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Important Questions: CBSE Class 12 Mathematics Chapter 1 Relations and Functions

Relations and Functions is the foundational chapter of CBSE Class 12 Mathematics, setting the stage for calculus, algebra, and applied mathematics topics. CBSE typically allocates 8-10 marks to this chapter across formats: 2-3 one-mark MCQs, one or two 2-mark questions, and a 3-mark or 5-mark long-answer problem. Mastering types of relations (reflexive, symmetric, transitive, equivalence), function composition, and inverse functions is non-negotiable for a 90+ score.

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Key takeaways

  • ✓Chapter 1 Relations and Functions carries 8-10 marks in CBSE Class 12 board exams, split between MCQs and descriptive questions.
  • ✓Equivalence relation questions (prove reflexive, symmetric, transitive) are near-guaranteed 3-mark or 5-mark problems every year.
  • ✓Composition of functions and finding inverse functions appear as 2-mark or 3-mark questions; master the domain-range reversal technique.
  • ✓One-one (injective) and onto (surjective) proofs are high-scoring 3-mark questions if you write proper algebraic steps.
  • ✓Binary operations and their properties (commutativity, associativity, identity, inverse) often appear as 2-mark or case-based questions.
  • ✓Common mistakes include confusing 'reflexive' with 'symmetric' and forgetting to verify all three properties for equivalence relations.
  • ✓Practice with NCERT Exemplar and past five years' board papers; CBSE repeats question structures with minor numerical tweaks.

Chapter Overview and Marks Weightage in CBSE Board Exam

CBSE Class 12 Mathematics Chapter 1 Relations and Functions appears in Unit I (Relations and Functions) of the syllabus. The 2025 board exam blueprints show this unit contributes 10 marks total, shared between this chapter and Chapter 2 (Inverse Trigonometric Functions). Expect approximately 8 marks directly from Chapter 1. The distribution is usually one 1-mark MCQ, one 2-mark VSA, one 3-mark short answer, and occasionally a 5-mark long-answer question or case-based integrated problem. High-frequency topics include proving a relation is an equivalence relation (always 3 or 5 marks), checking injectivity and surjectivity of functions (3 marks), finding the inverse of a bijective function (2 or 3 marks), and composition of functions (2 marks). Binary operations on sets appear less often but can be part of a case-based question. The chapter is theory-light but proof-heavy; every mark requires clear logical steps, so partial credit depends on systematic working.
  • Unit I total weightage: 10 marks (Relations & Functions + Inverse Trigonometric Functions)
  • Chapter 1 alone: ~8 marks across MCQ (1 m), VSA (2 m), SA (3 m), LA (5 m or case-based)
  • Must-prepare topics: equivalence relations, one-one and onto functions, inverse functions, composition
  • 2024 board paper pattern: 1 MCQ on function type, 1 three-mark equivalence proof, 1 two-mark composition question

1-Mark Questions: Multiple Choice and Very Short Answer

One-mark questions test definitional clarity and quick pattern recognition. CBSE draws these directly from NCERT Exercise 1.1 and 1.2. You must answer in under 30 seconds during the exam. Typical formats include identifying relation types from a given set of ordered pairs, determining whether a function is one-one or onto from a simple formula, or picking the correct composition result. No working is required, but mark your NCERT examples well because CBSE often reuses the same sets with minor changes.
  • Q1. Let A = {1,2,3} and R = {(1,1),(2,2),(3,3),(1,2)}. R is: (a) reflexive only (b) symmetric only (c) transitive only (d) equivalence. — Answer: (a) reflexive only. (R is reflexive because (1,1),(2,2),(3,3) are present. Not symmetric because (1,2) exists but (2,1) does not. Hence not equivalence.)
  • Q2. The function f: R → R defined by f(x) = 3x + 5 is: (a) one-one but not onto (b) onto but not one-one (c) bijective (d) neither. — Answer: (c) bijective. (Linear functions with non-zero slope are always bijective on R.)
  • Q3. If f(x) = x² and g(x) = √x, then (fog)(4) equals: (a) 2 (b) 4 (c) 16 (d) undefined. — Answer: (b) 4. ((fog)(4) = f(g(4)) = f(2) = 4.)
  • Q4. The number of equivalence relations on the set {1,2} is: (a) 1 (b) 2 (c) 3 (d) 4. — Answer: (b) 2. (The two are: {(1,1),(2,2)} and {(1,1),(2,2),(1,2),(2,1)}.)

