Algebra

Introduction

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields. In the context of JEE Main Mathematics, algebra forms a significant portion of the syllabus and is crucial for scoring well. This study note aims to break down the complex ideas of algebra into digestible sections, covering all the necessary concepts and nuances with examples.

Basic Concepts

Variables and Constants

• Variables: Symbols that represent numbers whose values can change. Typically denoted by letters such as $x$, $y$, and $z$.
• Constants: Fixed values that do not change. Examples include numbers like $5$, $-3$, or $\pi$.

Example

If $x + 5 = 9$, then $x$ is the variable, and $5$ and $9$ are constants.

Algebraic Expressions

An algebraic expression is a combination of variables, constants, and operators. For example, $3x + 2y - 5$ is an algebraic expression.

Types of Algebraic Expressions

1. Monomial: An expression with a single term, e.g., $5x$.
2. Binomial: An expression with two terms, e.g., $3x + 4$.
3. Polynomial: An expression with multiple terms, e.g., $2x^2 + 3x + 1$.

Simplification of Expressions

Simplification involves combining like terms and using the distributive property.

Example

Simplify $2x + 3x - 4 + 5$: $$ 2x + 3x - 4 + 5 = 5x + 1 $$

Tip

Always combine like terms to simplify expressions effectively.

Equations and Inequalities

Linear Equations

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. The general form is $ax + b = 0$.

Example

Solve $3x + 2 = 11$: $$ 3x + 2 = 11 \ 3x = 9 \ x = 3 $$

Quadratic Equations

Quadratic equations are second-degree equations of the form $ax^2 + bx + c = 0$.

Methods to Solve Quadratic Equations

1. Factoring: Express the quadratic equation as a product of two binomials.
2. Quadratic Formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
3. Completing the Square: Transform the equation to the form $(x + p)^2 = q$ and then solve for $x$.

Example

Solve $x^2 - 5x + 6 = 0$ by factoring: $$ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \ x = 2 \text{ or } x = 3 $$

Note

Always check for the discriminant $\Delta = b^2 - 4ac$ to determine the nature of the roots.

Inequalities

Inequalities are like equations but with inequality signs ($

<$, $>

$, $\leq$, $\geq$).

Solving Linear Inequalities

1. Isolate the variable on one side.
2. If you multiply or divide by a negative number, reverse the inequality sign.

Example

Solve $2x - 3 > 7$: $$ 2x - 3 > 7 \ 2x > 10 \ x > 5 $$

Common Mistake

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.

Functions and Graphs

Definition of a Function

A function is a relation between a set of inputs and a set of permissible outputs. Each input is related to exactly one output.

Types of Functions

1. Linear Function: $f(x) = mx + c$
2. Quadratic Function: $f(x) = ax^2 + bx + c$
3. Polynomial Function: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

Graphing Functions

Graphing involves plotting points on a coordinate plane to visualize the function.

Example

Graph $y = 2x + 3$:

• Plot points for different values of $x$.

• Draw a straight line through the points.

Tip

Use a graphing calculator or software for complex functions.

Progressions

Arithmetic Progression (AP)

A sequence where each term after the first is obtained by adding a constant difference, $d$, to the previous term.

• General Term: $a_n = a + (n-1)d$
• Sum of First $n$ Terms: $S_n = \frac{n}{2} [2a + (n-1)d]$

Example

Find the 5th term of the AP: $2, 5, 8, \ldots$: $$ a = 2, , d = 3 \ a_5 = 2 + (5-1) \cdot 3 = 2 + 12 = 14 $$

Geometric Progression (GP)

A sequence where each term after the first is obtained by multiplying the previous term by a constant ratio, $r$.

• General Term: $a_n = ar^{n-1}$
• Sum of First $n$ Terms: $S_n = a \frac{r^n - 1}{r - 1}$ for $r \neq 1$

Example

Find the 4th term of the GP: $3, 6, 12, \ldots$: $$ a = 3, , r = 2 \ a_4 = 3 \cdot 2^{4-1} = 3 \cdot 8 = 24 $$

Note

For $r = 1$, the sum of the first $n$ terms is simply $na$.

Complex Numbers

Definition

A complex number is of the form $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with $i^2 = -1$.

Operations

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Modulus and Conjugate

• Modulus: $|z| = \sqrt{a^2 + b^2}$
• Conjugate: $\overline{z} = a - bi$

Example

Find the modulus and conjugate of $3 + 4i$: $$ |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \ \overline{3 + 4i} = 3 - 4i $$

Tip

Use the conjugate to divide complex numbers.

Conclusion

Algebra is a broad and essential topic in JEE Main Mathematics. Understanding the basic concepts, equations, inequalities, functions, progressions, and complex numbers will provide a strong foundation for tackling more complex problems. Practice regularly and use these notes as a reference to strengthen your algebra skills.