RELATIONS AND FUNCTIONS 1.

61

COMPETENCY BASED QUESTIONS

IMCQ's-Very Short Answer Type-Assertion-Reason Based-Case Study Based Questions]

TypeI- Multiple Choice Questions:

1 Let A =|1, 2, 3} and R be relation on Agiven by R= ((1, 2)). Then R is :
in) reflexive, symmetric but not transitive (6) neither reflexive, symmetricnor transitive
(c) only transitive. (d) only symmetric
9. IfR=(x, y):x,y e Z, x+ y' s4) is a relation in set Z, then domain of Ris
(a) {0, 1, 2) (6) {2, -1,0, 1, 2} (c) {0, 1, 2} (d) {-1, 0, 1)
[CBSE 2022]
9. Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm iffl 1s
perpendicular to m for alll, m eL. Then R is
(a) reflexive (6) symmetric (c) transitive (d) none of these

A. Let Rbe the relation in the set Ngiven byR =(a, b) :a=b -2, b> 6). Then :
(a) (2, 4) e R (6) (3, 8)e R (c) (6, 8) e R (d) (8, 7) e R
[CBSE Sample Paper 2021]
relation R defined as
5. Consider the non-empty set consisting of children in a family and a
aRb ifa is brother of b. Then R is
(6) an equivalence relation
(a) symmetric but not transitive
(c) only symmetric (d) only transitive.
iffn divides m. Then R is
6. Let R be a relation on the set N of natural numbers defined by n Rm
(a) reflexive and symmetric (6) transitive and symmetric
(c) equivalence relation (d) reflexive, transitive but not symmetric
[NCERTEXEMPLAR]

7. Let A= (3, 5). Then number of reflexive

relations on A is
(c) 0 (d) 8 [CBSE 2023]
(a) 2 (6) 4
R={(1, 3), (2, 2), (3, 2)). Then minimum ordered
8. Let set X= (1,2, 3) anda relation R is defined in Xas symmetric are
pairs which should be added in relation R to make it reflexive and
(a) ((1, 1), (2, 3), (1, 2)) (b) (3, 3), (3, 1), (1, 2)}
(d) ((1, 1), (3, 3), (3, 1), (1, 2)} [CBSE 2022|
(c) {(1, 1), (3, 3), (3, 1), (2, 3))
((1, 1), (1, 2), (2, 2), (3,3)). Which of the following
9. A relation R in set A = (1, 2, 3) is defined as R= equivalence relation in A?
ordered pair in R shall be removed to make it an
(c) (2, 2) (d) (3, 3)
(a) (1, 1) (6) (1, 2)
[CBSE Sample Paper 2021|
relations on A containing (1, 3)is
10. IfA = (1, 3,5), then the number of equivalence CBSE 2020|
(c) 4 (d) 5
(a) 1 (b) 2
R:x>0} and let S =((x, y) e A x B:x' +y' =1) and
I. Let A=B= (x e R:-1<x<1l and C= lxe
S, ={(r,y) e AxC:x2+y² =1). Then
(a) S defines a function from A to B (6) S, defines a function from AtoC
(d) none of these
(c) S,defines a function from A to B
 1.62 ELEMENTS OF MATHEMATICS-(XI)

12. Iffis any function, then -ix) +f-x)) is always

2
(a) even (b) odd (c) neither even nor odd
13. Afunction f: R’ Rdefined byf(x) = 2 +r*is (d) one-0ne
(a) not one-one (b) one-one
(c) not onto (d) neither one-one nor onto
14. The function f: [0, ) ’ Rgiven by f(x) =is CBSE 2022)
x+1
(a) one-one and onto (6) one-one but not onto
(c) onto but not
one-one (d) neither one-one nor onto
15. The functionf:R’ R:f(x) = cos x
is
(a) one-0ne and into
(6) one-one and onto
(c) many one and into
(d) many one and onto
16. Let f be function on R given by f(x) =(),
where []represent greater integer function. Then fis
(a) one-0ne
(b) onto
(c) bijective
(d) neither one-one nor onto.
17. IfA ={1, 2,3,.., n), n> 2 and B= (a, b}.,
then the number of onto functions from A to B is
(a) "P (b) 2r 2 (c) 2 -1 (d) none of these
18. If the function f:R’Agiven by
fr) = is a surjection, then A is equal to
(a) R
(6) [0, 1] (c) (0, 1)
19. LetX= x:xe N) and the (d) (0, 1)
function f: N ’X is defined by f(x) =x',xe N. Then
(a) injective only (b) not bijective
this function is
(c) surjective only (d) bijective
20. If the set A contains 5 (CBSE 2022)
elements and the
one-one and onto mappings from set A to set Bset B contains 6
is elements, then the number of
(a) 720 (b) 120
(c) 0 (d) none of these
1 (CBSE Sumple Paper 2024
21. Let flx)= , then fo (fof))(x) is equal to
1-x
(a) xfor all xeR
(b) xfor all x e R-{1}
(c) x for all x e R-{0, 1}
(d) none of these
3x +2
22. If f(x) = then
5x-3