2-Mark Questions: Conceptual and Computational

Two-mark questions require you to show brief working. CBSE awards 1 mark for method and 1 for the correct final answer. Popular themes are checking one-one or onto for a simple function, computing (fog)(x) or (gof)(x), or identifying whether a given binary operation is commutative or associative. Write definitions first, then substitute. Do not skip steps or you lose the method mark even if your answer is correct. These questions come from NCERT Exercise 1.2 and 1.3.
  • Q5. Show that f: R → R given by f(x) = 2x − 3 is one-one. — Solution: Assume f(x₁) = f(x₂). Then 2x₁−3 = 2x₂−3 ⇒ 2x₁=2x₂ ⇒ x₁=x₂. Hence one-one. (Award 1 mark for assumption, 1 mark for conclusion.)
  • Q6. If f(x) = x + 1 and g(x) = 2x, find (fog)(x) and (gof)(x). — Solution: (fog)(x) = f(g(x)) = f(2x) = 2x+1. (gof)(x) = g(f(x)) = g(x+1) = 2(x+1) = 2x+2. (1 mark each composition.)
  • Q7. Let * be a binary operation on N defined by a*b = a + b. Is * commutative? — Solution: a*b = a+b, b*a = b+a. Since addition is commutative, a*b = b*a. Yes, commutative. (1 mark reasoning, 1 mark conclusion.)
  • Q8. Find the range of f: R → R, f(x) = x² + 1. — Solution: Since x² ≥ 0 for all x ∈ R, f(x) ≥ 1. Range = [1, ∞). (1 mark for inequality, 1 mark for interval notation.)

3-Mark Questions: Proofs and Short Answer Type

Three-mark questions are the backbone of this chapter. The most common is 'Prove that a given relation is an equivalence relation' — you must establish reflexivity, symmetry, and transitivity separately, each worth roughly 1 mark. Another frequent type is proving a function is bijective (1 mark one-one, 1 mark onto, 1 mark conclusion) or finding the inverse of a given bijective function. Examiners look for definitions stated explicitly, clear logical flow, and proper set notation. Missing any one property costs you marks. Practice NCERT Exemplar Problems 1.9 to 1.14 thoroughly.
  • Q9. Show that the relation R on the set of real numbers defined by R = {(a,b): a−b is an integer} is an equivalence relation. — Solution: (i) Reflexive: a−a = 0, an integer, so (a,a) ∈ R for all a. (ii) Symmetric: If (a,b) ∈ R, then a−b is an integer. Hence b−a = −(a−b) is also an integer, so (b,a) ∈ R. (iii) Transitive: If (a,b) ∈ R and (b,c) ∈ R, then a−b and b−c are integers. Adding, (a−b)+(b−c) = a−c is an integer, so (a,c) ∈ R. Hence R is reflexive, symmetric, and transitive — an equivalence relation. (1+1+1 marks)
  • Q10. Prove that f: N → N given by f(x) = x³ is one-one but not onto. — Solution: One-one: Assume f(x₁)=f(x₂) ⇒ x₁³=x₂³. Since cube function is strictly increasing on N, x₁=x₂. Hence one-one. (1 mark) Onto: 2 ∈ N (codomain), but there is no n ∈ N such that n³=2. Hence f is not onto. (1 mark) Conclusion: one-one but not onto. (1 mark)
  • Q11. If f: R → R is defined by f(x) = (3x+2)/5, find f⁻¹(x). — Solution: Let y = (3x+2)/5. Solve for x: 5y = 3x+2 ⇒ 3x=5y−2 ⇒ x=(5y−2)/3. Replace y with x: f⁻¹(x) = (5x−2)/3. (1 mark setup, 1 mark solving, 1 mark final answer.)
  • Q12. Check if the function f: R−{3} → R defined by f(x) = (2x+1)/(x−3) is one-one and onto. — Solution: One-one: f(x₁)=f(x₂) ⇒ (2x₁+1)/(x₁−3) = (2x₂+1)/(x₂−3). Cross-multiply and simplify: (2x₁+1)(x₂−3) = (2x₂+1)(x₁−3) ⇒ 2x₁x₂−6x₁+x₂−3 = 2x₁x₂−6x₂+x₁−3 ⇒ −6x₁+x₂ = −6x₂+x₁ ⇒ 7x₂=7x₁ ⇒ x₁=x₂. One-one. (1.5 marks) Onto: Let y ∈ R. Solve y=(2x+1)/(x−3): y(x−3)=2x+1 ⇒ yx−3y=2x+1 ⇒ x(y−2)=3y+1 ⇒ x=(3y+1)/(y−2). For y≠2, x exists. But y=2 gives denominator 0, so range = R−{2}. Not onto R. (1.5 marks)

