Mathematics Notes

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Mathematics Notes: Optimized for Effective Learning

Comprehensive Calculus Notes:

Our online Mathematics notes offer a comprehensive coverage of Calculus, ensuring a deep understanding of concepts like derivatives, integrals, and limits. These notes facilitate active learning and foster problem-solving skills.

Statistical Insights:

Our online Statistics notes equip students with the knowledge needed to analyze and interpret data effectively.

.Component and Defination of Mathematical Terms:

1.Sets
2.Complex Numbers
3.Quadratic Equations
3.Systems of Two Equations
4.Matrices

Determinants and Inverse Matrices

Groups

Sequence and Series

Permutations and Combinations

Introduction to Probability

Mathematical Induction

The Binomial Theorem

Fundamental Trigonometry

Trigonometric Identities

Trigonometric Functions

Solutions of Triangles

Inverse Trigonometric Functions

and Trigonometric equations

Orthogonal Trajectory

SETS

A set is a well-defined collection of distinct objects, considered as an object in its own right. The objects in a set are called elements or members of the set. Sets usually denote by capital letters, and we list their elements within curly braces, like this: A = {1, 2, 3, 4}.

Definition:

A set is a collection of distinct elements, typically written as: where are the elements of the set . If an element belongs to a set, we say it is a member of the set, and this is denoted by if $x$ is an element of $A$ .

Types of Sets:

Finite Set: A set with a limited number of elements.
Example:
Infinite Set: A set with an unlimited number of elements.
Example:
Subset: A set where all elements of one set are contained in another.
Example:
Universal Set: The set that contains all objects under consideration, usually denoted by .
Empty Set (Null Set): A set that contains no elements, denoted by
Example:
Power Set: The set of all subsets of a set, including the empty set and the set itself.
Example: Power set of is
Equal Sets: Two sets are equal if they have exactly the same elements.
Example:
Union of Sets: The set containing all elements from both sets. A={1,2,3) & B={2,4,6)
Example:
Intersection of Sets: The set containing only elements common to both sets.
Example:

Applications of Sets:

Engineering:

Sets are crucial in computer science, particularly in the development of algorithms, data structures, and databases. Engineers also use sets to model systems, solve optimization problems, and perform operations like union and intersection for network analysis and circuit design.

Mathematics:

In mathematics, sets form the foundation for various concepts such as probability, calculus, and geometry. Set theory is used in solving complex problems, formulating proofs, and understanding functions.

Economics:

Sets help in analyzing data, consumer behavior, and market segmentation. Economists use sets to categorize and analyze different populations and to study the overlap and intersection of markets.

Physics:

In physics, sets are applied in quantum mechanics and relativity theory. Sets describe physical quantities, particle systems, and the union of various physical states.

Computer Science:

In programming, sets are used to manage collections of data, solve problems related to databases, and execute operations such as searching, sorting, and filtering.

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Complex Numbers:

Definition, Operations, and Applications

Complex numbers extend the concept of real numbers to include solutions to equations that do not have real solutions, such as the square root of negative numbers. They are essential in various fields of mathematics and engineering.

Definition

A complex number is a number of the form:

$z = a + bi$

where:

$a$ is the real part of the complex number,
$b$ is the imaginary part,
$i$ is the imaginary unit, defined as $i = \sqrt{-1}$ .

For example, in the complex number $3 + 4i$ , $3$ is the real part and $4$ is the imaginary part.

Basic Operations

1. Addition: To add two complex numbers, add their real parts and imaginary parts separately.

$(a + bi) + (c + di) = (a + c) + (b + d)i$

2. Subtraction: To subtract two complex numbers, subtract their real parts and imaginary parts separately.

$(a + bi) - (c + di) = (a - c) + (b - d)i$

3. Multiplication: To multiply two complex numbers, use the distributive property and apply the fact that $i^2 = -1$ .

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

4. Division: To divide two complex numbers, multiply the numerator and the denominator by the conjugate of the denominator.

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

5. Conjugate: The conjugate of a complex number $a + bi$ is $a - bi$ . Conjugates are useful in division and finding magnitudes.

$Conjugate of (a + bi) = a - bi$

6. Magnitude: The magnitude (or modulus) of a complex number $a + bi$ is given by:

$|a + bi| = \sqrt{a^2 + b^2}$

7. Argument: The argument (or angle) of a complex number $a + bi$ is the angle $\theta$ such that:

$\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Polar Form of Complex Numbers

A complex number can be represented in the form

$z = a + bi$

, where $a$ is the real part of the complex number,
$b$ is the imaginary part,
$i$ is the imaginary unit, defined as $i = \sqrt{-1}$ .