(a) f-x)=fu) (b) fx) =-flz) 1

(c) ffx)) =-x
Type II-Very Short Answers Questions:
93 Let R be a
relation in N (natural numbers) given by a
94. If R= {(x, v) :x+2y = 8} 1s Rb if 2a + 36 =30. Find R.
a relation on N, then ICBSE2014C]
25. Let R={(a, a):ais a find the range of R.
prime number less than 10}. Find R.
26. A relation in a set A is called
relation, if each element of Aiis related to itselfonly.221l
Paper
(CBSE Sample.
 RELATIONS AND PUNCTIONS I.63

27. |fotA) , o(B3) =ln, then find the total numbey of relationn from Ato B

mnany relexive relationn are poxaible in n net. A whoHe nA) 3 ICISE umple Pajer 201211
28. How
the reason why the relation R= ((u, b) : a 64 0n the set t of renl numbers i# no.
9. State
reflexive. |CISE Sumple Iuper 200171
reason fo the relation Rin the set (1, 2,3) given by It 0, 2), (2, D) 1ot to be transtiv
30. State the CLSE 2011|
31. Showthat the relntion "le8N than or equal to" on I, in reflexive but not eyDmetre.
32. Whatis the maximum number of equivalenee relations on et A=|1, 2, 3).
the cquivalence relation in Lhe set A =(0, 1, 2, 3, 4, 5) given by R= u, b):2 divide
83. Let R be |CISE 2014CI
a -b)). Then find the equivalence class lol
relation Rin Adivides it into cquivalence classes A,, A.,, A, What is the value of
34. An equivalence |CBSE Sample laer 2021|
A, UA, UA, and A, A,nA,
or Which of the following does not represents afunction f:R, R? Give reaon (i) y =x (u) y X
function
O£ State whether relation R in the set of real numbers R defined a8 R-(a. b): u- b} is a
or not. Justify. (CBSE Sumple Puper 20021|
g What is the number of all one-one functions from the set A =la.6. cl to iteelf.
|CBSE Sample Paper 2010|
be a function from A to B. Show that
38. Let A=(1, 2, 3}and B =4, 5, 6, 7} and letf= (1,4), (2, 5), (3, 6)) |CBSE 2011|
fis one-one.
One-one and onto?
39. Is the functionf: (0,o) ’ R given by f(x) -= x+1
fis one-one ont0.
40. Let f: R’ R be defined as f(x)= 3x. Show that
number of one-one functions from A to B.
41. Ifo(A) = 4, o(B)= 3, then write the
=7, then find the number of injective functions that can be defined from set A to B
42. Ifo(A) = 5, o(B)
|CBSE 2008)
then find (fo g)(7).
43. If f() =x+7and g(x) =x-7,x eR,
44. Iff(x)= x*-1, findffc), x + , - 1.
+1 function defined as
greatest integer function defined as x) = x] and g be the modulus
45. Iff be the
|CBSE Sample Paper 20201
gl*) =|x|,then find the value of gofl- |CBSE 20191
46. Iff(x) =x + 1, find (fof) (x).
defined by gx) Find gof
f(x)= 3r2-5 andg: R ’ Rbe
T. Letf:R’Rbe defined by ICBSESample Paper 20171|
off
is an invertible function, then find the inverse
defined by f()=3x 4
une Tunctionf:R-> R CBSE 2010|

7:Qis given byf(a) =r2. then findf(16). ICBSE2012C, 08CI

f 1
2x-7
is an invertible function, find
50. Iff:R’ Rdefined as f(x)=
4
4 then findf (*).
51. Iff:R-{0) -> R-(0} is defined as fx) =
 1.64 ELEMENTS OF MATHEMATICS-N

52. Iff: R--1 ’R-(1) be defined asf x)= then findf-' (r).
I-1

53. Iff R’Ris given by f(x=+5. then findf- (x).

54. State with reason whether function f:(1.2.3. 4) ’ {10} where f= {(1, 10), (2, 10) (3
inverse 0r not.

Type III - Assertion-Reasoning Questions :

