- Relations and Functions Class 12 Important Questions with Solutions
- Relations and Functions Class 12 MCQs with Answers
- Relations and Functions Case Study Questions with Solutions
- Relations and Functions Class 12 PYQs with Solutions
- Download Class 12 Maths Chapter 1 Important Questions PDF
- Common Mistakes in Relations and Functions Important Questions
- How to Prepare Relations and Functions Class 12 Important Questions
- Related Class 12 Maths Study Resources
- FAQs on Class 12 Maths Chapter 1 Important Questions
Class 12 Maths Important Questions Chapter 1: Relations and Functions with Solutions
By Rohit RP
|
Updated on 12 Nov 2026, 16:08 IST
Class 12 Maths Chapter 1 important questions help students practise relations and functions for the CBSE board exam. This collection of Relations and Functions Class 12 important questions with solutions covers reflexive, symmetric, transitive and equivalence relations, along with one-one and onto functions. Practise the topic-wise questions, MCQs, assertion–reason questions and case studies below. Each original question includes an answer or a worked solution for revision.
The current CBSE Class 12 Mathematics (Code 041) syllabus lists reflexive, symmetric, transitive and equivalence relations, and one-one and onto functions in Chapter 1. The question bank below focuses on these topics.
| Important topic | What to practise |
| Reflexive relations | Check whether every element is related to itself. |
| Symmetric relations | Check whether (a,b) ∈ R implies (b,a) ∈ R. |
| Transitive relations | Check whether (a,b) ∈ R and (b,c) ∈ R imply (a,c) ∈ R. |
| Equivalence relations | Verify all three properties and identify equivalence classes where useful. |
| One-one functions | Prove injectivity or identify two different inputs with the same output. |
| Onto and bijective functions | Check the given codomain and determine whether every output has a preimage. |
Syllabus note: Composition of functions, inverse functions and binary operations are not listed in the core 2026–27 Chapter 1 entry. The questions on this page stay within the listed topics. See the official CBSE syllabus.
Relations and Functions Class 12 Important Formulas and Definitions
| Concept | Condition / Formula |
| Reflexive | For every a ∈ A, (a,a) ∈ R. |
| Symmetric | If (a,b) ∈ R, then (b,a) ∈ R. |
| Transitive | If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. |
| Equivalence relation | Reflexive + symmetric + transitive. |
| One-one (injective) function | f(a) = f(b) implies a = b. |
| Onto (surjective) function | Range(f) = codomain(f). |
| Bijective function | Both one-one and onto. |
| Equivalence class of a | [a] = {b ∈ A : (a,b) ∈ R}. |
One-one, not ontoOnto, not one-oneAB12abcAB123ab
Relations and Functions Class 12 Important Questions with Solutions
Important Questions on Reflexive, Symmetric and Transitive Relations
Question 1. Basic Is R = {(1,1), (2,2), (3,3), (1,2)} on A = {1,2,3} reflexive?
Solution: A relation on A is reflexive when (1,1), (2,2) and (3,3) all belong to R. All three pairs are present. The extra pair (1,2) does not change this property.
Answer: Yes, R is reflexive.
Question 2. Is R = {(1,1), (2,2), (1,2)} on A = {1,2} symmetric?
Solution: Symmetry requires that whenever (a,b) is in R, (b,a) is also in R. Here (1,2) ∈ R, but (2,1) ∉ R.
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Answer: R is not symmetric.
Question 3. Is R = {(1,2), (2,3), (1,3)} on A = {1,2,3} transitive?
Solution: The only possible two-step chain in R is (1,2) followed by (2,3). It requires (1,3), which is present. No other pair creates another two-step chain.
Answer: Yes, R is transitive.
Question 4. Give an example of a relation that is symmetric but not transitive.
Solution: On A = {1,2,3}, take R = {(1,2), (2,1), (2,3), (3,2)}. Each pair appears with its reverse, so R is symmetric. But (1,2) and (2,3) belong to R while (1,3) does not.
Answer: The given R is symmetric but not transitive.
Question 5. Consider R on the set of integers Z defined by aRb if a ≤ b. Determine whether R is reflexive, symmetric and transitive.
Solution: Reflexive: a ≤ a for every integer a. Symmetric: 2 ≤ 3 but 3 ≤ 2 is false, so R is not symmetric. Transitive: if a ≤ b and b ≤ c, then a ≤ c.
Answer: R is reflexive and transitive, but not symmetric.
Question 6. On A = {1,2,3}, is the empty relation R = ∅ reflexive, symmetric or transitive?
