India's #1 AI Tutorworksheet · Mathematics · Chapter 1

CBSE Class 12 Mathematics Chapter 1 Relations and Functions Worksheet with Answers

Relations and Functions form the gateway to higher mathematics in CBSE Class 12, accounting for approximately 10 marks in the Board examination. Mastery of equivalence relations, function types (injective, surjective, bijective), composition rules, and inverse function theorems is non-negotiable for scoring in both Board exams and competitive tests like JEE Main. This worksheet mirrors the latest CBSE pattern and NCERT exercise difficulty, offering structured practice across all question formats.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓The worksheet tests all NCERT-prescribed topics: types of relations, equivalence relations, types of functions, composition, and inverse functions.
✓Difficulty level is Medium-to-Hard, suitable for Board exam preparation and competitive entrance practice for JEE Main.
✓Suggested completion time is 90 minutes under exam conditions; 120 minutes for first-time learners with reference to NCERT.
✓Section E long-answer questions carry 5 marks each and demand multi-step proofs and composite function manipulation.
✓The case-study question integrates Relations and Functions with real-world scenarios, mirroring 2024-25 CBSE Board pattern.
✓Answer key provides concise working for MCQs and detailed step-by-step solutions for short and long answers.
✓Parents can download and print this worksheet for weekly revision or pre-Board mock practice at home.

Quick Chapter Recap – Relations and Functions

Chapter 1 of NCERT Class 12 Mathematics introduces Relations and Functions as foundational structures in algebra. A relation R from set A to set B is any subset of the Cartesian product A × B. Key properties of a relation on set A include reflexive (every element relates to itself), symmetric (if a relates to b then b relates to a), transitive (if a relates to b and b relates to c then a relates to c), and equivalence (simultaneously reflexive, symmetric, and transitive). A function f: A → B is a special relation where each element in domain A is associated with exactly one element in codomain B. Functions are classified as one-one or injective (distinct inputs yield distinct outputs), onto or surjective (every element in codomain is an image), and bijective (both one-one and onto). Composition of functions (fog)(x) = f(g(x)) obeys associativity but not commutativity. A function f is invertible if and only if it is bijective; the inverse function f⁻¹ satisfies fof⁻¹ = I_B and f⁻¹of = I_A where I denotes the identity function. Mastering these definitions, properties, and theorems is essential for solving Board problems efficiently and accurately.

Relation: subset of A × B; function: special relation with unique mapping.
Equivalence relation: reflexive + symmetric + transitive.
Injective (one-one): f(x₁) = f(x₂) ⟹ x₁ = x₂; Surjective (onto): range = codomain.
Bijective functions are invertible; inverse reverses the mapping.
Composition: (fog)(x) = f(g(x)); associative but generally not commutative.

Section A – Multiple Choice Questions (1 mark each)

Multiple-choice questions test conceptual clarity and the ability to eliminate incorrect options quickly. In CBSE Class 12 Mathematics Board exams, MCQs often appear in the objective section worth 20 marks. For Relations and Functions, MCQs probe understanding of relation types, function properties, domain-range identification, and composition outcomes. Students must verify each option systematically rather than guessing. Common traps include confusing reflexive with symmetric properties, misidentifying the range of composite functions, and overlooking the condition for invertibility (bijective nature). Practising these six MCQs sharpens pattern recognition and builds speed. Each question is designed to mirror Board exam style and difficulty, ensuring that concepts from NCERT exercises 1.1 to 1.4 are covered comprehensively. Students should aim to complete this section in 8–10 minutes, leaving ample time for descriptive answers in later sections.