10),(4, 10have
In the following quuestions a statement of assertion (A) is followed by a statement
Choose the correct option from the given options. of
a) Both Assertion (A) and Reason R) are true and Reason (R) is the
reason (R
Assertion A). correct explanation of
(b) Both Assertion (A) and Reason R) are true but Reason R) is not
Assertion (A). the correct
(c) Assertion (A) is true but Reason (R) is
explanation
ot
false.
(d) Assertion (A) is false but
Reason (R) is true.
1. Assertion : Arelation R={(1.1), (1.2).
(2.2), (2, 3), (3,3)) defined on the set A=(1, 2,3) is
Reason:A relation R on a non-empty set Ais said to be refleyiw.
2. Assertion : R= {<, y):
reflexive (a,a) eRfor all ae A
if
lx-y| = 0} defined on the set A =(3, 5, 7) is
Reason:A relation R on a non-empty set A is said to be symmetric.
(b, a) =R for all a, b e A. symmetric if for each (a, b) R. we have
3. Assertion : The
relation R= {(a, b):la -b|<2) defined on set A = {1, 2,
3,4, 5) is reflexive.
Reason:Arelation R on a non-empty set A is reflexive if (a, b) e
4. Assertion: If X= {0, 2, 4, 6, R ’ (6,a) eR for all a, be R.
8} and P is a relation on X
defined by
P={(0, 2), (4, 2), (4,6), (8, 6), (2, 4), (0, 4), then
Reason: The relation P has a subset of the form (a,relation P on set Xis a transitive relation.
5. Assertion: The b), (b, c), (a, c)}, where a, b,c e
relation Rin the set A= {1,2, 3, 4, 5, 6) X.
equivalence relation. defined as R= {(x. y): y is divisible by x1S at
Reason:Arelation R on the set Ais
equivalence relation if it is
6. Assertion: If the relation R defined in A={1, 2, 3, 4, 5} by a reflexive, symmetric and trans1t1ve
Reason:For the given relation, domain of R-l= Range of R. Rb, if la2-62| <7, thenRz K.
7. Assertion:
Domain and range of a relation R=
A=(1, 2, 3, 4, 5,6, 7, 8} are (x,y):xe A, ye A
(1, 2, 3, 4) and (2,4, 6, 8). and x-2v = 0} defined on
Reason:
respectively
Domain and range of a relation R are
la:aeA and (a, b) eR) and {b :be Aandrespectively
(a. b)eR}
the sets
8. Assertion: f:N’N, f(x) = 2x is a
bijection.
Reason:Afunction is said to be bijective if it is
one-one and onto.
9. Assertion : If
f:0,’R is defined as f(x) = sin x and >R is
defined as
gx) = cOs x, then f andg are
one-one.
ReasoOn:(f+g) is not one-one.
 RELATIONS AND FUNCTIONS
1.65

10. Assertion : The relation f: (1, 2, 3, 4} ’ {x, y, z, p} defined by f= {(1, x), (2, y), (3, z)} is a bijective
function. [CBSE Sumple Paper 2024]
Reason: The functionf:(1,
2, 3) ’ k, y, z, p) such that f= (1L,: x), (2, y), (3, z)} is one-one.
11. Assertion: The greatest integer function f: R’Rgiven by f(x)= [x] is not one-one.
Reason:Afunctionf:A’Bis said to be injective if f(a) =f(b) ’ a=b.
12. Assertion:f::Q’Q, f(u)=:1+2x for all x e Qis
bijective.
Reason:f()1s one-one and onto function
I-2
Aertion : For two sets A = R-(3} and B= R-(11., the functionf: A ’ Bdefined as f(x) = 1s
x-3
bijective.
Poason: A function f : A ’ Bis said to be onto if for all y eB. there exists x e A such that
flx)=y.

ANSWERS
Type I- Multiple Choice Questions
1. (c) 2. (b) 3. (b) 4. (c) 5. (d) 6. (d)
7. (b) 8. (c) 9. (6) 10. (6) 11. (6) 12. (a)

13. (d) 14. (6) 15. (c) 16. (d) 17. (b) 18. (d)
19. (d) 20. (c) 21. (c) 22. (a)

Type II- Very Short Answers Questions

23. R= {(3, 8), (6, 6), (9, 4), (12,2)} 24. {1, 2, 3} 25. R= ((2, 8), (3, 27), (5, 125), (7, 343)}
27. 2mn 28. 26
26. Identity
29. Sinceasa' is not true for 0 <a< 1 30. (1, 2) and (2, 1) e Rbut (1, 1) R
32. 5 33. {0, 2, 4) 34. A, UA, UA, = Aand A,n A, nA,=¢
35. y'=xdoes not represent a function, since each value of domain is not uniquely associated with a
value from co-domain.
36. Not a function, as /a is not defined fora e (- o, 0) 37. 3! =6
39. One-one but not onto 41. 0 42. P.= 2520 43. 7

3x-5
44 46. x+2 47.
45. 2 91 -30x + 26
4x +7
48. f- (x) = X+4 49. f- (16) =(4, 4) 50. f- (x)= 2
3

5l. f-(x) = -, for all xeR-{0} 52. f-l() = for allxeR-(1)

1-x

53. f-l(x) =(x- 5)3 54. No, sincefis many-one

Type III - Assertion-Reasoning Questions

1. (a) 4. (d) 5. (d)
2. (a) 3. (c)
6. (b) 9. (b) 10. (d)
7. (d) 8. (d)

12. (a) 13. (b)