Solution: It is not reflexive because (1,1), (2,2) and (3,3) are missing. It is symmetric because no pair violates the condition: there is no (a,b) in R requiring a reverse. The same reasoning shows that transitivity is satisfied: no two-step chain exists to violate it.
Answer: Not reflexive; symmetric and transitive.
Question 7. Let R = {(a,b) ∈ Z × Z : a − b is odd}. Is R symmetric? Is it transitive?
Solution: If a − b is odd, then b − a = −(a − b) is odd, so R is symmetric. For transitivity, take a = 1, b = 2, c = 3: 1 − 2 and 2 − 3 are odd, but 1 − 3 = −2 is even.
Answer: Symmetric, but not transitive.
Equivalence Relation Class 12 Important Questions
Question 8. Show that aRb if a − b is divisible by 3 is an equivalence relation on Z.
Solution: Reflexive: a − a = 0 is divisible by 3. Symmetric: if 3 divides a − b, it also divides b − a. Transitive: if 3 divides a − b and b − c, it divides their sum a − c.
Answer: R is an equivalence relation.
Question 9. On Z, define aRb if a and b have the same parity. Prove that R is an equivalence relation.
Solution: Every integer has the same parity as itself, so R is reflexive. If a has the same parity as b, b has the same parity as a, so R is symmetric. If a and b have the same parity and b and c have the same parity, a and c have the same parity, so R is transitive.
Answer: R is an equivalence relation. Its two classes are even and odd integers.
Question 10. On R, let aRb if |a − b| < 1. Is R an equivalence relation?
Solution: Reflexive: |a − a| = 0 < 1. Symmetric: |a − b| = |b − a|. Transitive fails: choose a = 0, b = 0.75 and c = 1.5. Both |0 − 0.75| and |0.75 − 1.5| are less than 1, but |0 − 1.5| is not.
Answer: R is not an equivalence relation because it is not transitive.
Question 11. On A = {1,2,3,4,5,6}, let aRb if 2 divides a − b. Find the equivalence classes [1] and [2].
Solution: [1] contains all elements of A with odd parity: {1,3,5}. [2] contains all elements with even parity: {2,4,6}. This is an equivalence relation because congruence modulo 2 is reflexive, symmetric and transitive.
Answer: [1] = {1,3,5}; [2] = {2,4,6}.
Question 12. On A = {1,2,3}, find the smallest equivalence relation containing the pairs (1,2) and (2,3).
Solution: Symmetry requires (2,1) and (3,2). Transitivity requires (1,3) and (3,1), and reflexivity requires (1,1), (2,2), (3,3). Every element is therefore related to every element.
Answer: R = A × A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}.
A = {1, 2, 3, 4, 5, 6}Odd class [1]Even class [2]1, 3, 52, 4, 6
One-to-One Functions Class 12 Important Questions
Question 13. Show that f: R → R, f(x) = 4x − 7, is one-one.
Solution: Suppose f(a) = f(b). Then 4a − 7 = 4b − 7, so 4a = 4b and a = b.
Answer: f is one-one.
Question 14. Is f: R → R, f(x) = x2, one-one?
Solution: f(2) = 4 and f(−2) = 4 although 2 ≠ −2. Two different inputs have the same output.
Answer: f is not one-one.
Question 15. Is f: [0,∞) → [0,∞), f(x) = x2, one-one?
Solution: Let f(a) = f(b), where a,b ≥ 0. Then a2 = b2. Since both inputs are non-negative, a = b.
Answer: Yes, f is one-one on the given domain.
Question 16. Let f: {1,2,3} → {a,b,c,d} be defined by f(1)=a, f(2)=b and f(3)=c. Is f one-one?
Solution: The three inputs have distinct images a, b and c. Therefore, equal outputs cannot come from different inputs.
Answer: Yes, f is one-one, but it is not onto because d has no preimage.
Onto and Bijective Functions Class 12 Important Questions
Question 17. Show that f: R → R, f(x) = 3x + 1, is onto.
Solution: For any y ∈ R, solve y = 3x + 1 to get x = (y − 1)/3. This x is real and satisfies f(x)=y.
Answer: f is onto.
Question 18. Is f: R → R, f(x)=x2, onto?
Solution: The output x2 is never negative. For example, y = −1 belongs to the codomain R but has no real preimage.
Answer: f is not onto.
Question 19. Is f: [0,∞) → [0,∞), f(x)=x2, bijective?
