MC0063 Discrete Mathematics Model Question Paper

Model Question Paper

Subject Code: MC0063

Subject Name: Discrete Mathematics

Credits: 4 Marks: 140

Part A (One mark questions)

1. If the number of elements in a set is not finite then the set is called an

a) finite set

b) collective set

c) Infinite set

d) arranged set

2. If A = {1,3,5} and B = {1,3,5,7} then A is a ……….. subset of B

a) smaller

b) proper

c) improper

d) normal

3. If A is the arithmetic mean between the extremes a and b then A =

a) 2

ba

b) 2

ba

c) 2

2ba

d) 2

2ba

4. The nth term of an arithmetic progression a + (a + d) + (a + 2d) + …. is

a) a + nd

b) a + (n–1)d

c) a + (n+1)d

d) 2a + (n+1)d

5. Combinatorics is the branch of discrete mathematics concerned with ………….

a) counting problems

b) abstract algebra

c) derivative problems

d) integrated problems

6. If the object A is chosen in m ways and B in n ways then either A or B is chosen in ………

ways

a) n

m

b) mn

c) m + n

d) m – n

7. A recurrence relation of the form )(.......22110 rfaCaCaCaC krkrrr where sCi ' are

constants, is called a ……………………….

a) Quadratic linear relation

b) Quadratic recurrence relation

c) Linear recurrence relation

d) Cubic recurrence relation

8. rrr aa 232 1 is a ………… order linear recurrence

a) second

b) first

c) third

d) fourth

9. The relation R between the sets nAAA ......,,, 21 is a subset of

a) nAAA ....21

b) nAAA ....21

c) nAAA ....21

d) nAAA ....21

10. A relation means …………….. on a set S.

a) dual relation

b) binary relation

c) reflexive relation

d) symmetric relation

11. Let S be a non-empty set, then the operation on S is said to be associative if for all a, b,

cS we have

a) cbacba )()(

b) bccb

c) )()( bacb

d) cba

12. Let (A,) be an algebraic system where is a binary operation on A. Then (A,) is called a

semigroup if it satisfies the

a) closure law

b) associative law

c) reflexive law

d) closure and associative law

13. If for any ring R, a.b = b.a for all a, bR then R is said to be a ……………..

a) integer ring

b) commutative ring

c) cyclic ring

d) non-commutative ring

14. A commutative ring is said to be an integral domain if it has no ………………..

a) zero divisors

b) inverse

c) multiples

d) identity

15. Reasoning is a special kind of thinking called as …………………

a) inferring

b) logics

c) bijective

d) contradictive

16. The basic unit of our objective language is called a …………………….

a) prime divisor

b) prime statement

c) bijective statement

d) statement

17. A set is a collection of ………………..

a) well defined objects

b) undefined objects

c) objects

d) only numbers

18. Two sets A and B are said to be …………., if A is a subset of B and B is a subset of A.

a) proper subsets

b) void

c) equal

d) unequal

19. “Any non-empty subset of the set of all positive integers contains a smallest element”. This

principle is called as

a) well ordering principle

b) ordering principle

c) grouping principle

d) ungrouping principle

20. If nd , then d

nis called the ………………… to d

a) addition conjugate

b) subtractive conjugate

c) divisor conjugate

d) multiplicative conjugate

21. A language L can be considered as a subset of the free …………… on an alphabet.

a) group

b) Monoid

c) ring

d) vector

22. If y = aaab then y=

a) 2

b) 3

c) 5

d) 4

23. Boolean Algebra is an algebra of …………….

a) logic

b) sets

c) rings

d) groups

24. If S is a poset and a, b are in S such that a > b and there is no c in S such that a > c and c >

b then we say that

a) b covers a

b) a covers b

c) a uncovers b

d) b uncovers a

25. A finite-state machine is an abstract model of a machine with a primitive …………………….

a) circuit

b) internal memory

c) structure

d) external memory

26. A finite machine M consists of a finite set Q of …………..

a) input symbols

b) output symbols

c) states

d) functions

27. Suppose that nn Zxx 21......... and n

n Zyyy 21 ............. then the distance between x and y is

a) }:,....,1{),( jj yxnjyx

b) }:,....,1{),( jj yxnxyx

c) }:,....,1{),( jj yxnjyx

d) }:,....,1{),( jj yxnyyx

28. The probability of the string nZe 2 having exactly k 1’s is ………..

a) nk ppk

n)1(

b) knk ppk

n

)1(

c) knk ppp

n

)1(

d) knk ppk

n

)1(

29. The number of elements in the power set P(A) is ………….

a) A2

b) A

2

c) A

2

d) A

2

30. Let and be any two fuzzy subsets of a set S. Then is said to be contained in if

……………..

