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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-01-CT
Relations and Functions — Full Chapter Test
Chapter: Relations and Functions
Topic: Complete chapter — all topics
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • This is a full-length test covering the whole chapter — every topic is included.
  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
On $A=\{1,2,3\}$, the relation $R=A\times A$ is called the:
  • A.empty relation
  • B.universal relation
  • C.identity relation
  • D.void relation
2.
The function $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^3$ is:
  • A.one-one but not onto
  • B.onto but not one-one
  • C.bijective
  • D.neither
3.
If $f(x)=x+2$ and $g(x)=3x$, then $(g\circ f)(1)=$
  • A.$5$
  • B.$9$
  • C.$6$
  • D.$3$
4.
The number of relations that can be defined on a set with $4$ elements is:
  • A.$16$
  • B.$256$
  • C.$2^{16}$
  • D.$4^{4}$
5.
$f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2$ is:
  • A.one-one
  • B.onto
  • C.bijective
  • D.neither one-one nor onto
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Define a reflexive relation and give one example on the set $\{1,2,3\}$.
7.
Show that $f:\mathbb{R}\to\mathbb{R},\ f(x)=2x+5$ is one-one.
8.
If $f(x)=x^2$ and $g(x)=x+1$, find $(f\circ g)(2)$.
9.
Let $R=\{(1,2),(2,1),(1,1),(2,2)\}$ on $\{1,2\}$. State, with reason, whether $R$ is symmetric.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Check whether the relation $R$ on $\mathbb{R}$ given by $a\,R\,b \iff a\le b$ is reflexive, symmetric and transitive.
11.
Show that $f:\mathbb{R}\to\mathbb{R},\ f(x)=2x+3$ is a bijection.
12.
If $f(x)=2x+1,\ g(x)=x^2$, find $(g\circ f)(x)$ and $(f\circ g)(x)$.
13.
On $A=\{1,2,3,4\}$, let $R=\{(a,b):a+b\text{ is even}\}$. Determine the type of relation.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Prove that the relation $R$ on $\mathbb{Z}$ defined by $a\,R\,b \iff (a-b)$ is divisible by $5$ is an equivalence relation. Hence write its equivalence classes.
15.
Show that $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2$ is neither one-one nor onto, and give a restriction of domain and codomain that makes it bijective.

Answer Key

Section A — Multiple Choice Questions
  1. (B) universal relation
  2. (C) bijective
  3. (B) $9$
  4. (C) $2^{16}$
  5. (D) neither one-one nor onto
Section B — Short Answer (2 marks)
  1. A relation $R$ on $A$ is reflexive if $(a,a)\in R$ for every $a\in A$. Example: $R=\{(1,1),(2,2),(3,3)\}$.
  2. One-one.
  3. $9$.
  4. Yes, $R$ is symmetric.
Section C — Short Answer (3 marks)
  1. Reflexive and transitive, but not symmetric (hence not an equivalence relation).
  2. Bijective.
  3. $(g\circ f)(x)=(2x+1)^2$; $(f\circ g)(x)=2x^2+1$.
  4. Equivalence relation; classes $\{1,3\}$ and $\{2,4\}$.
Section D — Long Answer (5 marks)
  1. Equivalence relation; equivalence classes $[0],[1],[2],[3],[4]$.
  2. Neither on $\mathbb{R}$; bijective as $f:[0,\infty)\to[0,\infty)$.
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