Solution: It is one-one because non-negative inputs with equal squares are equal. For any y ≥ 0 in the codomain, x = √y is in the domain and satisfies f(x)=y. So the function is also onto.
Answer: Yes, f is bijective.
Question 20. Let f: {1,2,3} → {a,b} be given by f(1)=a, f(2)=b, f(3)=a. Is f onto? Is it one-one?
Solution: Both a and b are images of elements of the domain, so f is onto. But f(1)=f(3)=a even though 1 ≠ 3, so it is not one-one.
Answer: Onto but not one-one.
Question 21. Show that f: R → R, f(x)=5 − 2x, is bijective.
Solution: One-one: if 5 − 2a = 5 − 2b, then a=b. Onto: for any y ∈ R, take x=(5 − y)/2, which is real and gives f(x)=y.
Answer: f is bijective.
Relations and Functions Class 12 MCQs with Answers
Choose the correct option. These original MCQs focus on concepts from the current Chapter 1 syllabus.
Question 22. Which condition is necessary for a relation R on A to be reflexive?
(A) (a,b) ∈ R for all a,b (B) (a,a) ∈ R for every a ∈ A (C) R = ∅ (D) R must be symmetric
Answer: (B). Every element must be related to itself.
Question 23. The relation R = {(1,2),(2,1)} on A={1,2} is:
(A) Reflexive only (B) Symmetric only (C) Transitive only (D) Equivalence
Answer: (B). Each pair has its reverse. Reflexivity fails because (1,1) and (2,2) are missing, and transitivity fails because (1,2) and (2,1) require (1,1).
Question 24. Which function is one-one on R?
(A) f(x)=x2 (B) f(x)=|x| (C) f(x)=2x+1 (D) f(x)=5
Answer: (C). If 2a+1=2b+1, then a=b.
Question 25. The function f: R → R defined by f(x)=x2 is:
(A) One-one and onto (B) One-one only (C) Onto only (D) Neither one-one nor onto
Answer: (D). f(1)=f(−1), and negative real numbers are not in the range.
Question 26. A relation is an equivalence relation if it is:
(A) Reflexive and symmetric only (B) Symmetric and transitive only (C) Reflexive, symmetric and transitive (D) Onto
Answer: (C). All three relation properties are required.
Question 27. An onto function f: A → B has:
(A) Range smaller than B (B) Range equal to B (C) Different outputs for every input (D) No preimage for any b ∈ B
Answer: (B). Onto means every codomain element is reached.
Question 28. If f: {1,2} → {a,b} is given by f(1)=a and f(2)=b, the function is:
(A) Neither one-one nor onto (B) One-one but not onto (C) Onto but not one-one (D) Bijective
Answer: (D). The inputs have distinct outputs and every codomain element is used.
Relations and Functions Assertion–Reason Questions
Answer choices: (A) Both Assertion and Reason are true, and Reason correctly explains Assertion. (B) Both are true, but Reason does not explain Assertion. (C) Assertion is true, Reason is false. (D) Assertion is false, Reason is true.
Question 29.
Assertion: The identity relation I = {(a,a): a ∈ A} on a non-empty set A is reflexive.
Reason: A relation is reflexive if (a,a) ∈ R for every a ∈ A.
Answer: (A). Both statements are true and the Reason gives the definition that proves the Assertion.
Question 30.
Assertion: f: R → R, f(x)=x2, is one-one.
Reason: A function is one-one if equal outputs imply equal inputs.
Answer: (D). The Assertion is false because f(1)=f(−1); the Reason is the correct definition of one-one.
Question 31.
Assertion: If a relation is an equivalence relation, it is symmetric.
Reason: An equivalence relation must be reflexive, symmetric and transitive.
Answer: (A). Both statements are true and the Reason explains the Assertion.
Relations and Functions Case Study Questions with Solutions
Case Study 1: Students in the Same House
A school has students A = {P,Q,R,S}. Students P and R belong to the Blue House; Q and S belong to the Green House. Define a relation R on A: xRy if x and y belong to the same house.
Question 32. Is the relation reflexive?
Solution: Every student belongs to the same house as themselves, so (x,x) ∈ R for every x ∈ A.
Answer: Yes, R is reflexive.
Question 33. Is the relation symmetric?
Solution: If x is in the same house as y, then y is in the same house as x.
Answer: Yes, R is symmetric.
Question 34. Is R an equivalence relation? Give its equivalence classes.
