+3} = -i$, where $n$ is an integer.

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2. Algebra of Complex Numbers (बीजगणित)

Let $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ be two complex numbers.

• Addition: $z_1 + z_2 = (a_1 + a_2) + i(b_1 + b_2)$
• Subtraction: $z_1 - z_2 = (a_1 - a_2) + i(b_1 - b_2)$
• Multiplication: $z_1 z_2 = (a_1a_2 - b_1b_2) + i(a_1b_2 + a_2b_1)$
• Division: $\frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{z_2 \bar{z_2}} = \frac{(a_1a_2 + b_1b_2) + i(a_2b_1 - a_1b_2)}{a_2^2 + b_2^2}$ (where $z_2 \neq 0$)

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3. Properties of Algebraic Operations

1. Closure Law: $z_1 + z_2$ and $z_1 z_2$ are complex numbers.
2. Commutative Law: $z_1 + z_2 = z_2 + z_1$ and $z_1 z_2 = z_2 z_1$
3. Associative Law: $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$
4. Distributive Law: $z_1(z_2 + z_3) = z_1z_2 + z_1z_3$
5. Additive Identity: $0 + i0$ is the additive identity ($z + 0 = z$).
6. Multiplicative Identity: $1 + i0$ is the multiplicative identity ($z \cdot 1 = z$).
7. Additive Inverse: $-z = -a - ib$ is the additive inverse of $z = a + ib$.
8. Multiplicative Inverse: $z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - ib}{a^2 + b^2}$ (for $z \neq 0$).

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4. Conjugate and Modulus of a Complex Number

Let $z = a + ib$.

• Conjugate ($\bar{z}$ - संयुग्मी): $\bar{z} = a - ib$
• Modulus ($|z|$ - मापांक): $|z| = \sqrt{a^2 + b^2}$

Important Properties:

• $z \bar{z} = |z|^2$
• $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$
• $\overline{z_1 z_2} = \bar{z_1} \cdot \bar{z_2}$
• $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$ (where $z_2 \neq 0$)
• $|z_1 z_2| = |z_1| |z_2|$
• $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ (where $z_2 \neq 0$)
• $|z| = |\bar{z}| = |-z| = |-\bar{z}|$

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5. Argand Plane and Polar Representation (आर्गंड तल और ध्रुवीय रूप)

• Argand Plane: A plane representing complex numbers as points $(a, b)$.
• Polar Form (ध्रुवीय रूप): $z = r(\cos\theta + i\sin\theta)$
  • Where $r = |z| = \sqrt{a^2 + b^2}$ (Modulus)
  • $\theta = \arg(z)$ (Argument / कोणांक)

Finding Argument ($\theta$):

1. First, find the acute angle $\alpha$ where $\tan\alpha = \left|\frac{b}{a}\right|$.
2. Determine the quadrant in which $z = a + ib$ lies:
   • 1st Quadrant ($+ve, +ve$): $\theta = \alpha$
   • 2nd Quadrant ($-ve, +ve$): $\theta = \pi - \alpha$
   • 3rd Quadrant ($-ve, -ve$): $\theta = -(\pi - \alpha)$ or $\theta = -\pi + \alpha$
   • 4th Quadrant ($+ve, -ve$): $\theta = -\alpha$

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6. Quadratic Equations (द्विघात समीकरण)

A quadratic equation in variable $x$ is of the form: $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$ and $a \neq 0$.

• Discriminant ($D$ or $\Delta$): $D = b^2 - 4ac$
• Quadratic Formula (श्रीधराचार्य सूत्र): $x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Nature of Roots (मूलों की प्रकृति):

1. If $D > 0$: Two distinct real roots.
2. If $D = 0$: Two equal (coincident) real roots.
3. If $D < 0$: Two complex conjugate roots.
   • Roots are given by $x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}$.