A complex number $z$ can be expressed in polar form as:

$z = r \left( \cos \theta + i \sin \theta \right)$

where $r$ is the magnitude (or modulus) of the complex number, given by

$r = |z| = \sqrt{x^2 + y^2}$

and $\theta$ is the argument (or angle) of the complex number, measured from the positive $x$ -axis, given by

$\theta = \arg(z) = \tan^{-1} \left( \frac{y}{x} \right)$

Euler’s Form of Complex Numbers

Euler’s formula relates complex exponentials to trigonometric functions:

Alternatively, the polar form can also be written using Euler’s formula as:

$z = r e^{i \theta}$

This is a more compact and elegant way of expressing the polar form. Here,

$e^{i \theta}$

represents the complex exponential, and $\theta$ is the argument of the complex number. This form is particularly useful for multiplication, division, and finding powers and roots of complex numbers.

Examples 1:

Convert

$z = 3 + 4i$

into Polar & Euler’s For

To convert into polar form:

Magnitude:

$r = \sqrt{3^2 + 4^2} = 5$

The argument $\theta$ is:

$\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$

Thus, the polar form is:

$z = 5 \left(\cos{53.13^\circ} + i \sin{53.13^\circ}\right)$

In Euler’s form:

$z = 5 e^{i 53.13^\circ}$

Examples 2:

Convert

$z = -1 + i$

into Polar & Euler’s For

To convert into polar form:

Magnitude:

$r = \sqrt{-1^2 + 1^2} = \sqrt{2}$

Argument:

$\theta = \tan^{-1}(1 / -1) = 135^\circ \, \text{or} \, \pi=\ - \frac{\pi}{4}$

Polar form:

$z = \sqrt{2} (\cos{135^\circ} + i \sin{135^\circ})$

Euler’s Form

$z = \sqrt{2} e^{i 135^\circ}$

Derivation of Euler’s Form:

Euler’s form of a complex number provides a powerful and elegant way to represent complex numbers using exponential functions. Here’s a step-by-step explanation of Euler’s form and how it is derived from the polar form using the expansions of cosine and sine.

Euler’s Form of a Complex Number

Euler’s form of a complex number $z$ is given by:

$z = r e^{i \theta}$

where:

$r$ is the magnitude of the complex number.
$\theta$ is the argument (angle) of the complex number.
$e^{i\theta}$ is the exponential function involving the imaginary unit $i$ .

Derivation from Polar Form

Start with the Polar Form:The polar form of a complex number $z$ is expressed as:

$z=r(cos⁡θ+isin⁡θ)$

where:

- $r$ is the magnitude of the complex number.
- $\theta$ is the argument or angle.

Use Euler’s Formula:Euler’s formula states that:

$z = cos⁡θ+isin⁡θ$

is equal

$z = e^{i\theta}$

This is a fundamental result in complex analysis and connects exponential functions with trigonometric functions.

Substitute Euler’s Formula into the Polar Form:

To express the polar form in terms of Euler’s formula, substitute Euler’s formula into the polar form expression:

$z=r(cos⁡θ+isin⁡θ)$

can be written as:

$z = r e^{i\theta}$

This substitution uses the fact that

$e^{i\theta}$

can be expanded as

$cos{\theta} + i \sin{\theta}$

Expansion of Cosine and Sine

To see this more clearly, let’s expand

$e^{i\theta}$

using its Taylor series:

- Taylor Series for $e^{i\theta}$
  :

$e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!}$

Expanding

$(i\theta)^n$

$i^n\theta^n$

So:

$e^{i\theta} = \sum_{n=0}^{\infty} \frac{i^n\theta^n}{n!}$

Separate Real and Imaginary Parts:

To separate the real and imaginary parts, recall that:

$i^n = \begin{cases} 1 \quad \text{if } n \mod 4 = 0 \\ i \quad \text{if } n \mod 4 = 1 \\ -1 \quad \text{if } n \mod 4 = 2 \\ -i \quad \text{if } n \mod 4 = 3 \end{cases}$

Grouping terms:

Real Part:

$\text{Re}(e^{i\theta}) = \sum_{\text{n even}} \frac{(-1)^{n/2} \theta^n}{n!} = \cos{\theta}$

Imaginary Part:

$\text{Im}(e^{i\theta}) = \sum_{\text{n odd}} \frac{(-1)^{(n-1)/2} \theta^n}{n!} = \sin{\theta}$