5-Mark Questions and Case-Based Problems

Five-mark questions integrate multiple concepts: you might be asked to prove a relation is an equivalence relation (3 marks) and then find the equivalence class of a given element (2 marks), or prove a function is bijective (3 marks) and find its inverse (2 marks). CBSE introduced case-based integrated problems in 2021; a typical structure presents a real-world scenario (e.g. student-course assignment, set of employees), defines a relation, and asks you to verify properties and answer sub-questions. Allocate 8-10 minutes per 5-mark question. Show every step — even obvious ones — because mark schemes reward process over final answers.
  • Q13. (a) Show that the relation R on Z defined by R={(a,b): a−b is divisible by 5} is an equivalence relation. (3 marks) (b) Find the equivalence class of 0 and 3. (2 marks) — Solution: (a) Reflexive: a−a=0, divisible by 5. Symmetric: If a−b is divisible by 5, then b−a = −(a−b) is also divisible by 5. Transitive: If a−b=5k and b−c=5m, then a−c=(a−b)+(b−c)=5(k+m), divisible by 5. Hence equivalence relation. (b) [0] = {...,−10,−5,0,5,10,...} = all multiples of 5. [3] = {...,−7,−2,3,8,13,...} = all integers of form 5n+3.
  • Q14. Let A = R−{3}, B = R−{1}. Consider f: A → B given by f(x) = (x−2)/(x−3). (a) Prove f is one-one and onto. (3 marks) (b) Find f⁻¹. (2 marks) — Solution: (a) One-one: f(x₁)=f(x₂) ⇒ (x₁−2)/(x₁−3) = (x₂−2)/(x₂−3). Cross-multiply: (x₁−2)(x₂−3)=(x₂−2)(x₁−3) ⇒ x₁x₂−3x₁−2x₂+6 = x₁x₂−3x₂−2x₁+6 ⇒ −3x₁−2x₂=−3x₂−2x₁ ⇒ x₁=x₂. One-one. Onto: Let y ∈ B. Solve y=(x−2)/(x−3): y(x−3)=x−2 ⇒ yx−3y=x−2 ⇒ x(y−1)=3y−2 ⇒ x=(3y−2)/(y−1). Since y≠1, x is well-defined and x≠3 (check). Hence onto. (b) From (a), f⁻¹(y)=(3y−2)/(y−1). Replace y with x: f⁻¹(x)=(3x−2)/(x−1).
  • Q15. (Case-based) A school assigns students to houses. Let S = {s₁,s₂,s₃,s₄,s₅} be five students and H = {Red, Blue, Green} be three houses. Relation R ⊆ S×S is defined by (sᵢ,sⱼ) ∈ R if sᵢ and sⱼ are in the same house. (a) Show R is an equivalence relation. (2 marks) (b) If s₁,s₂ are in Red, s₃ in Blue, s₄,s₅ in Green, list all elements of R. (2 marks) (c) How many equivalence classes exist? (1 mark) — Solution: (a) Reflexive: every student is in the same house as themselves. Symmetric: if sᵢ is in same house as sⱼ, then sⱼ is in same house as sᵢ. Transitive: if sᵢ,sⱼ same house and sⱼ,sₖ same house, then sᵢ,sₖ same house. Hence equivalence. (b) R = {(s₁,s₁),(s₁,s₂),(s₂,s₁),(s₂,s₂),(s₃,s₃),(s₄,s₄),(s₄,s₅),(s₅,s₄),(s₅,s₅)}. (c) Three equivalence classes: {s₁,s₂}, {s₃}, {s₄,s₅}.