Q1. Let R = {(a, b): a, b ∈ ℤ, a² + b² ≤ 4}. Then R is (a) reflexive (b) symmetric (c) transitive (d) equivalence relation.
Q2. If f: ℝ → ℝ is given by f(x) = 3x − 2, then f is (a) one-one only (b) onto only (c) bijective (d) neither one-one nor onto.
Q3. The number of bijective functions from set A to itself, where n(A) = 5, is (a) 25 (b) 52 (c) 120 (d) 3125.
Q4. If f(x) = x² and g(x) = sin x, then (fog)(x) equals (a) sin(x²) (b) (sin x)² (c) sin²x (d) x² sin x.
Q5. The function f: ℝ − {3} → ℝ defined by f(x) = (x + 2)/(x − 3) is (a) one-one and onto (b) one-one but not onto (c) onto but not one-one (d) neither one-one nor onto.
Q6. Which of the following relations on ℤ is an equivalence relation? (a) R = {(a,b): a ≥ b} (b) R = {(a,b): a divides b} (c) R = {(a,b): a − b is divisible by 5} (d) R = {(a,b): |a − b| ≤ 2}.

Section B – Fill in the Blanks (1 mark each)

Fill-in-the-blank questions demand precise recall of definitions, theorems, and standard results from NCERT Class 12 Mathematics Chapter 1. These questions appear frequently in CBSE sample papers and Board exams as part of the objective or very-short-answer section. Students must write exact mathematical terms or expressions without any ambiguity; partial answers receive zero marks. Common topics include properties of identity and constant functions, conditions for invertibility, formulae for the number of relations and functions between finite sets, and results on composition and inverse. For instance, knowing that the number of functions from a set with m elements to a set with n elements is n^m is a direct application tested repeatedly. Similarly, recognising that the inverse of a bijective function f is unique and denoted f⁻¹ is fundamental. This section trains students to internalise key formulae and definitions, building a mental repository that accelerates problem-solving in Sections D and E. Aim to complete these five blanks in 5 minutes, ensuring accurate spelling and notation.

Q7. A relation R on set A is called _______ if (a, a) ∈ R for all a ∈ A.
Q8. The function f: ℝ → ℝ given by f(x) = c (constant) is called a _______ function.
Q9. If f: A → B and g: B → C are both one-one and onto, then gof is _______.
Q10. The total number of relations from a set A with 3 elements to a set B with 4 elements is _______.
Q11. A function f is invertible if and only if f is _______.

Section C – True or False (1 mark each)

True/false questions assess depth of conceptual understanding and the ability to spot subtle errors in mathematical statements. CBSE Class 12 Mathematics Board exams occasionally include such questions in the objective section, and they are a staple in school-level formative assessments. For Relations and Functions, these statements test edge cases: whether the empty relation is symmetric, whether composition of functions is commutative, whether every function has an inverse, and whether the identity relation is the smallest equivalence relation. Students must justify their answer mentally or in rough work, even if the answer sheet requires only 'True' or 'False'. A common pitfall is accepting a statement that holds in special cases as universally true; for example, fog = gof holds only when f and g satisfy specific commutativity conditions, not in general. This section sharpens critical thinking and prepares students for assertion-reason questions in competitive exams. Allocate 5 minutes to this section, cross-checking each statement against NCERT definitions and theorems before marking your answer.

Q12. Every reflexive and symmetric relation is transitive. (True / False)
Q13. If f and g are bijective, then fog is also bijective. (True / False)
Q14. The relation R = {(a, b): a ≤ b} on ℕ is an equivalence relation. (True / False)
Q15. The identity function I_A: A → A is always invertible. (True / False)
Q16. Composition of functions is commutative, i.e., fog = gof for all functions f and g. (True / False)

Section D – Short Answer Questions (3 marks each)

Short-answer questions in CBSE Class 12 Mathematics typically carry 3 marks and require clear working across 4–6 lines. For Relations and Functions, these problems demand verification of relation properties, determination of function types, finding inverses, and solving composition equations. The CBSE marking scheme awards 1 mark for correct method or approach, 1 mark for intermediate steps, and 1 mark for the final answer and conclusion. Students must write all steps; skipping justification leads to mark deduction. Common question patterns include 'Show that the given relation is an equivalence relation', 'Prove that f is one-one and onto', 'Find fog and gof', and 'Determine the inverse function f⁻¹'. Each question in this section is drawn from NCERT exercises 1.1 to 1.4 and past Board papers, ensuring alignment with exam difficulty. Practise writing concise yet complete solutions, using standard notation (∀, ∃, ⟹, ∴) correctly. Allocate approximately 25–30 minutes for all five questions, maintaining neat presentation for partial credit even if the final answer is incorrect.