Hence, combining these:

$e^{i\theta} = \cos{\theta} + i \sin{\theta}$

WorkSheets On Complex Number

Convert Cartesian form into Polar & Euler’s Form

Convert $z = 2 + 3i$ into polar form.
Convert $z=−4+5i$ into Euler’s form.
Find the polar form of $z=5+12i$ .
Convert $z=−6+8i$ to Euler’s form.
Express $z=1+i$ in polar form.
Find the Euler’s form of $z=−3+4i$ .
Convert $z=−2−3i$ into polar form.
Convert $z=7−5i$ to Euler’s form.
Express $z=8+6i$ in polar form.
Find the polar form of $z=9+7i$ .
Convert $z=−3+3i$ to Euler’s form.
Express $z=4−4i$ in polar form.
Find the polar form of $z=−5−2i$ .
Convert $z=6+2i$ into Euler’s form.
Express $z=−2+6i$ in polar form.

Convert Polar & Euler’s form into Cartesian Form

$z = 4(\cos 30^\circ + i \sin 30^\circ)$ , convert to Cartesian form.
Convert $z = 6e^{i\frac{\pi}{6}}$ into Cartesian form.
Convert $z = 5(\cos 60^\circ + i \sin 60^\circ)$ to Cartesian form.
Convert $z = 7e^{i\frac{\pi}{4}}$ to Cartesian form.
Convert $z = 3(\cos 45^\circ + i \sin 45^\circ)$ to Cartesian form.
Convert $z = 2e^{i\frac{\pi}{3}}$ to Cartesian form.
Convert $z = 8(\cos 90^\circ + i \sin 90^\circ)$ to Cartesian form.
Convert $z = 10e^{i\frac{\pi}{2}}$ to Cartesian form.
Convert $z = 12(\cos 135^\circ + i \sin 135^\circ)$ to Cartesian form.
Convert $z = 4e^{i\frac{\pi}{4}}$ to Cartesian form.
Convert $z = 9(\cos 120^\circ + i \sin 120^\circ)$ to Cartesian form.
Convert $z = 5e^{i\frac{\pi}{3}}$ to Cartesian form.
Convert $z = 6(\cos 150^\circ + i \sin 150^\circ)$ to Cartesian form.
Convert $z = 7e^{i\frac{5\pi}{6}}$ to Cartesian form.
Convert $z = 8(\cos 180^\circ + i \sin 180^\circ)$ to Cartesian form.

Applications

1. Engineering:

Complex numbers are used in electrical engineering to analyze AC circuits, where they represent impedance and phase relationships. They simplify the analysis of alternating current (AC) signals and circuits.

2. Control Systems:

In control theory, complex numbers are used to describe system dynamics and stability. The poles and zeros of transfer functions are complex numbers that help in designing and analyzing control systems.

3. Signal Processing:

Complex numbers are employed in signal processing to handle frequency domain analysis. Fourier transforms and filters often use complex numbers to analyze and manipulate signals.

4. Physics:

Complex numbers are essential in quantum mechanics, where they describe wave functions and probability amplitudes. The Schrödinger equation, a fundamental equation in quantum mechanics, uses complex numbers.

5. Mathematics:

Complex numbers are used in various branches of mathematics, including complex analysis, which studies functions of complex variables. They also appear in solutions to polynomial equations and many areas of applied mathematics.

6. Computer Graphics:

In computer graphics, complex numbers are used for transformations and rotations. They help in manipulating images and performing geometric transformations efficiently.

Complex numbers provide a powerful framework for solving problems that cannot be addressed with real numbers alone, making them an indispensable tool in mathematics and engineering.

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Introduction to Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable $x$ , with the general form:

$ax^2 + bx + c = 0$ where $a$ , $b$ , and $c$ are constants, and $a \neq 0$ . The term “quadratic” comes from “quad,” meaning square, since the highest exponent of the variable is $2$ .

Uses and Applications of Quadratic Equations

Quadratic equations are used in various fields, including:

Physics: To describe projectile motion, the trajectory of an object can be modeled using quadratic equations.

Engineering: In structural engineering, quadratic equations help calculate loads, tensions, and forces.

Economics: They are used in cost, profit, and revenue functions, particularly in finding maximum or minimum values.

Biology: Quadratic equations help model population growth, the spread of diseases, or other natural phenomena.
Geometry: These equations are used in determining areas, and for working with parabolas and ellipses.