How CBSE Frames Questions from This Chapter

CBSE follows a predictable pattern for Relations and Functions. Around 60% of questions are lifted or adapted from NCERT Exercises 1.1, 1.2, 1.3, and 1.4, with numbers or set elements changed slightly. For equivalence relation proofs, the 2018, 2019, 2022, and 2024 papers all used modular arithmetic (a−b divisible by k) or integer difference conditions — master that template. One-one and onto questions appear in two flavours: algebraic functions (polynomial, rational) where you solve f(x₁)=f(x₂), and finite-set mappings where you check cardinality and list image elements. Composition questions are computational (just substitute) but watch the domain: (fog) is defined only where g(x) lands in the domain of f. Inverse function problems always start with 'Let y = f(x), solve for x, then swap' — CBSE awards method marks even if algebraic manipulation has a small error. Case-based questions (introduced 2021) typically have 3-4 sub-parts worth 1+1+2 or 2+2 marks; read the scenario carefully because all information is relevant. Binary operation questions are rare (last appeared 2020) but when they do, expect commutativity and associativity checks. The mark scheme is granular: for a 3-mark equivalence proof, you get 1 mark per property (reflexive, symmetric, transitive) only if you write the definition, show the algebra, and state the conclusion. Skipping any part costs you that mark.
  • Equivalence relation template: state R, prove reflexive (definition + example), symmetric (if-then logic), transitive (chain rule). 3 marks = 1+1+1.
  • One-one proof: write f(x₁)=f(x₂), manipulate to show x₁=x₂. State conclusion 'hence one-one'. 1.5-2 marks.
  • Onto proof: take arbitrary y in codomain, solve f(x)=y for x, verify x is in domain. State 'hence onto'. 1.5-2 marks.
  • Inverse function: write y=f(x), solve for x in terms of y, replace y with x. Always verify f(f⁻¹(x))=x as a check.
  • Composition: (fog)(x) = f(g(x)). Compute inside-out. State domain restrictions explicitly.

Common Mistakes Students Make and How to Avoid Them

Every year, CBSE examiners publish common errors in the board exam report. For Relations and Functions, the top mistake is confusing reflexive and symmetric properties: students write (a,a) ∈ R when proving symmetry, or they assume (a,b) ∈ R implies (b,a) ∈ R without showing the reverse implication. Always label which property you are proving and use the correct logical form. Another frequent error is claiming a function is onto without finding a pre-image: saying 'f covers all real numbers' is not a proof — you must show that for every y in the codomain, there exists an x such that f(x)=y. In composition questions, many students compute (fog)(x) correctly but then write (gof)(x) as the same thing, forgetting composition is generally not commutative. For inverse functions, a classic blunder is forgetting to verify that the domain of f becomes the range of f⁻¹ and vice versa; you can lose a mark if you do not state this explicitly. In case-based questions, students often skip part (a) because it looks easy, then realise part (c) depends on (a)'s conclusion — answer every sub-part. Finally, in MCQs, do not guess: eliminate obviously wrong options (e.g. if a relation is not reflexive, it cannot be an equivalence relation) and use quick counter-examples to test the remaining choices.
  • Mistake 1: Writing 'R is reflexive' without showing (a,a) ∈ R for all a. Always exhibit the element or give a general proof.
  • Mistake 2: Proving symmetry by assuming (a,b) and (b,a) both exist. Symmetry means if (a,b) ∈ R, then (b,a) must follow — prove the implication.
  • Mistake 3: Confusing one-one and onto. One-one: distinct inputs give distinct outputs. Onto: every output has a pre-image. Not the same.
  • Mistake 4: In composition, writing fog = gof. Only true for special pairs (e.g. inverse functions). Always compute both if asked.
  • Mistake 5: Finding f⁻¹ but not checking domain/range swap. State 'Domain of f⁻¹ = Range of f' to secure full marks.
  • Mistake 6: In equivalence class questions, listing only a few elements. Use set-builder notation or ellipsis to show the infinite set.
  • Mistake 7: Losing marks for poor notation. Write ∀, ∃, ⇒, ⇔ symbols correctly; examiners deduct 0.5 marks for ambiguous statements.