Q17. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Check whether R is reflexive, symmetric, and transitive.
Q18. Show that the function f: ℝ → ℝ defined by f(x) = 4x + 3 is bijective. Find its inverse.
Q19. If f(x) = x² and g(x) = √x, find the domain of fog and gof.
Q20. Let R be a relation on ℤ defined by R = {(a,b): 2 divides a − b}. Prove that R is an equivalence relation.
Q21. If f: ℝ → ℝ is given by f(x) = (3x − 2)/5, show that f is invertible and find f⁻¹.

Section E – Long Answer Questions (5 marks each)

Long-answer questions carry 5 marks in the CBSE Class 12 Mathematics Board exam and assess the ability to integrate multiple concepts, construct rigorous proofs, and solve multi-step problems. For Relations and Functions, typical 5-mark questions include proving properties of composite and inverse functions, establishing equivalence classes, solving functional equations, and demonstrating bijectivity through detailed verification. The CBSE marking scheme allocates marks as follows: 1 mark for stating relevant definitions or theorems, 2 marks for logical intermediate steps, 1 mark for correct computation, and 1 mark for clear conclusion. Students must present solutions in a structured manner: write 'To Prove' or 'To Find', show all algebraic manipulations, cite properties explicitly (e.g., 'since f is onto, range(f) = codomain(f)'), and box or underline the final answer. These three questions cover higher-order thinking skills (HOTS) and are ideal for students targeting 95+ in Boards or preparing for JEE Main. Allocate 35–40 minutes for Section E, ensuring every claim is justified and no step is skipped.

Q22. Let f: A → B and g: B → C be two functions. Prove that if gof is onto, then g is onto. Is the converse true? Justify.
Q23. Show that the relation R on the set A = {x ∈ ℤ: 0 ≤ x ≤ 12} given by R = {(a, b): |a − b| is a multiple of 4} is an equivalence relation. Find the equivalence classes [0], [1], and [2].
Q24. If f: ℝ − {3/5} → ℝ is defined by f(x) = (3x + 2)/(5x − 3), show that fof(x) = x for all x in the domain. Hence prove that f = f⁻¹.

Section F – Case Study Question (4 marks)

Case-study questions were introduced in the CBSE Class 12 Mathematics Board exam from the 2020-21 session and carry 4 marks, typically structured as a passage followed by four sub-questions (1 + 1 + 1 + 1 or 1 + 1 + 2 marking). These integrate real-world scenarios with mathematical modelling, testing comprehension, application, and analytical skills simultaneously. For Relations and Functions, a case study might describe a database relationship (student-course enrolment), a family tree (parent-child relation), or a coding scheme (encryption as a bijective function). Students must read the passage carefully, extract relevant data, map it to mathematical definitions (relation, function, type), and solve the sub-questions. The 2024-25 CBSE sample paper for Class 12 Mathematics includes one case study from each unit; Relations and Functions case studies often test equivalence partitions or inverse mappings. This question mirrors that pattern, providing a realistic application of Chapter 1 concepts. Allocate 10–12 minutes for reading, understanding, and answering all four parts accurately. Partial marking is generous if working is shown clearly.

