📐 Class 12 Mathematics (JEE) - Chapter-wise Study Material

1️⃣ Relations and Functions

Types of Relations: Reflexive, Symmetric, Transitive
Equivalence Relations
Types of Functions: One-One, Onto, Inverse

        Example Q: Prove that the relation R = {(a, b) | a - b is divisible by 3} is an equivalence relation.
      

2️⃣ Integrals

Indefinite Integrals – Basic formulas
Integration by Parts and Substitution
Definite Integrals – Properties and Applications
Area under Curves

        Example Q: Evaluate ∫20 (x² + 1) dx

        Solution: = [x³/3 + x]20 = (8/3 + 2) = 14/3

3️⃣ Differential Equations

Order and Degree of Differential Equations
General and Particular Solutions
Methods of Solving – Variable Separation, Linear

        Example Q: Solve dy/dx = 3x²

        Solution: Integrate both sides: y = x³ + C

4️⃣ Vector Algebra

Types of Vectors and Operations
Dot Product and Cross Product
Applications in Geometry

        Example Q: Find the angle between vectors A = i + 2j and B = 3i + 4j.

        Solution: Use dot product formula.

5️⃣ 3D Geometry

Direction Cosines and Ratios
Equation of a Line and Plane
Angle Between Lines and Planes
Shortest Distance between Two Skew Lines

Example Q: Find the angle between line x/1 = y/2 = z/3 and x/2 = y/3 = z/4.

6️⃣ Probability

Classical and Conditional Probability
Bayes’ Theorem
Independent and Dependent Events
Binomial Distribution Basics

        Example Q: A die is rolled. What is the probability of getting a prime number?

        Solution: Primes: 2, 3, 5 → P = 3/6 = 0.5