Blueprint: Question Type Distribution (2025 Pattern)

The CBSE Class 12 Mathematics 2025 question paper follows a two-term merged pattern with 38 questions total, split into Sections A to E. Section A has 18 MCQs (1 mark each), Section B has 5 MCQs with multiple correct or assertion-reason (1 mark each), Section C has 6 VSAs (2 marks each), Section D has 4 short answers (3 marks each), and Section E has 3 long answers (5 marks each) plus 2 case-based integrated units (4 marks each). For Relations and Functions, expect exactly 1 question in Section A (MCQ on relation type or function property), 0 or 1 in Section C (2-mark composition or binary operation), 1 in Section D (3-mark equivalence or bijection proof), and occasionally 1 in Section E (5-mark combined problem). The chapter never appears in case-based questions alone but may be integrated with sets or real-world mappings. Total assured marks: 6-8. Maximum possible: 10 if a 5-marker appears. Difficulty split: 40% easy (direct NCERT), 40% moderate (NCERT Exemplar), 20% challenging (unfamiliar sets or multi-step proofs). Time allocation: spend 1 minute per MCQ, 3 minutes per 2-marker, 5 minutes per 3-marker, 8 minutes per 5-marker. Do not exceed these limits or you will rush calculus questions worth more marks.
  • Section A (MCQs): 1 question, 1 mark. Topics: identify relation type, function injectivity/surjectivity, composition value.
  • Section B (Case MCQs): 0-1 question. Assertion-Reason or scenario-based function property.
  • Section C (VSA): 0-1 question, 2 marks. Compute (fog)(x), check commutativity of binary operation, find range.
  • Section D (SA): 1 question, 3 marks. Prove equivalence relation, prove bijection, find inverse.
  • Section E (LA): 0-1 question, 5 marks. Combined equivalence + equivalence class, or bijection + inverse + verification.
  • Total Chapter 1 marks: 6-10 out of 80. Weightage ~8-12%, moderate but foundational.

Practice Strategy: 30-Day Revision Plan for Chapter 1

Start 30 days before your board exam. Days 1-5: Read NCERT Chapter 1 thoroughly, understand definitions of reflexive, symmetric, transitive, one-one, onto, bijective, and composition. Solve all examples and check answers. Days 6-10: Complete NCERT Exercise 1.1 (relations), 1.2 (functions), 1.3 (composition), 1.4 (binary operations). Write solutions in a notebook as if you are in the exam; do not look at the answer key until you have attempted every question. Days 11-15: Solve NCERT Exemplar Chapter 1 (all exercises). Focus on long-answer problems and pay attention to the step-by-step proofs — these are your templates. Days 16-20: Collect previous five years' CBSE board papers (download from cbse.nic.in or cbseacademic.nic.in) and solve all Chapter 1 questions under timed conditions. Compare your answers with the official marking scheme PDFs. Days 21-25: Identify weak areas (e.g. proving transitivity, finding inverse of rational functions) and redo only those question types from NCERT and Exemplar. Make a one-page formula sheet: definitions, key results (e.g. f is bijective iff f⁻¹ exists), and common pitfalls. Days 26-28: Take two full-length mock tests that include Chapter 1 along with other chapters. Simulate exam conditions (3 hours, no phone, no breaks). Days 29-30: Revise your one-page formula sheet, re-read the common-mistakes section on this page, and solve 5-6 rapid-fire MCQs and 2-3 three-mark questions to keep your speed sharp. On exam day, tackle Chapter 1 questions early if they appear in Section C or D — they are high-scoring and less time-variable than calculus problems.
  • Week 1: NCERT Chapter 1 theory + all solved examples. Master definitions verbatim.
  • Week 2: NCERT Exercises 1.1, 1.2, 1.3, 1.4. Write every proof in full sentences, no shortcuts.
  • Week 3: NCERT Exemplar + previous 5 years' board papers. Time yourself: 3 min for 2-mark, 5 min for 3-mark, 8 min for 5-mark.
  • Week 4: Targeted weak-area practice + two full mocks + one-page revision sheet. Sleep well the night before the exam.
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Frequently asked questions

