Summary of "Manzil 2025: BASIC MATHS in One Shot: All Concepts & PYQs Covered | JEE Main & Advanced"

This extensive lecture covers fundamental and advanced concepts of inequalities, modulus (absolute value) functions, irrational equations, and related algebraic methods essential for JEE Main and Advanced preparation. The instructor uses a step-by-step approach, emphasizing conceptual clarity, problem-solving techniques, and application of previous years’ questions (PYQs). The session also includes discussions on Quadratic Functions, factorization, and the Wavy Curve Method for solving inequalities.

Main Ideas, Concepts, and Lessons

1. Inequalities and Their Representations

Definition: Expressions involving signs like >, <, ≥, ≤.
Solution representation:
- Interval notation (using brackets and parentheses).
- Number line representation using dark circles (included points) and hollow circles (excluded points).
- Use of union (∪) and intersection (∩) for combined solution sets.
Rules for solving inequalities:
- Adding/subtracting the same number on both sides doesn’t change inequality.
- Multiplying/dividing both sides by a positive number keeps inequality sign same.
- Multiplying/dividing both sides by a negative number reverses inequality sign.
Cross multiplication:
- Allowed only when the denominator’s sign is known (positive).
- If denominator sign is unknown or variable, avoid cross multiplication; instead, bring all terms to one side and solve using Wavy Curve Method.

2. Wavy Curve Method (WCM)

Used to solve inequalities involving rational expressions.
Steps:
- Convert inequality so one side equals zero.
- Find roots of numerator and denominator (critical points).
- Plot these roots on a number line.
- Test signs of the expression in intervals defined by these roots.
- Determine solution set based on inequality sign.
Important points:
- Roots of numerator get dark circles if equality is included.
- Roots of denominator always get hollow circles (excluded).
- Alternate signs between roots depend on the net power (odd/even) of factors.
- Net power odd → sign changes at root; even → sign does not change.

3. Modulus (Absolute Value) Functions and Equations

Definition: |x| = x if x ≥ 0, and -x if x < 0.
Removing modulus:
- Split into cases based on sign of the expression inside modulus.
- For unknown sign, consider two cases: expression ≥ 0 and expression < 0.
Multiple modulus expressions:
- Find roots where each modulus expression equals zero.
- Plot roots on number line, divide into intervals.
- Solve for each interval considering sign changes.
Modulus equations:
- If |A| = |B|, then A = B or A = -B.
- Solve both equations and check solutions against initial conditions.
Modulus inequalities:
- |A| ≥ α → A ≥ α or A ≤ -α.
- |A| ≤ α → -α ≤ A ≤ α.
Initial conditions:
- Always check the domain or sign conditions before solving.
- Intersection of initial condition and solution set gives final answer.

4. Quadratic Functions and Factorization

Factorized quadratics can be solved by splitting the middle term.
Non-factorizable quadratics are solved using the discriminant (D = b² - 4ac).
If D < 0 → no real roots; quadratic is always positive or always negative depending on leading coefficient.
Quadratic expressions with positive leading coefficient and negative discriminant are always positive → useful in inequalities and modulus removal.

5. Irrational Equations (Equations with Square Roots)

Key points:
- Remove square roots by squaring both sides.
- Squaring can introduce extraneous solutions; always recheck solutions in original equation.
- Both sides must be non-negative to square inequalities safely.
Defining domain (GD/BD):
- Identify where the expression under the root is ≥ 0.
- Take intersection of all such domain conditions.
Solution method:
- Define domain conditions.
- Square both sides (if allowed).
- Solve resulting polynomial/rational equations.
- Recheck solutions in original equation.
Option rejection method:
- Used to quickly eliminate extraneous solutions by testing values.

6. Additional Concepts and Tips

Interval notation and number line representation are crucial for clarity.
Sign analysis and testing values in intervals help confirm solutions.
Handling common factors in numerator and denominator is important for simplification and correct sign determination.