7 pages · 6 hand-drawn diagrams · 8 boxed key results · Class 12 Maths Chapter 1, 2026-27 NCERT
- CBSE Weightage: 4 to 6 marks (one short answer plus one long answer on equivalence relation or bijection proof)
- JEE Main Weightage: 2 to 3% (1 to 2 questions per shift on relations, function-classification, and invertibility)
- JEE Main Weightage: Not in JEE Main syllabus (Maths-only chapter; relevant for JEE and CUET)
Chapter 1 Relations and Functions Handwritten Notes PDF
Student Pulse - Relations and Functions Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
- Relations and Functions Class 12 Maths Notes
- Relations and Functions Class 12 Maths NCERT Solutions
- Relations and Functions Class 12 Maths Formula Sheet
Relations and Functions Handwritten Notes: Complete Diagram List for Class 12 Maths
| Page | Diagram | What It Shows |
|---|---|---|
| 1 | Empty & universal relation boxes | The two trivial-relation extremes with the R = φ and R = A × A shorthand |
| 2 | Partition into equivalence classes | Set X split into disjoint Ai blocks with the [0], [1], [2] mod-3 example |
| 3 | One-one vs many-one arrow diagrams | Two circle pairs showing distinct images (one-one) and merging arrows (many-one) |
| 4 | Modulus function not-onto sketch | The V-shape y = |x| with the “Range = [0, ∞)” annotation showing why it fails onto |
| 5 | Composition g ˆ f flow diagram | Three-circle map A → B → C with x → f(x) → g(f(x)) arrows |
| 7 | Summary learn table | Six-row condition table for reflexive, symmetric, transitive, equivalence, empty, universal |
Quick Tip: Copy the summary learn table (page 7) and the composition flow diagram (page 5) on the back of your answer sheet during revision; CBSE has asked at least one labelled diagram or summary-style table from the Class 12 Maths Handwritten Notes PDF in every Board paper since 2022.
Relations and Functions Video Walkthrough
Source: Magnet Brains on YouTube
Relations and Functions Class 12 Chapter 1 NCERT Exercise-Wise Snapshot
| Sub-Topic | NCERT Qs | Typical CBSE Marks |
|---|---|---|
| Types of Relations (empty, universal, identity) | 4 | 1 to 2 |
| Reflexive, Symmetric, Transitive Properties | 8 | 3 to 5 |
| Equivalence Relations and Equivalence Classes | 6 | 3 to 5 |
| One-one (Injective) and Onto (Surjective) Functions | 8 | 2 to 3 |
| Composition of Functions | 3 | 1 to 2 |
| Invertible Functions and Inverse Computation | 3 | 3 to 5 |
How the Class 12 Maths Handwritten Notes PDF on the Class 12 Maths Handwritten Notes PDF Help You
- Hand-Sketched Visual Anchors: Every key arrow diagram, one-one vs many-one mappings, composition flow, partition into equivalence classes, is drawn freehand so the reader can redraw it from memory in under 60 seconds.
- Boxed Key Results: Every definition that CBSE asks verbatim (injectivity test, onto condition, equivalence triple, invertibility iff bijective) lives inside a hand-drawn rectangle, so your eye locks onto the testable statement.
- 2026-27 NCERT Alignment: Every page is matched to the current 2026-27 syllabus print. Note: binary operations are NO LONGER in the rationalised Class 12 NCERT, so the notebook drops them entirely and doubles down on equivalence-class proofs and invertibility, the two highest-yield topics for Boards.
- Solved Function Examples: Four pre-classified examples (linear bijection f(x)=2x on R, squaring f(x)=x2, integer 2x on N, modulus |x|) so you have a template for every CBSE classification question.
Relations and Functions Handwritten Notes Concept-Flavoured Mnemonics
Remember — RST = Equivalence: Reflexive + Symmetric + Transitive = Equivalence relation. Miss any ONE and it's NOT equivalence. CBSE has burned this exact distinction in 4 of the last 5 Board papers.
Remember — Injectivity test: Always start with f(x1) = f(x2) and try to derive x1 = x2 . If you succeed, one-one. If you find a counter-example like f(1) = f(−1) for f(x) = x2, NOT one-one.
Remember — Onto test: Pick an arbitrary y in the codomain and SOLVE for x. If x exists in the domain for every y, onto. If the equation has no solution in the domain (like y < 0 for |x|), NOT onto.
Remember — Invertible iff Bijective: You do NOT need to find the inverse formula to prove invertibility. Showing one-one AND onto is enough. This is the single biggest time-saver on a 5-mark Board question.