Answer Key with Explanations

The answer key below provides correct answers and concise explanations for all questions in Sections A through F. For MCQs and objective questions, the rationale is given in one or two sentences. For short-answer questions (Section D), key steps are outlined with intermediate results. For long-answer questions (Section E), a structured proof or solution is presented with logical flow and necessary justifications. Cross-check your solutions step-by-step, awarding yourself marks according to CBSE guidelines: full marks only if method and answer are both correct, partial marks for correct approach even if a calculation error led to a wrong final answer. Use this answer key not just to tick right or wrong, but to understand why an answer is correct and to learn the standard format expected in Board exams. If you scored below 70 percent on this worksheet, revisit NCERT examples and solved exercises before attempting another practice set. For targeted doubt-clearing and step-by-step solutions to every NCERT problem, students can try CBSETUTOR.ai, which offers 24×7 AI tutoring with photo-upload problem solving at a flat ₹999/month for Classes 6–12, with a 3-day free trial to experience personalised learning support.

Section A Answers: Q1.(b) symmetric; Q2.(c) bijective; Q3.(c) 120; Q4.(b) (sin x)²; Q5.(a) one-one and onto; Q6.(c) R = {(a,b): a − b divisible by 5}.
Section B Answers: Q7. reflexive; Q8. constant; Q9. bijective (or one-one and onto); Q10. 2¹² = 4096; Q11. bijective (or one-one and onto).
Section C Answers: Q12. False (reflexive + symmetric ≠ transitive in general); Q13. True (composition of bijections is bijective); Q14. False (reflexive, not symmetric); Q15. True (identity is always bijective); Q16. False (composition not generally commutative).
Section D Answers: Q17. Reflexive: yes; Symmetric: yes; Transitive: no (since (1,2) and (2,1) in R but no (1,1) issue, check (2,1),(1,2) gives (2,2) present, however (1,2),(2,3) not in R so transitive fails if 3 not related). Q18. One-one and onto shown by algebra; f⁻¹(x) = (x − 3)/4. Q19. Domain of fog = ℝ; domain of gof = [0, ∞). Q20. Reflexive: a − a = 0 divisible by 2; Symmetric: if 2|(a − b) then 2|(b − a); Transitive: if 2|(a − b) and 2|(b − c) then 2|(a − c). Q21. f is bijective; f⁻¹(x) = (5x + 2)/3.
Section E Answers: Q22. Proof: Let c ∈ C. Since gof is onto, ∃ a ∈ A such that g(f(a)) = c. Put b = f(a) ∈ B, then g(b) = c. Hence g is onto. Converse false: g onto does not imply gof onto (f may not be onto). Q23. Reflexive, symmetric, transitive verified; [0] = {0,4,8,12}, [1] = {1,5,9}, [2] = {2,6,10}. Q24. Compute fof(x) = f((3x+2)/(5x−3)) = (3((3x+2)/(5x−3))+2)/(5((3x+2)/(5x−3))−3) = simplify to x; hence f = f⁻¹.
Section F Answers: Q25(i). (c) equivalence relation; Q25(ii). (c) 200 (if all in one dept, though typically distributed); Q25(iii). (b) onto (since codomain = range if all depts have ≥1 student); Q25(iv). (a) always equal (same dept ⟹ same code).

How to Use This Worksheet Effectively for Board Exam Preparation

To maximise learning from this Relations and Functions worksheet, follow a structured approach. First, attempt the worksheet under timed conditions—set a 90-minute timer and work without referring to notes or NCERT solutions. This simulates Board exam pressure and identifies time-management gaps. After completion, self-evaluate using the answer key, awarding marks strictly according to CBSE guidelines (no half-marks for incomplete working in objective questions). Note every error and classify it: conceptual misunderstanding, calculation mistake, or misreading the question. For conceptual errors, revisit the corresponding NCERT section—read the theory, study solved examples, and rework similar exercise problems. For calculation errors, practice mental math and algebraic manipulation drills. Second, reattempt only the questions you got wrong, after a one-day gap, to ensure the concept has solidified. Third, use this worksheet as a revision tool one week before your Board or school exam—complete it again and aim for 95 percent-plus accuracy within 75 minutes. Finally, discuss challenging long-answer questions (Section E) with classmates or teachers to explore alternative solution methods and deepen understanding. Consistent weekly practice with such worksheets builds confidence, speed, and accuracy, all essential for scoring 95-plus in CBSE Class 12 Mathematics.

