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CBSE · Class 12 · Mathematics

Relations and Functions

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The number of relations on a set with 1 element is:
  1. 2
  2. 3
  3. 1
  4. 4
2.
The number of bijective functions from a set with 6 elements to itself is:
  1. 720
  2. 46,656
  3. 64
  4. 36
3.
If f(x) = −3x − 2 and g(x) = −2x + 8, then (g ∘ f)(x) is:
  1. 6x + 12
  2. 6x − 16
  3. 6x − 26
  4. −5x + 6
4.
The inverse of f: R → R, f(x) = 8x + 9 is f−1(x) =
  1. x/8 − 9
  2. (x + 9)/8
  3. (x − 9)/8
  4. 8x − 9
5.
A relation R on A is reflexive when:
  1. (a, b) ∈ R for all a, b
  2. (a, a) ∈ R for every a ∈ A
  3. (a, b) ∈ R implies (b, a) ∈ R
  4. R is empty
6.
The relation R = {(1, 1), (1, 3), (2, 2), (3, 1), (3, 3)} on A = {1, 2, 3} is:
  1. an equivalence relation
  2. reflexive and symmetric, but not transitive
  3. reflexive only
  4. symmetric and transitive, but not reflexive
7.
The function f: R → [0, ∞), f(x) = x2 is:
  1. neither one-one nor onto
  2. one-one but not onto
  3. onto but not one-one
  4. one-one and onto
8.
Assertion (A): The relation R = {(a, b) : a − b > 3} on Z is an equivalence relation.
Reason (R): A relation is an equivalence relation when it is reflexive, symmetric and transitive.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Determine whether the relation R = {(1, 1), (1, 4), (2, 2), (3, 2), (3, 3)} on A = {1, 2, 3, 4} is reflexive, symmetric and transitive.
10.
R = {(a, b) : a − b is divisible by 4} is an equivalence relation on A = {0, 1, 2, …, 8}. Find the equivalence class [0].
11.
If f(x) = 3x + 5 and g(x) = x2 + 7, find (f ∘ g)(x) and (g ∘ f)(x).
12.
Find the inverse of f(x) = (4x − 1)/(2x − 2), x ≠ 1.
13.
A has 4 elements and B has 5 elements. Find the number of (a) relations from A to B, (b) functions from A to B, (c) one-one functions from A to B.
14.
f: {1, 2, 3, 4} → {a, b, c, d, e} is given by f = {(1, d), (2, d), (3, e), (4, a)}. Is f one-one? Is f onto?
15.
If f(x) = x2 − 3 and g(x) = −2x + 6, find (f ∘ g)(1) and (g ∘ f)(1).
16.
A relation R on A = {1, 2, …, 7} is defined by R = {(x, y) : y = 2x}. Write R in roster form and find its domain and range.
Section C — Word problems4 × 4 = 16 marks
17.
Show that R = {(a, b) : |a − b| is a multiple of 3} is an equivalence relation on A = {1, 2, …, 12}, and write all its equivalence classes.
18.
Let f(x) = 5x − 6 and g(x) = x2 − 4. Find (g ∘ f)(x) and (f ∘ g)(x). Is g ∘ f = f ∘ g? Find (g ∘ f)(2) and (f ∘ g)(2).
19.
Show that f: R − {−7/4} → R − {1}, f(x) = (4x − 5)/(4x + 7) is one-one and onto, and find f−1.
20.
A = {1, 2} and B has 2 elements. Find the number of (a) reflexive relations on A, (b) symmetric relations on A, (c) onto functions from B to A, (d) bijections from A to A.
AmbifulWorksheets
Practice that never repeats
CBSE · Class 12 · Mathematics

Answer key

Relations and Functions · AW-12M-RAF-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(a) 22^(1 × 1) = 21
2(a) 7206! = 6 × 5 × 4 × 3 × 2 × 1
3(a) 6x + 12g(f(x)) = −2(−3x − 2) + 8
4(c) (x − 9)/8y = 8x + 9 ⇒ x = (y − 9)/8
5(b) (a, a) ∈ R for every a ∈ A—
6(a) an equivalence relationreflexive: every (a, a) is in R; symmetric; transitive
7(c) onto but not one-onef(2) = f(−2), but every non-negative value is reached
8(d) A is false, but R is true.A is false: (a, a) ∉ R because 0 > 3 is false, so R is not reflexive. R is true.
9transitive onlynot reflexive: (4, 4) ∉ R; not symmetric: (1, 4) ∈ R but (4, 1) ∉ R; transitive
10{0, 4, 8}the elements that leave remainder 0 on division by 4
11(f ∘ g)(x) = 3x2 + 26; (g ∘ f)(x) = 9x2 + 30x + 32f(g(x)) = 3(x2 + 7) + 5; g(f(x)) = (3x + 5)2 + 7
12f−1(x) = (2x − 1)/(2x − 4), x ≠ 2y(2x − 2) = 4x − 1 ⇒ x(2y − 4) = 2y − 1
13(a) 220 = 1,048,576; (b) 625; (c) 1202^(4 × 5); 54; 5 × 4 × 3 × 2
14Not one-one; not ontof(1) = f(2) = d; b, c have no pre-image
15(f ∘ g)(1) = 13; (g ∘ f)(1) = 10g(1) = 4, f(1) = −2
16R = {(1, 2), (2, 4), (3, 6)}; domain {1, 2, 3}; range {2, 4, 6}list every pair that satisfies the rule
17Reflexive: |a − a| = 0 is a multiple of 3. Symmetric: |a − b| = |b − a|. Transitive: a − c = (a − b) + (b − c). Classes: {1, 4, 7, 10}, {2, 5, 8, 11}, {3, 6, 9, 12}group the numbers by their remainder on division by 3
18(g ∘ f)(x) = 25x2 − 60x + 32; (f ∘ g)(x) = 5x2 − 26; not equal; (g ∘ f)(2) = 12, (f ∘ g)(2) = −6substitute one function into the other
19f(x1) = f(x2) gives (48)(x1 − x2) = 0, so x1 = x2 (one-one); every y ≠ 1 has the pre-image x = (7y + 5)/(−4y + 4) (onto). f−1(y) = (7y + 5)/(−4y + 4)cross-multiply: ad − bc = 48 ≠ 0
20(a) 4; (b) 8; (c) 2; (d) 2(a) 2^(n2 − n); (b) 2^(n(n + 1)/2); (c) 22 minus the maps that miss some element (inclusion–exclusion); (d) 2!