Solution: The relation is reflexive and symmetric. It is also transitive: if x shares a house with y and y shares a house with z, then x and z share that house. The two classes are the Blue and Green House groups.
Answer: R is an equivalence relation. Its classes are {P,R} and {Q,S}.
Case Study 2: Assigning Students to Projects
Three students A = {1,2,3} are assigned to project groups B = {X,Y}. Define f: A → B by f(1)=X, f(2)=Y and f(3)=X.
Question 35. Is f a function?
Solution: Each student is assigned to exactly one group. Two different students may belong to the same group without violating the definition of a function.
Answer: Yes, f is a function.
Question 36. Is f one-one?
Solution: f(1)=f(3)=X even though 1 ≠ 3.
Answer: No, f is not one-one.
Question 37. Is f onto?
Solution: X is the image of students 1 and 3, while Y is the image of student 2. Every element of B has a preimage.
Answer: Yes, f is onto.
Relations and Functions Class 12 PYQs with Solutions
Previous-year questions help students understand how CBSE has assessed relation properties and function types. Use Class 12 Maths Previous Year Question Papers and compare solutions with the official marking schemes where available.
Editorial note: The 42 questions above are original practice questions, not authenticated CBSE PYQs. Do not label them as previous-year questions. Add actual PYQs here only after checking their year, paper/set and source. See CBSE Class XII sample papers and marking schemes for verified official practice material.
Download Class 12 Maths Chapter 1 Important Questions PDF
Students can practise this question bank online, including the topic-wise solutions, MCQs and case studies. An offline PDF should contain a question-only section followed by matching worked solutions.
Common Mistakes in Relations and Functions Important Questions
| Common mistake | How to correct it |
| Assuming reflexive means symmetric | Check the definitions independently. |
| Using only one example to prove transitivity | Give a general proof or check all relevant finite cases. |
| Ignoring the codomain in onto questions | State both domain and codomain before testing the range. |
| Assuming one-one implies onto | Test onto separately; Diagram 1 shows why. |
| Stating that a relation is an equivalence relation without proof | Verify reflexive, symmetric and transitive properties one by one. |
How to Prepare Relations and Functions Class 12 Important Questions
- Revise the definitions and conditions in the formula table.
- Solve each worked question before reading the solution.
- Practise both proofs and counterexamples.
- Remember to state the domain and codomain for function questions.
- Attempt MCQs and case studies under a time limit.
- Use NCERT Solutions for Class 12 Maths Chapter 1 to review methods you find difficult.
These Class 12 Maths Chapter 1 important questions with solutions provide practice on relation properties, equivalence relations and one-one and onto functions. Work through the proofs, MCQs and case studies, then revise questions you answered incorrectly. For board preparation, combine the question bank with NCERT exercises and verified CBSE pap
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FAQs on Class 12 Maths Chapter 1 Important Questions
What are the important topics in Relations and Functions Class 12?
The current CBSE Chapter 1 syllabus covers reflexive, symmetric, transitive and equivalence relations, as well as one-one and onto functions. Practise checking relation properties and proving whether a function has the stated property.
How do you prove that a relation is an equivalence relation?
Show that the relation is reflexive, symmetric and transitive on the given set. Give a separate logical argument for each property. If even one property fails, the relation is not an equivalence relation.
How do you check whether a function is one-one and onto?
To prove one-one, show that f(a)=f(b) implies a=b. To prove onto, show that every element of the stated codomain has a preimage in the domain. Always specify the domain and codomain.
Are NCERT questions enough for Class 12 Maths Chapter 1?
Start with the NCERT examples and exercises. Then practise extra questions, MCQs and official sample papers to test whether you can apply the same concepts in different forms.
Where can I find Relations and Functions Class 12 PYQs with solutions?
Use authenticated Class 12 Maths previous-year papers and their marking schemes. Check the examination year and paper/set for each question instead of relying on an unsourced “PYQ” label.
Can I download Relations and Functions important questions with solutions as a PDF?
This online question bank includes worked solutions for revision. A PDF download should be added once a matching file is published and checked.
Does Chapter 1 carry eight marks in CBSE Class 12 Maths?
No fixed eight-mark allocation is guaranteed for Chapter 1 alone. CBSE assigns eight marks to the wider Relations and Functions unit, which also includes Inverse Trigonometric Functions. CBSE states that there is no fixed chapter-wise weightage.
How should I revise Relations and Functions before an exam?
Revise definitions, solve relation-property proofs, practise one-one and onto questions, and check mistakes carefully. For every proof, write the reasoning rather than only the final conclusion.
