Class 12 · Chapter 1 · Relations and Functions unit (8 of 80 marks)

Relations and Functions Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 42 board-style questions.

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42 questions
13 multiple choice, 3 assertion–reason, 9 very short answer, 9 short answer, 5 long answer, 3 case study
About 12 hours to master

Relations and Functions unit: 8 of 80 theory marks (Relations and Functions and Inverse Trigonometric Functions).

NCERT textbook: Chapter 1 on ncert.nic.in (PDF)

Revision notes

Relations and Functions — revision notes

1. Relations

A relation $R$ on a set $A$ is a subset of $A \times A$ . For $A$ with $n$ elements there are $2^{n^{2}}$ relations.

Reflexive: $(a, a) \in R$ for every $a \in A$ .
Symmetric: $(a, b) \in R \Rightarrow (b, a) \in R$ .
Transitive: $(a, b), (b, c) \in R \Rightarrow (a, c) \in R$ .
Equivalence: all three. Empty relation: symmetric and transitive (vacuously), not reflexive on a non-empty set. Universal relation $A \times A$ : equivalence.

2. Equivalence classes

$[a] = {x \in A : x R a}$ . Two classes are either identical or disjoint, and together they cover $A$ (a partition). Example: “ $a - b$ divisible by $m$ ” on $Z$ has $m$ classes (remainders $0, 1, \dots, m - 1$ ).

3. Functions

One-one (injective): $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ . Test: solve $f (x_{1}) = f (x_{2})$ ; or find two different inputs with the same output to disprove. A strictly increasing/decreasing function is one-one.
Onto (surjective): range $=$ codomain. Test: for arbitrary $y$ in the codomain, solve $y = f (x)$ and check that $x$ lies in the domain.
Bijective: one-one and onto.
Counting ( $| A | = m$ , $| B | = n$ ): functions $n^{m}$ ; one-one functions $n (n - 1) \dots (n - m + 1)$ if $m \leq n$ , else $0$ ; onto functions from a $3$ -set to a $2$ -set: $2^{3} - 2 = 6$ .

Worked example 1

On ${1, \dots, 6}$ , $a R b ⟺ 3 ∣ (a - b)$ : classes ${1, 4}, {2, 5}, {3, 6}$ .

Worked example 2

$f : R \to R$ , $f (x) = 4 - 3 x$ : $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ (one-one); for any $y$ , $x = \frac{4 - y}{3}$ works (onto). Bijective.

Worked example 3

$f : N \to N$ , $f (n) = n^{2} + 1$ : one-one (as $n > 0$ ), not onto ( $1$ and $3$ have no pre-image).

Common errors

Checking reflexivity for only the elements that appear in $R$ instead of all of $A$ .
Showing onto by solving for $x$ but not checking that $x$ lies in the domain (e.g. $x \in N$ , $x \neq 2$ ).
Forgetting that the answer depends on the domain and codomain: $x^{2}$ is not one-one on $R$ but is one-one on $[0, \infty)$ .
Giving an “example” instead of a proof when asked to show a property holds; a single counter-example is enough only to show it fails.

Board-exam tips

Write each of reflexive, symmetric, transitive under its own heading with a one-line reason.
For a “not” answer, always give a specific counter-example pair.
Composite and inverse functions are not in the current syllabus — do not spend time on $g \circ f$ or $f^{- 1}$ .

Topics in this chapter: Types of relations · Counting relations and functions · Equivalence relations and equivalence classes · One-one and onto functions.

Formulas for this chapter: Chapter 1 on the Class 12 formula sheet (free PDF and A3 poster).

Relations and Functions Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 215 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceTypes of relations

Which statement is true of the relation $R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 3)}$ on the set ${1, 2, 3, 4}$ ?

Show worked answer

All four pairs $(a, a)$ are in $R$ (reflexive); $(2, 3) \in R$ but $(3, 2) \notin R$ (not symmetric); every chain such as $(2, 3), (3, 3)$ gives a pair already in $R$ (transitive) A1
Correct option: (a) It is reflexive and transitive but not symmetric.

Q2·1 mark·Multiple choiceOne-one and onto functions

Which describes $f (x) = 7 - 2 x$ as a function from $R$ to $R$ ?

Q3·1 mark·Assertion–reasonOne-one and onto functions

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The function $f : R \to R$ , $f (x) = x^{2}$ , is one-one.

Reason (R): The function $f : R \to R$ , $f (x) = x^{2}$ , is not onto.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerEquivalence relations

On the set of integers $Z$ , $R = {(a, b) : a - b is divisible by 5}$ . Show that $R$ is reflexive and symmetric.

Show worked answer

Answer: Shown

Reflexive: $a - a = 0 = 5 \times 0$ , so $(a, a) \in R$ for every integer $a$ A1
Symmetric: if $a - b = 5 m$ , then $b - a = 5 (- m)$ , so $(a, b) \in R \Rightarrow (b, a) \in R$ A1

Q5·2 marks·Short answerOne-one and onto functions

Show that $f : R \to R$ , $f (x) = x^{3} + 2$ , is one-one.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerTypes of relations

On $A = {1, 2, 3, \dots, 10}$ , $R = {(a, b) : a + b = 11}$ . Determine whether $R$ is reflexive, symmetric and transitive.

Show worked answer

Answer: Symmetric only

Not reflexive: $(a, a) \in R$ needs $2 a = 11$ , which no $a \in A$ satisfies (e.g. $(1, 1) \notin R$ ) A1
Symmetric: $a + b = 11 \Rightarrow b + a = 11$ A1
Not transitive: $(1, 10) \in R$ and $(10, 1) \in R$ , but $(1, 1) \notin R$ A1

Q7·3 marks·Short answerOne-one and onto functions

Prove that the function $f (x) = \frac{3 x - 2}{x + 1}$ , from $R - {- 1}$ to $R - {3}$ , is a bijection.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerEquivalence relations

On $A = {1, 2, 3, \dots, 14}$ , $R = {(a, b) : a - b is divisible by 3}$ . Show that $R$ is an equivalence relation. Write the equivalence class $[1]$ and state how many equivalence classes there are.

Show worked answer

Answer: Equivalence relation;

[1] = {1, 4, 7, 10, 13}

;

3

classes

Reflexive: $a - a = 0$ is divisible by $3$ B1
Symmetric: $a - b = 3 m \Rightarrow b - a = 3 (- m)$ B1
Transitive: $a - b = 3 m$ and $b - c = 3 n$ M1
$\Rightarrow a - c = 3 (m + n)$ , so $(a, c) \in R$ : $R$ is an equivalence relation A1
$[1] = {1, 4, 7, 10, 13}$ ; there are $3$ classes (remainders $0$ , $1$ , $2$ on division by $3$ ) A1

Q9·5 marks·Long answerOne-one and onto functions

(a) Show that $f : R \to R$ , $f (x) = 4 x + 3$ , is one-one and onto. (b) Show that $g : R \to R$ , $g (x) = x^{2} + 1$ , is neither one-one nor onto.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyOne-one and onto functions

Cinema seats. Five friends (Asha, Bina, Chetan, Dev and Esha) form the set $A$ , and the six seats $s_{1}, s_{2}, \dots, s_{6}$ of a row form the set $B$ . Each friend gets a different seat, so the seating is a one-one function $f : A \to B$ .

Can $f$ be onto? Give a reason. [1 mark]
How many one-one functions from $A$ to $B$ are there? [1 mark]
If the row had only five seats, how many one-one functions would there be, and would each be onto? [2 marks]

Show worked answer

No: $5$ friends can fill at most $5$ of the $6$ seats, so one seat has no pre-image A1
$6 \times 5 \times 4 \times 3 \times 2 = 720$ A1
$5 \times 4 \times 3 \times 2 \times 1 = 120$ one-one functions M1
Each fills all five seats, so each is also onto (a bijection) A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can test a relation for reflexive, symmetric and transitive and tell whether a function is one-one or onto.

Read first: 1. Relations; 3. Functions 13 practice questions · checkpoint: 4 questions, 7 marks, pass 80%

Practise step 1

Step 2

Board standard

You can prove equivalence relations, write equivalence classes, count relations and functions, and prove one-one or onto properly.

Read first: 1. Relations; 2. Equivalence classes; 3. Functions; Worked examples 1-3 14 practice questions · checkpoint: 4 questions, 11 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write the 5-mark relation and function proofs in full, with a heading and reason for each property and counter-examples where needed.

Read first: 2. Equivalence classes; 3. Functions; Board-exam tips 6 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle bijection proofs on restricted domains, tricky counting of relations and the hardest assertion-reason items.

Read first: 3. Functions; Common errors 6 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 42 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Assertion–reasonTypes of relations

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The relation $R = {(x, y) : x + y = 10}$ on $N$ is symmetric.

Reason (R): For all natural numbers $x, y$ , $x + y = y + x$ .

(a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c)Assertion (A) is true but Reason (R) is false.
(d)Assertion (A) is false but Reason (R) is true.

Q2·2 marks·Very short answerTypes of relations

Five seats in a row are numbered $1$ to $5$ . On $A = {1, 2, 3, 4, 5}$ define $a R b$ if seats $a$ and $b$ are the same seat or next to each other, i.e. $| a - b | \leq 1$ . Determine whether $R$ is reflexive, symmetric or transitive.

Q3·2 marks·Very short answerEquivalence relations and equivalence classes

On $Z$ , $a R b ⟺ | a | = | b |$ . Show that $R$ is an equivalence relation and write the equivalence class of $- 3$ .

Worked solution·2 marks·Very short answerOne-one and onto functions

For $f : R - {2} \to R - {k}$ , $f (x) = \frac{x + 1}{x - 2}$ , find the value of $k$ for which $f$ is onto.

Full worked solution Free, no sign-in

$y = \frac{x + 1}{x - 2} \Rightarrow x = \frac{2 y + 1}{y - 1}$ , defined for all $y \neq 1$ (and then $x \neq 2$ ) M1
$y = 1$ is never attained, so $k = 1$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original combinatorics problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S59 · Problem S10 · Problem S58 · Problem S09. All 22 →

Knowledge organiser: Relations and Functions on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Relations and Functions

Checking reflexivity only for elements that appear in R. Fix: check (a, a) for every a in the set A and say so.
Proving onto by solving for x without checking that x lies in the domain. Fix: after x = (4 − y)/3, write 'x ∈ ℝ for every y ∈ ℝ' (or check x ∈ ℕ, x ≠ 2, as needed).
Showing a property holds with an example instead of a proof. Fix: use general a, b, c for 'holds'; use one specific counter-example pair only for 'does not hold'.
Forgetting that one-one and onto depend on the domain and codomain. Fix: copy the domain and codomain into your first line and use them.
Transitive proofs that skip the algebra linking the two conditions. Fix: write 'aRb and bRc ⇒ … ⇒ aRc' with the working on each arrow.
Spending time on composition or inverse of functions. Fix: they are out of the current syllabus; focus on types of relations, equivalence classes and one-one/onto.

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Relations and Functions in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 1, 38
Sample paper 2 (Board standard): questions 1, 21, 36
Sample paper 3 (Above board): questions 1, 21, 38
Sample paper 4 (Above board): questions 1, 21, 36
Sample paper 5 (95+ challenge): questions 1, 21, 38

Want it against the clock? Take a timed 30-mark chapter test on Relations and Functions, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on sets, relations and functions.

Relations and Functions: common questions

What is in CBSE Class 12 Maths Chapter 1, Relations and Functions?

The topics are: Types of relations, Counting relations and functions, Equivalence relations and equivalence classes, One-one and onto functions. This page has revision notes for each, and 42 board-style practice questions with step-marked solutions.

How many marks is Relations and Functions worth?

It is in the Relations and Functions unit, which carries 8 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Relations and Functions?

Checking reflexivity only for elements that appear in R. Fix: check (a, a) for every a in the set A and say so. Proving onto by solving for x without checking that x lies in the domain. Fix: after x = (4 − y)/3, write 'x ∈ ℝ for every y ∈ ℝ' (or check x ∈ ℕ, x ≠ 2, as needed).

Are the Relations and Functions important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Relations and Functions — revision notes

1. Relations

2. Equivalence classes

3. Functions

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Relations and Functions Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Relations and Functions

Relations and Functions in our sample papers

Relations and Functions: common questions

Work through the Route to 95 with your progress saved