Quadratic Expressions and Quadratic Equations

Method of Solving Quadratic Expressions $ax^2+bx+c$ , $a \neq 0$

1.Factorisation Method:

The goal is to express the quadratic expression in the form of two binomials:

$ax^2 + bx + c = (px + q)(rx + s)$

where $p, q, r,$ and $s$ are constants, and the equation is solved by finding appropriate factors of $a \times c$ .

2. Completing the Square Method:

To solve the quadratic expression, we can complete the square as follows:

$ax^2 + bx + c = a\left( x^2 + \frac{b}{a}x \right) + c$

The next step is to add and subtract a term to make a perfect square trinomial:

$x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2$

After simplifying, we can rewrite it as:

$\left( x + \frac{b}{2a} \right)^2 - \text{constant}$

3. Quadratic Formula:

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This method can be used for any quadratic equation to find its roots directly.

Factorisation Based on the Value of $a$ and the Sign of $c$

In solving quadratic equations, it is important to consider the value of $a$ and the sign of $c$ .

CASE 1:

If $a = 1$ , we check the sign of $c$ .

If $c$ is positive, both factors have the same sign as $b$ , and their sum equals $b$ .

Example:

$x^2 + 7x + 12 = 0$

First, find factors of $12$ :

$(1, 12), (2, 6), (3, 4)$

Since $3 + 4 = 7$ , we split $7x$ into $3x$ and $4x$ , so the equation becomes:

$x^2 + 3x + 4x + 12 = 0$

Now, factor:

$x(x + 3) + 4(x + 3) = 0$

Thus:

$(x + 3)(x + 4) = 0$

Setting each factor equal to zero:

$x + 3 = 0 \quad \text{or} \quad x + 4 = 0$

So, the solutions are:

$x = -3 \quad \text{and} \quad x = -4$

Similarly we can solve $x^2 - 7x +12 =0$

Case II:

if $c$ is negative, factors subtracted and have different signs, the large factor keep the sign of $b$ .

for example $x^2 - x -12$ , in this case we need to find the factors of $12$ , which when subtracted give us $-1$ , so again the difference of $4$ and $3$ is $1$ , but we put $-$ sign with $4$ nd split $-x$ as $-4x and + 3x$ . now the equation becomes

$x^2 -4x + 3x -12$

$(x -4)(x +3)$

$x-4 =0$ and $x+3=0$

therefore the solution of the above equation are $+4$ and $-3$ .

If $a \neq 0$

we first multiply $a$ and $c$ and the factorize the value of $ac$ , repeat the case I and case II accordingly.

Quadratic Equation Questions

1. Solve the quadratic equation: $x^2 - 4x - 5 = 0$ .
2. Solve the quadratic equation: $2x^2 + 3x - 2 = 0$ .
3. Find the roots of $x^2 - 9 = 0$ .
4. Factor and solve: $x^2 + 6x + 9 = 0$ .
5. Solve for $x$ : $3x^2 - 2x - 8 = 0$ .
6. What are the solutions to $x^2 + 4x + 4 = 0$ ?
7. Solve by completing the square: $x^2 - 6x + 5 = 0$ .
8. Use the quadratic formula to solve: $x^2 - 7x + 10 = 0$ .
9. Solve the equation: $4x^2 - 12x + 9 = 0$ .
10. Solve $x^2 + 8x + 16 = 0$ by factoring.
11. Solve $x^2 - 2x - 3 = 0$ using the quadratic formula.
12. Find the roots of $5x^2 - 20x + 15 = 0$ .
13. Solve $3x^2 + 2x - 8 = 0$ by completing the square.
14. Factor and solve $x^2 - 16 = 0$ .
15. Solve $6x^2 - x - 1 = 0$ using the quadratic formula.
16. Solve for $x$ : $x^2 + 5x + 6 = 0$ .
17. Find the solutions to $x^2 - 25 = 0$ .
18. Solve $7x^2 + 14x + 7 = 0$ by factoring.
19. What are the roots of $9x^2 - 24x + 16 = 0$ ?
20. Solve $2x^2 - 10x + 12 = 0$ using the quadratic formula.

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Methods Of Solving System of Two Linear Equations

solving a system of two linear equations. There are three main methods to solve such systems:

Substitution,
Elimination, and
Graphical Method.
Matrix Method

Substitution Method

Explanation:
In the substitution method, we solve one equation for one variable and substitute that expression into the other equation to find the value of the second variable.

Steps:

Solve one equation for one variable in terms of the other (e.g., solve for
$𝑥 or 𝑦$ .
Substitute the expression into the other equation.
Solve for the second variable.
Use this value to find the first variable.