Mathematics (Class 12) — Free Notes (CBSE & HBSE)
Free Class 12 Maths notes with NCERT solutions, sample papers 2026, previous year question papers (PYQs), chapter-wise revision and MCQ practice — Calculus, Vectors, 3D Geometry, Probability, LPP. CBSE/HBSE + JEE Main aligned.
Free, chapter-wise Mathematics notes for Class 12 on Siksha Sarovar, aligned to NCERT and both CBSE and Haryana Board (HBSE). Covers 13 chapters with explanations, worked examples and board-pattern practice questions.
Chapters covered (13)
- Relations and Functions — Relations and Functions forms the theoretical backbone of Class 12 Maths. CBSE typically allocates 8 marks and HBSE 10 marks. Expect questions on identifying relation types…
- Inverse Trigonometric Functions — Inverse Trigonometric Functions carries 5-8 marks in CBSE. Students must know the principal value branch domains/ranges for all six inverse trig functions and be able to simplify…
- Matrices — Matrices carries 8-10 marks in CBSE and HBSE. Questions span operations (addition, multiplication), transpose, symmetric/skew-symmetric decomposition, elementary operations, and…
- Determinants — Determinants is one of the most heavily tested chapters, carrying 10-12 marks in CBSE. Topics include properties of determinants, cofactor expansion, adjoint, inverse, and solving…
- Continuity and Differentiability — Continuity and Differentiability is one of the largest chapters carrying 10-12 marks in CBSE. Topics include continuity checks, differentiation of composite/implicit/parametric…
- Application of Derivatives — Application of Derivatives carries 8-10 marks. CBSE tests rate of change, increasing/decreasing functions, tangents and normals, and maxima/minima. The 5-mark question usually…
- Integrals — Integrals is the largest chapter in Class 12, carrying 12-15 marks in CBSE. Topics include indefinite integrals (standard forms, substitution, by parts, partial fractions) and…
- Application of Integrals — Application of Integrals carries 5-6 marks in CBSE. One 5-mark question always asks for area under a curve (typically parabola, ellipse, or area between two curves). Drawing a…
- Differential Equations — Differential Equations carries 8-10 marks in CBSE. Questions cover order/degree, formation by eliminating constants, variable separable method, homogeneous equations, and linear…
- Vector Algebra — Vector Algebra carries 8 marks in CBSE. Topics include types of vectors, vector operations, dot and cross products. Questions typically involve finding unit vectors, dot/cross…
- Three Dimensional Geometry — Three Dimensional Geometry carries 10-12 marks in CBSE. Questions cover direction cosines/ratios, equations of lines and planes (vector and Cartesian), angle between lines/planes,…
- Linear Programming — Linear Programming carries 5-6 marks in CBSE as one 5-mark question (or LA). Students must formulate the LPP, solve graphically, find the feasible region, and optimise the…
- Probability — Probability is the final chapter, carrying 8-10 marks in CBSE. Conditional probability, Bayes theorem, and binomial distribution are most frequently tested. Every year CBSE…
Relations and Functions: Types of Relations
Types of Relations
A relation R on a set A is a subset of A × A. We write aRb or (a,b) ∈ R.
| Relation Type | Definition | Example on A={1,2,3} |
|---|---|---|
| Empty | R = ∅ | No pair related |
| Universal | R = A×A | Every pair related |
| Reflexive | (a,a) ∈ R ∀ a ∈ A | {(1,1),(2,2),(3,3),…} |
| Symmetric | (a,b) ∈ R ⇒ (b,a) ∈ R | {(1,2),(2,1)} |
| Transitive | (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R | {(1,2),(2,3),(1,3)} |
| Equivalence | Reflexive + Symmetric + Transitive | See below |
Checking Relations Step-by-Step
Step 1 — Reflexive: Check if (a,a) is in R for EVERY element a of A. Step 2 — Symmetric: For every (a,b) in R, verify (b,a) is also in R. Step 3 — Transitive: For every (a,b) and (b,c) in R, verify (a,c) is in R.
Equivalence Relations
R is an equivalence relation iff it is reflexive, symmetric AND transitive simultaneously.
Equivalence Class of a ∈ A: [a] = {x ∈ A : xRa}
- Equivalence classes partition the set A into disjoint subsets.
- Every element belongs to exactly one equivalence class.
Classic Example: R on ℤ defined by aRb iff (a – b) is divisible by n is an equivalence relation. The equivalence classes are {0,n,2n,…}, {1,n+1,2n+1,…}, …, {n-1,2n-1,…}.
Important Tips
- Empty relation is symmetric and transitive but NOT reflexive (if A ≠ ∅).
- Universal relation is always an equivalence relation.
- Identity relation I = {(a,a) : a ∈ A} is always an equivalence relation.
CBSE Tip: CBSE frequently asks: "Is the relation R = {(a,b): a ≤ b} on ℝ an equivalence relation?" — Answer: Reflexive ✓, NOT symmetric (2 ≤ 3 but 3 ≰ 2), transitive ✓. So NOT equivalence.
Frequently asked questions
Are these Mathematics notes free?
Yes — all Mathematics notes on Siksha Sarovar are free to read, no account required.
Do the notes follow CBSE and HBSE?
Yes. Notes are NCERT-aligned and include both CBSE and Haryana Board (HBSE) exam guidance, important questions and MCQs.
Can I prepare for board exams here?
Yes — each chapter includes key concepts, formulas, important questions and practice MCQs for board exam revision.