a) Sxxx )()(

b) Sxxx )()(

c) Sxxx )()(

d) Sxxx )()(

31. An edge having the same vertex as both its end vertices is called a ……………….

a) self edge

b) self loop

c) self line

d) self curve

32. A graph is also called a ………….

a) 2 – complex

b) 1 – complex

c) 3 – complex

d) 4 – complex

33. A tree without any edge is called as a ………………

a) group tree

b) self tree

c) branch

d) null tree

34. A connected graph without circuits is called a …………….

a) unique tree

b) vertex

c) decision tree

d) mail tree

35. A given connected graph G is an Euler graph all the vertices of G are of …………….

a) odd degree

b) even degree

c) odd values less than 100

d) even values greater than 50

36. A simple graph with n vertices and k components can have atmost …………….. edges

a) 2

)1()( knkn

b) 2

)1()( knkn

c) 2

)1()( knkn

d) 2

)1( kn

37. A graph G is said to be a …………………….. if there exists some geometric representation

of G which can be drawn on a plane such that no two of its edges intersect.

a) non-planar graph

b) planar graph

c) line graph

d) null graph

38. A drawing of a geometric representation of a graph on any surface such that no edges

intersect is called an …………………

a) isomorphism

b) homomorphism

c) embedding

d) epimorphism

39. The entries along the principal diagnol of the adjacency matrix are all ……….

a) unity

b) non-zeroes

c) real numbers

d) zeroes

40. Two graphs G1 and G2 are isomorphic if and only if their incidence matrices I(G1) and I(G2)

differ only by …………. of rows and columns

a) combinations

b) shuffling

c) permutations

d) cyclic permutation

Part B (Two mark questions)

41. If A = {2, 3, 4}, B = {4, 5, 6} and C = {6, 7} then )( BCA

a) {(2,7) (3,7) (7,4)}

b) {(2,7) (3,3) (4,7)}

c) {(7,2) (3,7) (4,7)}

d) {(2,7) (3,7) (4,7)}

42. The nth term of 1 + 3 + 5 + 7 + ……….

a) 2n

b) 2n + 1

c) 2n – 1

d) 1 – 2n

43. In how many ways can a lady wear five rings on the fingers (not the thumb) of her right

hand?

a) 6620

b) 6720

c) 6520

d) 6700

44. If x = 2.52 then 52.2

a) 0

b) 1

c) 2

d) 3

45. The least upper bound of a set A is called ………………..

a) Infimum

b) Supremum

c) greatest element

d) least element

46. Square roots of unity is an abelian group with respect to

a) division

b) addition

c) multiplication

d) subtraction

47. A finite integral domain is a ………….

a) subfield

b) vector

c) field

d) ring

48. The class of variables which are quantified stand for only those objects that are members of

a particular set and is called the……………….

a) universe

b) discourse

c) universe of discourse

d) injective of discourse

49. If )(, SAthennS

a) (n – 2)!

b) n!

c) (n – 3)!

d) (n – 1)!

50. If nd and dn then …………..

a) nd

b) nd

c) nd

d) all the three a), b) and c)

51. A grammer in which there are no restrictions on its productions is called a ………………

a) type – 1 grammar

b) type – 0 grammar

c) type – 2 grammar

d) grammar

52. The length of the chain of the sequence of elements a0, a1, a2, ……,an is ……….

a) n – 1

b) n

c) n – 2

d) n – 3

53. Let A = {a, b}, if L1 consists of all words beginning with an a and followed by zero or more

b’s, then the language over A is

a) .....},,,{ 221 baabaL

b) .....},,,{ 221 ababaL

c) .....},,,{ 21 ababaL

d) .....},,,{ 22221 babaaL

54. If nZyx 2, , then d(x, y) = 0 exactly when

a) x = y

b) x > y

c) x < y

d) x y

55. ( ) =

a)

b)

c)

d) +

56. State whether true(T) or false(F)

(i) A tree is a connected graph without any circuits

(ii) A single vertex in a graph G is not a sub-graph of G

a) (i) F (ii) T

b) (i) T (ii) T

c) (i) F (ii) F

d) (i) T (ii) F

57. The number of internal vertices in a binary tree is ……………

a) 12

1

p

n

b) pn

2

1

c) 12

1

p

n

d) 12

1

p

n

58. State T or F

i) An Euler graph G is arbitrarily traceable from vertex v in G if and only if every circuit in G

contains v

ii) If a graph has exactly has exactly two vertices of odd degree, then there exists a path joining

these two vertices.