Remember — Finite set shortcut: If X is a finite set and f: X → X , then f is one-one IFF f is onto. So proving either property automatically gives the other, and the function is bijective on the spot.
Relations and Functions Handwritten Notes: Color Legend for the Notebook
| Colour | Box Type | What's Inside |
|---|---|---|
| Orange | Definition box | NCERT one-line definitions: relation, reflexive, symmetric, transitive, equivalence, injective, surjective, bijective, invertible |
| Blue | Key result box | Every theorem-style result. f(x1) = f(x2) ⇒ x1 = x2 , g ∘ f = IX , f invertible ⇔ f bijective |
| Yellow | MCQ-trap callout | The 6 facts CBSE / JEE re-use as MCQ distractors: f(x) = x2 on R is neither one-one nor onto, mod 2 is equivalence on Z, etc. |
| Red | Common-mistake box | Most students confuse "domain" with "codomain" or skip proving symmetric in equivalence proofs; the red box shows the correct sequence right next to the wrong one |
Relations and Functions Top 6 Key Results for Class 12 Maths Quick Recall
| Concept | Definition / Test |
|---|---|
| Reflexive Relation | (a, a) ∈ R for every a ∈ A |
| Symmetric Relation | (a1, a2) ∈ R ⇒ (a2, a1) ∈ R |
| Transitive Relation | (a1, a2), (a2, a3) ∈ R ⇒ (a1, a3) ∈ R |
| One-one (Injective) | f(x1) = f(x2) ⇒ x1 = x2 |
| Onto (Surjective) | For every y ∈ Y , there exists x ∈ X such that f(x) = y |
| Invertibility Criterion | f is invertible ⇔ f is bijective (one-one and onto) |
Relations and Functions: Last 24-Hour Revision Card for Class 12 Maths
- Relation definition: R from A to B is a subset of A × B. If (a, b) is in R, write a R b. CBSE asks the definition every 2 to 3 years.
- Three property tests: Reflexive (a, a) in R; Symmetric (a, b) implies (b, a); Transitive (a, b) and (b, c) implies (a, c). Apply each one in this order.
- Equivalence relation: A relation is equivalence iff it's reflexive, symmetric AND transitive. Standard CBSE example: R = (a, b) : 2 (a - b) on the integers.
- Function classification: One-one (distinct images), Onto (range = codomain), Bijective (both), Invertible (iff bijective). Memorise this four-step decision ladder.
- Composition: (g ∘ f)(x) = g(f(x)) . NOT commutative in general. Associative: h ∘ (g ∘ f) = (h ∘ g) ∘ f .
- Inverse function recipe: Step 1, write y = f(x). Step 2, solve for x in terms of y. Step 3, swap to write f-1(y) . Step 4, verify g ∘ f = IX and f ∘ g = IY .
- One solved equivalence proof: Be ready to write out the three-step proof (R, S, T) for a "divides" or "congruent mod n" relation in under 5 minutes.
Watch Out: Students routinely skip the symmetric check when proving an equivalence relation, especially when the relation looks “obviously” symmetric. CBSE deducts 1 mark every time. Always explicitly write the line: “Suppose (a, b) is in R. Then... so (b, a) is in R.”
Relations and Functions Class 12 Maths Self-Assessment Quick Quiz
Q1. Let R = (1, 1), (2, 2), (3, 3), (1, 2) on A = 1, 2, 3 . Then R is:
Q2. The function f: R → R defined by f(x) = x2 is:
Q3. If f: N → N is given by f(x) = 2x , then f is:
Q4. The number of one-one functions from 1, 2, 3 to itself is:
Q5. If f(x) = cos x and g(x) = 3x , then g ∘ f(x) equals:
Class 12th Maths Chapter 1 Relations and Functions PYQ Snapshot (2026 to 2021)
| Topic | Asked in | Typical Marks |
|---|---|---|
| Prove that a given relation (mod n, divides, congruent) is an equivalence relation | CBSE 2025, 2024, 2022, JEE Main 2025, 2023 | 3 to 5 |
| Show a function is bijective and find its inverse (e.g. f(x) = 4x + 3 on a restricted set) | CBSE 2025, 2023, JEE Main 2024, 2022 | 3 to 5 |
| Classify a function as one-one / onto / bijective on a given domain | CBSE 2024, 2022, JEE Main 2025, 2023 | 2 to 3 |
Best Way to Use These Relations and Functions Handwritten Notes
- Pass 1 (30 minutes): Flip through the 6 hand-drawn diagrams and the colour-coded legend without reading prose. The point is to imprint the visual layout so you can later "see" where every definition and test sits.