Attempt under timed conditions (90 minutes) to simulate exam environment and build speed.
Self-evaluate strictly using the answer key; award partial marks only if steps are shown.
Classify every mistake: conceptual, calculation, or careless; revisit NCERT for conceptual gaps.
Reattempt incorrect questions after a gap to reinforce learning and test retention.
Use the worksheet for weekly revision and pre-exam mock practice to track improvement.
Discuss Section E (long-answer) solutions with peers or mentors to learn alternative approaches.
Maintain a separate error log noting question number, error type, and corrective action taken.

Boost Your Confidence with CBSETUTOR.ai's 24×7 Personalised Support

While worksheets and NCERT exercises form the backbone of CBSE Class 12 Mathematics preparation, many students encounter doubts at odd hours—late evenings after coaching, early mornings before school tests, or weekends when teachers are unavailable. CBSETUTOR.ai bridges this gap by offering round-the-clock AI-powered tutoring for Classes 6 to 12 at a flat ₹999 per month, covering all subjects. Simply upload a photo of any Relations and Functions problem—whether from this worksheet, NCERT exercises, or past Board papers—and receive step-by-step solutions with conceptual explanations in seconds. The AI tutor identifies your mistake patterns, suggests targeted practice, and adapts to your learning pace. Unlike generic video lectures, CBSETUTOR.ai provides personalised, on-demand help exactly when you need it, making it ideal for self-paced learners and students in cities or towns with limited access to quality coaching. Parents appreciate the transparent, affordable pricing—one subscription works for every class, eliminating the need for multiple expensive tuition centres. Start with a 3-day free trial to experience how instant doubt resolution and adaptive practice can transform your child's confidence and performance in Mathematics. Thousands of CBSE students already rely on CBSETUTOR.ai to clarify tough concepts, verify solutions, and stay ahead in their Board exam preparation journey.

24×7 AI tutor available for instant doubt-clearing anytime, anywhere—no waiting for the next class.
Photo-upload feature: snap a Relations and Functions problem and get step-by-step solutions in seconds.
Flat ₹999/month for Classes 6–12, all subjects; one price, unlimited questions, maximum value.
Personalised learning: AI identifies weak areas and recommends targeted NCERT exercises and practice.
Ideal for students in Tier-2 and Tier-3 cities with limited access to expert Mathematics coaching.
3-day free trial—no credit card required—lets students and parents evaluate effectiveness risk-free.
Trusted by thousands of CBSE families for Board exam prep, NCERT mastery, and competitive exam practice.

Frequently asked questions

What is the difficulty level and suggested time for this Relations and Functions worksheet?+

This worksheet is Medium-to-Hard difficulty, aligned with CBSE Board exam standards and suitable for students targeting 90-plus marks. The suggested completion time is 90 minutes under exam conditions. First-time learners may take up to 120 minutes with NCERT reference.

How many marks does Chapter 1 Relations and Functions carry in the CBSE Class 12 Maths Board exam?+

Relations and Functions typically carries 8–10 marks in the CBSE Class 12 Mathematics Board exam, distributed across one 5-mark long-answer question, one or two 3-mark short-answer questions, and objective questions (MCQs or VSAQs) worth 2–3 marks.

Are the questions in this worksheet taken directly from NCERT exercises?+

No, the questions are original and designed to mirror NCERT exercise difficulty and Board exam pattern. They cover all topics from NCERT exercises 1.1 to 1.4—types of relations, equivalence, function types, composition, and inverse—ensuring comprehensive practice beyond textbook problems.

What is an equivalence relation and how do I prove it?+

An equivalence relation on set A is a relation that is simultaneously reflexive (every element relates to itself), symmetric (if a relates to b then b relates to a), and transitive (if a relates to b and b relates to c then a relates to c). To prove, verify all three properties separately using definitions and set elements.