a) i) F ii)T

b) i) T ii)F

c) i) T ii)T

d) i) F ii)F

59. State T or F

i) A Jordan curve is a continuous non-self-intersecting curve whose origin and terminus donot

coincide

ii) A planar graph is denoted by K3, 3

a) i) F ii)T

b) i) T ii)F

c) i) T ii)T

d) i) F ii)F

60. State T or F

i)The reduced incidence matrix of a tree is singular

ii) Each largest non-separable subgraph is called a block.

a) i) F ii)T

b) i) T ii)F

c) i) T ii)T

d) i) F ii)F

Part C (Four mark questions)

61. The value of cA with respect to the universal set of reals if )2,(),1( A is

a) [–2, 1]

b) [2, 1]

c) [–2, ]

d) [–, 1]

62. The number of divisors of 9504 is

a) 42

b) 43

c) 44

d) 48

63. The language for the grammar ),},{},1,0{( SSVVG NT with the set of productions

0,11: SSS is

a) L(G)={0, 110, 11110, 1010110,….}

b) L(G)={0, 101, 10110, 1010100,….}

c) L(G)={1, 110, 11110, 1010110,….}

d) L(G)={0, 110, 11110, 1111110,….}

64. In a Boolean algebra for all a, b B, )( baa

a) a

b) – a

c) ab

d) a2

65. A grammar G is said to be context-sensitive if the productions are of the form

a) A

b) A

c)

d) A

66. The word c = 1010110 is transmitted through a binary symmetric. If p = 0.02 is the

probability of incorrect receipt of a signal, then the probability that c is received as……….

r = 1011111 is

a) 0.0036

b) 0.000036

c) 0.00036

d) 0.000306

67. State whether true(T) or false(F)

(i) Truth values 0 and 1 are logic formulae.

(ii) If v is a logic variable, then v is not a logic formula.

a) (i) T (ii) F

b) (i) F (ii) T

c) (i) T (ii) T

d) (i) F (ii) F

68. State whether true(T) or false(F)

(i) Every graph is its own subgraph.

(ii) A single edge in G, together with its end vertices, is also a sub-graph of G.

a) (i) T (ii) F

b) (i) F (ii) T

c) (i) T (ii) T

d) (i) F (ii) F

69. State whether true(T) or false(F)

(i) The number of labeled trees with n vertices is )2( nn .

(ii) Every disconnected graph has atleast one spanning tree.

a) (i) T (ii) F

b) (i) F (ii) T

c) (i) T (ii) T

d) (i) F (ii) F

70. State whether true(T) or false(F)

(i) Km,n is not Hamiltonian when m + n is odd.

(ii) If G is a Hamiltonian graph, then for every proper subset S of V(G), we must have

SSGc )(

a) (i) T (ii) F

b) (i) F (ii) T

c) (i) T (ii) T

d) (i) F (ii) F

71. State whether true(T) or false(F)

(i) Every tree is not a Bipartite graph

(ii) A Bipartite graph contains self loops

a) (i) T (ii) F

b) (i) F (ii) T

c) (i) T (ii) T

d) (i) F (ii) F

72. The graph for the adjacency matrix

010

101

010

is

a) b)

c) d)

73. State T or F

A digraph D is said to be an arborescence if it satisfy the following two conditions:

i) D contains no circuit (neither a directed circuit nor a semi-circuit).

ii) There exists exactly one vertex v of zero in-degree (this vertex v is called the root of the

arborescence).

a) i) F ii)T

b) i) T ii)F

c) i) T ii)T

d) i) F ii)F

o

o o

o

o o

o

o o

o o

74. The compound proposition pqpp )]([ is a

a) contradiction

b) tautology

c) neither (a) nor (b)

d) predicate

75. If RR : is a homomorphism then )( a

a) )( 2a

b) )(a

c) )(a

d) a

Answer Keys

Part - A Part - B Part - C

Q. No. Ans. Key Q. No. Ans. Key Q. No. Ans. Key Q. No. Ans. Key

1 c 21 b 41 d 61 a

2 b 22 d 42 c 62 d

3 b 23 a 43 b 63 d

4 b 24 b 44 c 64 a

5 a 25 b 45 b 65 d

6 c 26 c 46 c 66 c

7 c 27 c 47 c 67 a

8 c 28 b 48 c 68 c

9 c 29 b 49 b 69 a

10 b 30 c 50 a 70 c

11 a 31 b 51 b 71 d

12 d 32 b 52 b 72 a

13 b 33 d 53 c 73 c

14 a 34 c 54 a 74 b

15 a 35 b 55 b 75 c

16 b 36 b 56 d

17 a 37 b 57 a

18 c 38 c 58 c

19 a 39 d 59 d

20 c 40 c 60 a