- Pass 2 (45 minutes): Read the blue key-result boxes plus every yellow MCQ-trap. Cover the right side of each page and recall the test from the concept name (e.g. read "Symmetric" and force yourself to say the (a, b) implies (b, a) condition out loud).
- Pass 3 (last 24 hours): Skim only the summary learn table on page 7 and re-sketch the composition flow and the equivalence-class partition from memory.
- Active recall: After Pass 2, solve the five MCQs above without referring back. If you score below 4, return to Pass 2; if 4 to 5, you are exam-ready on this chapter.
- Relations and Functions Class 12 Maths NCERT Book PDF
- Relations and Functions Class 12 Maths NCERT Exemplar Solutions
Relations and Functions Weightage Compared Across Class 12 Maths Chapters
Ch 1 Relations and Functions
5 marks
Ch 2 Inverse Trigonometric Functions
4 marks
Ch 3 Matrices
6 marks
Ch 4 Determinants
6 marks
Ch 5 Continuity and Differentiability
8 marks
Ch 6 Application of Derivatives
6 marks
Ch 7 Integrals
9 marks
Ch 8 Application of Integrals
4 marks
Ch 9 Differential Equations
5 marks
Ch 10 Vector Algebra
5 marks
Ch 11 Three Dimensional Geometry
5 marks
Ch 12 Linear Programming
5 marks
Ch 13 Probability
6 marks
More Relations and Functions Maths Class 12 Resources
- Relations and Functions Class 12 Maths NCERT Solutions
- Relations and Functions Class 12 Maths Notes
- Relations and Functions Class 12 Maths Formula Sheet
- Relations and Functions Class 12 Maths NCERT Book PDF
- Relations and Functions Class 12 Maths NCERT Exemplar Book PDF
- Relations and Functions Class 12 Maths NCERT Exemplar Solutions
NCERT Handwritten Notes for Class 12 Maths: All Chapters
| Chapter | Resource |
|---|---|
| Chapter 2 | Inverse Trigonometric Functions Handwritten Notes |
| Chapter 3 | Matrices Handwritten Notes |
| Chapter 4 | Determinants Handwritten Notes |
| Chapter 5 | Continuity and Differentiability Handwritten Notes |
| Chapter 6 | Application of Derivatives Handwritten Notes |
| Chapter 7 | Integrals Handwritten Notes |
| Chapter 8 | Application of Integrals Handwritten Notes |
| Chapter 9 | Differential Equations Handwritten Notes |
| Chapter 10 | Vector Algebra Handwritten Notes |
| Chapter 11 | Three Dimensional Geometry Handwritten Notes |
| Chapter 12 | Linear Programming Handwritten Notes |
| Chapter 13 | Probability Handwritten Notes |
Exercise-wise Breakdown of the Relations and Functions Chapter
| Exercise | Topic Tested |
|---|---|
| Exercise 1.1 | Empty, universal, reflexive, symmetric, transitive relations |
| Exercise 1.2 | Injective, surjective, bijective functions; composition |
| Miscellaneous Exercise | Mixed concepts; bijection-invertibility and counting |
PDF Download Formats and Languages for the Relations and Functions Chapter
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
- NCERT-faithful: Every definition, theorem and exercise on the relations and functions class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The relations and functions class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The relations and functions class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The relations and functions class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this relations and functions class 12 pdf - only the exercise numbers differ.
Important Questions and Previous Year Trends for the Relations and Functions Chapter
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
- relations and functions class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The relations and functions class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching relations and functions class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
How the Relations and Functions Notes Pair with NCERT Solutions and the Formula Sheet
| Resource | Use it for | When |
|---|---|---|
| Relations and Functions Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| relations and functions class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| relations and functions class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
- The relations and functions class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The relations and functions class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The relations and functions class 12 formulas reference sheet is the same A4 file students sometimes refer to as relations and functions class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board relations and functions class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 relations and functions solutions, class 12 relations and functions ncert solutions, ncert class 12 relations and functions solutions - the same files cover the request.
Reference Books and State-Board Mapping
| Reference | How it maps to Relations and Functions Class 12 |
|---|---|
| RD Sharma Class 12 Relations and Functions | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Relations and Functions | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo relations and functions class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa relations and functions class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board relations and functions class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Relations and Functions | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Relations and Functions Notes Page Most Effectively
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the relations and functions class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The relations and functions class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
This Collegedunia NCERT Class 12 Mathematics page is reviewed against every CBSE board paper release.
The Collegedunia Class 12 Maths team checks every formula on this page against the printed NCERT 2026-27 edition.
Class 12 Maths Handwritten Notes PDF - Frequently Asked Questions
Ques. Where can I download the Class 12 Maths Handwritten Notes PDF for free?
Comments