How do I determine if a function is one-one (injective)?+

A function f: A → B is one-one if distinct inputs produce distinct outputs. Formally, prove f(x₁) = f(x₂) implies x₁ = x₂. Alternatively, check that f is strictly monotonic (always increasing or always decreasing) or use the horizontal line test graphically.

What is the difference between onto (surjective) and into functions?+

A function f: A → B is onto (surjective) if every element in codomain B is the image of at least one element in domain A, i.e., range(f) = B. If range(f) is a proper subset of B, the function is into. To prove onto, show for every y ∈ B there exists x ∈ A such that f(x) = y.

How do I find the inverse of a function?+

First, verify the function is bijective (both one-one and onto); only bijective functions have inverses. Then solve y = f(x) for x in terms of y to get x = f⁻¹(y). Replace y with x in the final expression to write f⁻¹(x). Verify by checking f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

What is the composition of functions and is it commutative?+

The composition (fog)(x) means f(g(x))—apply g first, then f to the result. Composition is associative but not commutative: in general fog ≠ gof. Commutativity holds only for special pairs, such as f and its inverse f⁻¹, or the identity function with any function.

Should I memorise all proofs for equivalence relations and function properties?+

Yes, you must be able to reproduce standard proofs for reflexive, symmetric, transitive properties, and proofs that composition of bijections is bijective, and that a bijective function is invertible. CBSE awards full marks only when logical steps and justifications are clearly written, not for stating results without proof.

How is the case-study question in Section F marked in the Board exam?+

Case-study questions carry 4 marks, typically split as 1 + 1 + 1 + 1 or 1 + 1 + 2 across four sub-questions. Marks are awarded for correct answer with working. Read the passage carefully, extract data, apply Relations and Functions concepts, and show all steps to ensure partial credit even if one sub-question is wrong.

Can I use this worksheet for JEE Main preparation as well?+

Absolutely. Relations and Functions is a core topic in JEE Main Mathematics, often appearing in multiple-choice questions and integer-type questions. The worksheet's Section E long-answer questions and case study develop problem-solving skills and conceptual depth essential for JEE, beyond rote Board exam practice.

Where can I find step-by-step solutions if I am stuck on a worksheet question?+

The answer key in Section G provides concise solutions with key steps. For detailed, personalised explanations, upload a photo of the question to CBSETUTOR.ai's 24×7 AI tutor, available at ₹999/month for all classes with a 3-day free trial. The AI breaks down each step and highlights common mistakes, helping you learn independently.

Related resources

Important Questions: CBSE Class 12 Mathematics Chapter 1 Relations and Functions Class 12 Mathematics Chapter 1 Relations and Functions — Formulas & Key Points Relations and Functions for Class 12: The Complete CBSE Guide (2026-27)Inverse Trigonometric Functions for Class 12: The Complete CBSE Guide (2026-27)AI Tutor for Class 12: The Smart Alternative to Tuition AI Tutor for Class 12 Accountancy: Learn Faster with Instant Help Class 9 Coordinate Geometry Chapter 3: Cartesian System & Plotting Points CBSE Class 9 Mathematics Chapter 1 Number Systems Worksheet with Answers

Keep learning — related guides

Class 12Mathematics

Class 12 Mathematics Chapter 13 Probability — Formulas & Key Points

Class 12Mathematics

CBSE Class 12 Mathematics Chapter 13 Probability Worksheet with Answers

Class 12Mathematics

Important Questions: CBSE Class 12 Mathematics Chapter 13 Probability

Class 12Mathematics

Important Questions: CBSE Class 12 Mathematics Chapter 12 Linear Programming

Class 12Mathematics

CBSE Class 12 Mathematics Chapter 12 Linear Programming Worksheet with Answers

Class 12Mathematics

Class 12 Mathematics Chapter 12 Linear Programming — Formulas & Key Points

Ready to give your Class 12 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 12 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →