CBSE Important Questions Class 12 Maths Chapter 1 Relations and Functions

Relations and Functions study ordered pairs, mappings, equivalence relations, one-one functions, onto functions and inverse functions.
CBSE Important Questions Class 12 Maths Chapter 1 help students practise Relations and Functions through MCQs, proofs and case-based questions.

Class 12 Maths Chapter 1 begins with the idea that a relation connects elements through ordered pairs. The chapter then moves into proof-based checks for reflexive, symmetric and transitive relations. Students also learn how to identify equivalence relations, one-one functions, onto functions, bijective functions, composition and inverse functions. For the 2026-27 CBSE exam, this chapter is important because small changes in domain, co-domain or relation definition can change the full answer. The questions below follow the 80-mark Maths paper pattern and use simple steps for better board-style presentation.

Key Takeaways

• Equivalence Relation: A relation is an equivalence relation only when it is reflexive, symmetric and transitive.
• Function Type: A function is bijective only when it is both one-one and onto.
• Invertibility: A function is invertible only when there exists an inverse function from the co-domain to the domain.
• Composition: For two functions f and g, (g ∘ f)(x) means g(f(x)). First apply f, then apply g.

CBSE Important Questions Class 12 Maths Chapter 1 Structure 2026-27

Section A: MCQs from CBSE Important Questions Class 12 Maths Chapter 1

Section A has 20 one-mark questions. Relations and Functions Class 12 MCQs usually test definitions, examples, counterexamples and quick function checks.

Q1. A relation R in a set A is called reflexive if:

1. If (a,b) belongs to R, then (b,a) belongs to R
   b. (a,a) belongs to R for every a in A
   c. If (a,b) and (b,c) belong to R, then (a,c) belongs to R
   d. R is an empty relation

Answer: b. (a,a) belongs to R for every a in A

A reflexive relation must relate every element to itself.

Q2. A relation which is reflexive, symmetric and transitive is called:

1. Empty relation
   b. Universal relation
   c. Equivalence relation
   d. Inverse relation

Answer: c. Equivalence relation

An equivalence relation class 12 question always checks all three properties.

Q3. The empty relation on a non-empty set is:

1. Always reflexive
   b. Always symmetric
   c. Always universal
   d. Always one-one

Answer: b. Always symmetric

There is no pair that violates symmetry. It is not reflexive on a non-empty set.

Q4. What is a universal relation on set A?

1. A relation with no ordered pair
   b. A relation with only one ordered pair
   c. A relation containing all possible ordered pairs of A
   d. A relation containing only equal pairs

Answer: c. A relation containing all possible ordered pairs of A

For example, if A = {1, 2}, then the universal relation is {(1,1), (1,2), (2,1), (2,2)}.

Q5. Let R = {(1,1), (2,2), (1,2)} on A = {1,2}. Which property fails?

1. Reflexive
   b. Symmetric
   c. Both reflexive and symmetric
   d. Transitive

Answer: b. Symmetric

(1,2) belongs to R, but (2,1) does not belong to R.

Q6. The relation R = {(a,b): a = b} on a set A is:

1. Reflexive only
   b. Symmetric only
   c. Transitive only
   d. Equivalence relation

Answer: d. Equivalence relation

Equality is reflexive, symmetric and transitive.

Q7. A function f from X to Y is one-one if:

1. Every element of Y has a preimage
   b. Different elements of X have different images
   c. Range is always empty
   d. Domain and co-domain are equal

Answer: b. Different elements of X have different images

One one and onto functions are tested through image comparison.

Q8. A function f from X to Y is onto if:

1. Range of f is equal to Y
   b. Domain of f is equal to Y
   c. f(x) = 0 for all x
   d. f is constant

Answer: a. Range of f is equal to Y

Onto means every element of the co-domain is covered.

Q9. The function f from R to R defined by f(x) = 2x is:

1. One-one but not onto
   b. Onto but not one-one
   c. One-one and onto
   d. Neither one-one nor onto

Answer: c. One-one and onto

For every real number y, x = y/2 gives f(x) = y.

Q10. The function f from N to N defined by f(x) = 2x is:

1. One-one but not onto
   b. Onto but not one-one
   c. Bijective
   d. Neither one-one nor onto

Answer: a. One-one but not onto

Odd natural numbers are not images of any natural number x.

Q11. The function f from R to R defined by f(x) = x² is:

1. One-one and onto
   b. One-one but not onto
   c. Onto but not one-one
   d. Neither one-one nor onto

Answer: d. Neither one-one nor onto

f(1) = 1 and f(-1) = 1, so it is not one-one. Negative real numbers are not images.

Q12. If f from X to Y is both one-one and onto, then f is called:

1. Constant
   b. Bijective
   c. Empty
   d. Universal

Answer: b. Bijective

A bijective function is both injective and surjective.

Q13. If f maps A to B and g maps B to C, then g ∘ f maps:

1. A to C
   b. C to A
   c. B to A
   d. C to B

Answer: a. A to C

Since f maps A to B and g maps B to C, their composition maps A to C.

Q14. If f(x) = x + 1 and g(x) = 2x, then (g ∘ f)(x) equals:

1. 2x + 1
   b. 2x + 2
   c. x + 3
   d. 2x - 1

Answer: b. 2x + 2

(g ∘ f)(x) = g(f(x))
= g(x + 1)
= 2(x + 1)
= 2x + 2

Q15. A function is invertible if:

1. It is only one-one
   b. It is only onto
   c. It is bijective
   d. It is constant

Answer: c. It is bijective

Invertible functions class 12 questions often require proving one-one and onto.

Q16. If f from X to Y is invertible, then its inverse maps:

1. Y to X
   b. X to Y
   c. X to X only
   d. No set to any set

Answer: a. Y to X

The inverse reverses the direction of the original function.

Q17. Let R on Z be defined by “a is related to b if 2 divides a - b”. Then R is:

1. Reflexive only
   b. Symmetric only
   c. Transitive only
   d. Equivalence relation

Answer: d. Equivalence relation

Divisibility by 2 satisfies reflexive, symmetric and transitive properties.

Q18. If R is an equivalence relation on A, it divides A into:

1. Equal functions
   b. Equivalence classes
   c. Empty domains
   d. Non-functions

Answer: b. Equivalence classes

Equivalence classes form mutually disjoint partitions of the set.

Q19. Assertion: The relation “is parallel to” on the set of all lines in a plane is an equivalence relation.

Reason: It is reflexive, symmetric and transitive.

1. Both Assertion and Reason are true, and Reason 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

Answer: a. Both Assertion and Reason are true, and Reason explains Assertion

Every line is parallel to itself in this relation, and the relation is symmetric and transitive.

Q20. Assertion: The function f from R to R given by f(x) = x³ is one-one.

Reason: If x₁³ = x₂³, then x₁ = x₂ for real numbers.

1. Both Assertion and Reason are true, and Reason 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

Answer: a. Both Assertion and Reason are true, and Reason explains Assertion

The cube function is injective on R.

Section B: Very Short Answer Questions from Relations and Functions Class 12 Important Questions

Section B has 2-mark VSA questions. Answers should show the key test or calculation in minimum steps.

Q21. Define an empty relation and universal relation.

An empty relation in a set A is a relation where no element of A is related to any element of A.

So, R has no ordered pair.

A universal relation in A is a relation where every element of A is related to every element of A.

For example, if A = {1, 2}, then the universal relation is {(1,1), (1,2), (2,1), (2,2)}.

Q22. Check whether R = {(1,1), (2,2), (1,2)} on A = {1,2} is reflexive.

Yes, R is reflexive.

For reflexivity, (a,a) must belong to R for every element a in A.

Here, (1,1) belongs to R and (2,2) belongs to R.

So, R is reflexive.

Q23. Show that f from R to R given by f(x) = 3x is one-one.

Let f(x₁) = f(x₂).

Then, 3x₁ = 3x₂.

Dividing both sides by 3, x₁ = x₂.

Hence, different inputs have different images.

Therefore, f is one-one.

Q24. Find (f ∘ g)(x) if f(x) = x² and g(x) = x + 2.

(f ∘ g)(x) = f(g(x))

= f(x + 2)

= (x + 2)²

= x² + 4x + 4

So, (f ∘ g)(x) = x² + 4x + 4.

Q25. State the condition for a function to be invertible.

A function f from X to Y is invertible if it is one-one and onto.

It means every element of Y has exactly one preimage in X.

The inverse function maps Y back to X.

Section C: Short Answer Questions from Class 12 Maths Chapter 1 Relations and Functions

Section C has 3-mark questions. These answers need clear property checks or step-by-step function testing.

Q26. Show that the relation R on Z defined by “2 divides a - b” is an equivalence relation.

For reflexive property:

a - a = 0.

Since 2 divides 0, (a,a) belongs to R.

For symmetric property:

Let (a,b) belong to R.

Then 2 divides a - b.

So, 2 also divides b - a.

Hence, (b,a) belongs to R.

For transitive property:

Let (a,b) and (b,c) belong to R.

Then 2 divides a - b and 2 divides b - c.

So, 2 divides (a - b) + (b - c).

That means 2 divides a - c.

Hence, (a,c) belongs to R.

Therefore, R is an equivalence relation.

Q27. Check whether f from N to N given by f(x) = x² is one-one and onto.

Let f(x₁) = f(x₂).

Then, x₁² = x₂².

Since x₁ and x₂ are natural numbers, x₁ = x₂.

So, f is one-one.

To check onto, take 2 in N.

There is no natural number x such that x² = 2.

So, 2 has no preimage in N.

Therefore, f is one-one but not onto.

Q28. Show that the relation R in R defined by “a is related to b if a ≤ b” is reflexive and transitive but not symmetric.

For reflexive property:

a ≤ a for every real number a.

So, (a,a) belongs to R.

For transitive property:

Let (a,b) and (b,c) belong to R.

Then, a ≤ b and b ≤ c.

So, a ≤ c.

Hence, (a,c) belongs to R.

It is not symmetric because 2 ≤ 3 is true, but 3 ≤ 2 is false.

Therefore, R is reflexive and transitive but not symmetric.

Q29. Find (g ∘ f)(x) and (f ∘ g)(x) if f(x) = 2x + 1 and g(x) = x².

First,

(g ∘ f)(x) = g(f(x))

= g(2x + 1)

= (2x + 1)²

= 4x² + 4x + 1

Now,

(f ∘ g)(x) = f(g(x))

= f(x²)

= 2x² + 1

Hence, (g ∘ f)(x) = 4x² + 4x + 1 and (f ∘ g)(x) = 2x² + 1.

Q30. Show that f from R to R defined by f(x) = x³ is one-one.

Let f(x₁) = f(x₂).

Then, x₁³ = x₂³.

Taking cube root on both sides, x₁ = x₂.

Thus, equal images imply equal preimages.

Hence, f is one-one.

Q31. Prove that the modulus function f from R to R, f(x) = |x|, is neither one-one nor onto.

To check one-one:

f(1) = |1| = 1.

f(-1) = |-1| = 1.

Since 1 and -1 are different inputs with the same image, f is not one-one.

To check onto, take -2 in R.

There is no real value of x for which |x| = -2.

So, -2 has no preimage.

Therefore, f is neither one-one nor onto.

Section D: Long Answer Questions from CBSE Class 12 Maths Relations and Functions

Section D has 5-mark questions. These questions need complete proofs, correct notation and proper conclusion.

Q32. Show that the relation R in A = {1,2,3,4,5}, defined by “|a - b| is even”, is an equivalence relation. Also find the equivalence classes.

For reflexive property:

For every a in A, |a - a| = 0.

Since 0 is even, (a,a) belongs to R.

So, R is reflexive.

For symmetric property:

Let (a,b) belong to R.

Then, |a - b| is even.

But |b - a| = |a - b|.

So, |b - a| is also even.

Hence, (b,a) belongs to R.

So, R is symmetric.

For transitive property:

If (a,b) belongs to R and (b,c) belongs to R, then a and b have the same parity.

Also, b and c have the same parity.

Therefore, a and c have the same parity.

So, |a - c| is even.

Hence, (a,c) belongs to R.

Therefore, R is an equivalence relation.

Equivalence classes:

[1] = {1, 3, 5}

[2] = {2, 4}

Q33. Check whether f from R to R defined by f(x) = 3 - 4x is one-one and onto.

To check one-one:

Let f(x₁) = f(x₂).

Then, 3 - 4x₁ = 3 - 4x₂.

Subtracting 3 from both sides:

-4x₁ = -4x₂.

Dividing by -4:

x₁ = x₂.

So, f is one-one.

To check onto:

Take any real number y.

Let y = 3 - 4x.

Then, 4x = 3 - y.

So, x = (3 - y)/4.

Since x is a real number for every real number y, every y has a preimage.

So, f is onto.

Therefore, f is one-one and onto.

Q34. Let f from R to R and g from R to R be given by f(x) = cos x and g(x) = 3x². Find (g ∘ f)(x) and (f ∘ g)(x). Show that g ∘ f is not equal to f ∘ g.

First,

(g ∘ f)(x) = g(f(x))

= g(cos x)

= 3cos²x

Now,

(f ∘ g)(x) = f(g(x))

= f(3x²)

= cos(3x²)

To compare, take x = 0.

(g ∘ f)(0) = 3cos²0 = 3

(f ∘ g)(0) = cos 0 = 1

Since (g ∘ f)(0) is not equal to (f ∘ g)(0), g ∘ f is not equal to f ∘ g.

Q35. Let f from N to Y be defined by f(x) = 4x + 3, where Y contains all natural numbers of the form 4x + 3. Show that f is invertible and find f⁻¹.

Given f(x) = 4x + 3.

Let y = 4x + 3.

Then, x = (y - 3)/4.

Define g from Y to N by g(y) = (y - 3)/4.

Now,

(g ∘ f)(x) = g(f(x))

= g(4x + 3)

= (4x + 3 - 3)/4

= x

So, g ∘ f is the identity function on N.

Also,

(f ∘ g)(y) = f(g(y))

= f((y - 3)/4)

= 4((y - 3)/4) + 3

= y - 3 + 3

= y

So, f ∘ g is the identity function on Y.

Hence, f is invertible and f⁻¹(y) = (y - 3)/4.

Section E: Case Study-Based Questions from CBSE Class 12 Maths Chapter 1

Section E has 4-mark case study questions. These questions usually test a definition through sub-parts.

Q36. Case Study: Roll Numbers as a Function

A school assigns roll numbers to 50 students of Class 12. Let A be the set of all 50 students and N be the set of natural numbers. Define f from A to N by:

f(x) = roll number of student x.

Answer the following questions.

Q36(a). Is f one-one?

Yes, f is one-one.

No two students can have the same roll number.

Q36(b). Is f onto N?

No, f is not onto N.

For example, 51 is a natural number, but it is not the roll number of any student in the class.

Q36(c). If the co-domain is changed to {1,2,3,...,50}, is f bijective?

Yes, f becomes bijective.

Every roll number from 1 to 50 is assigned to exactly one student.

Q37. Case Study: Parallel Lines as a Relation

Let L be the set of all lines in the XY plane. Define a relation R on L as:

Line L₁ is related to line L₂ if L₁ is parallel to L₂.

Answer the following questions.

Q37(a). Is R reflexive?

Yes, R is reflexive.

Every line is parallel to itself.

Q37(b). Is R symmetric?

Yes, R is symmetric.

If L₁ is parallel to L₂, then L₂ is parallel to L₁.

Q37(c). Is R an equivalence relation?

Yes, R is an equivalence relation.

It is reflexive, symmetric and transitive.

Q37(d). Find the set of all lines related to y = 2x + 4.

All lines related to y = 2x + 4 must be parallel to it.

Parallel lines have the same slope.

Since the slope of y = 2x + 4 is 2, all related lines are of the form:

y = 2x + c, where c is any real number.

Q38. Case Study: Composition of Functions

A coding club uses two operations on a number x. First, function f doubles the number and adds 1. Second, function g squares the input.

Let f(x) = 2x + 1.

Let g(x) = x².

Answer the following questions.

Q38(a). Find (g ∘ f)(x).

(g ∘ f)(x) = g(f(x))

= g(2x + 1)

= (2x + 1)²

= 4x² + 4x + 1

Q38(b). Find (f ∘ g)(x).

(f ∘ g)(x) = f(g(x))

= f(x²)

= 2x² + 1

Q38(c). Are g ∘ f and f ∘ g equal?

No, g ∘ f and f ∘ g are not equal.

(g ∘ f)(x) = 4x² + 4x + 1

but

(f ∘ g)(x) = 2x² + 1.

Q38(d). Find (g ∘ f)(1) and (f ∘ g)(1).

(g ∘ f)(1) = 4(1)² + 4(1) + 1 = 9.

(f ∘ g)(1) = 2(1)² + 1 = 3.

So, (g ∘ f)(1) = 9 and (f ∘ g)(1) = 3.

Important Questions Class 12 Maths Chapter-Wise