How many marks does Chapter 1 Relations and Functions carry in CBSE Class 12 board exam 2025?+
Chapter 1 typically carries 8-10 marks out of 80 in the CBSE Class 12 Mathematics board exam. This includes 1-2 MCQs (1 mark each), one VSA or short-answer question (2-3 marks), and one long-answer or case-based question (3-5 marks). Unit I (Relations and Functions) is allocated 10 marks total, shared with Chapter 2 (Inverse Trigonometric Functions).
What is the most important topic in Relations and Functions for board exams?+
Equivalence relations are the highest-priority topic. CBSE asks a 3-mark or 5-mark question almost every year requiring you to prove a relation is reflexive, symmetric, and transitive. The second most important is proving a function is one-one and onto (bijective), which appears as a 2-mark or 3-mark question. Master these two proof templates and you secure 5-8 marks guaranteed.
How do I prove a relation is an equivalence relation step-by-step?+
Write three sub-proofs in order. (1) Reflexive: Show (a,a) ∈ R for every element a in the set; use the definition of R to verify this. (2) Symmetric: Assume (a,b) ∈ R and prove (b,a) ∈ R by reversing the condition. (3) Transitive: Assume (a,b) ∈ R and (b,c) ∈ R, then show (a,c) ∈ R by combining the two conditions. State 'Hence R is reflexive, symmetric, and transitive, so R is an equivalence relation.' Each property is worth 1 mark in a 3-mark question.
What is the difference between one-one and onto functions?+
A function f: A → B is one-one (injective) if distinct inputs always produce distinct outputs: f(x₁) = f(x₂) implies x₁ = x₂. It is onto (surjective) if every element in the codomain B is the image of at least one element in A: for every y ∈ B, there exists x ∈ A such that f(x) = y. A function that is both one-one and onto is called bijective, and only bijective functions have inverses.
How do I find the inverse of a function?+
First verify the function is bijective (one-one and onto). Then write y = f(x), solve the equation for x in terms of y (i.e. x = some expression in y), and finally replace y with x to get f⁻¹(x). For example, if f(x) = (2x+3)/5, write y = (2x+3)/5, solve: 5y = 2x+3 ⇒ x = (5y−3)/2, so f⁻¹(x) = (5x−3)/2. Always verify f(f⁻¹(x)) = x as a check.
What are the most common mistakes in Chapter 1 questions?+
Top mistakes: (1) confusing reflexive and symmetric properties in equivalence proofs, (2) forgetting to verify all three properties (reflexive, symmetric, transitive) and losing 1 mark, (3) claiming a function is onto without showing a pre-image exists for every element in the codomain, (4) mixing up composition order (fog is not the same as gof), (5) not stating domain and range when finding an inverse, and (6) using poor notation (missing ∀, ∃ symbols). Always write definitions explicitly and conclude each step.
Is NCERT enough for scoring full marks in Relations and Functions, or do I need reference books?+
NCERT textbook plus NCERT Exemplar is sufficient for 90% of board exam questions. CBSE directly adapts problems from these two sources. For the remaining 10% (unfamiliar contexts or multi-step proofs), solve the previous five years' board papers available on cbse.nic.in. Additional reference books like RD Sharma are helpful for extra practice but not mandatory if you master every NCERT and Exemplar exercise.
How much time should I spend on a 3-mark equivalence relation proof in the exam?+
Allocate 5-6 minutes maximum. Write one paragraph each for reflexive, symmetric, and transitive (about 3 lines each). Do not spend more than 6 minutes or you will run out of time for calculus and vector questions later. Practice writing these proofs in under 5 minutes at home so you build speed and confidence.
Can I score marks if I prove only two out of three properties for an equivalence relation?+
Partially. If the question is worth 3 marks and you correctly prove two properties (e.g. reflexive and symmetric) but omit transitive, you will receive 2 marks. However, you will lose the third mark and cannot conclude 'R is an equivalence relation' because all three properties are necessary. Always attempt all three to secure full marks.
What is a binary operation and does it appear in board exams?+
A binary operation * on a set A is a rule that assigns to each ordered pair (a,b) ∈ A×A a unique element a*b in A. Examples include addition, multiplication, and modular operations. Binary operation questions (checking commutativity, associativity, identity element, inverse element) appear occasionally as 2-mark VSAs or as part of case-based questions. They were more common in pre-2020 papers but are less frequent now. Still, revise NCERT Exercise 1.4 to be safe.

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Hi! I'm your CBSETUTOR.ai — an AI tutor that has ingested every NCERT book for Class 6 to 12. To get started, tell me which class you're in and which subject you'd like help with today (e.g. "Class 9, Physics").