Discrete Mathematics MCQS

1. A collection of graph is:

 A. row and coloumn
 B. Equation
 C. vertices and columns
 D.  None of above
Ans: C
2.  The number of edges in a complete graph with ‘n’ vertices is equal to:
 A. 2n-1
 B. n(n-1)
 C. n^2
 D. n(n-1)/2
Ans: D
3. Error correcting code is a _____:
 A. hamming code
 B. gray code
 C. error deducting code
 D. none of above
Ans: A
4. ordered collection of objects is:
 A. Relation
 B. set
 C. proposition
 D. Function
Ans: B
5. A function is a Domain of:
 A. it is set of natural numbers for which a function is defined
 B. the maximal set of numbers for which a function is defined
 C. the maximal set of numbers which a function can take values
 D. none of above
Ans: B
6. Range of a function is :
 A. the maximal set of numbers for which a function is defined
 B. the maximal set of numbers which a function can take values
 C. it is set of natural numbers for which a function is defined
 D. none of above
Ans: B
7.  In an undirected graph the number of nodes with odd degree must be:
 A. odd
 B. prime
 C. even
 D. zero
Ans: C
8. What is the cardinality of the set of odd positive integers less than 10?
 A. 5
 B. 10
  C. 3
 D. 20
Ans: A
9. If P then Q is called _________ statement
A. Conjunction
B. disjunction
C. conditional
D. bi conditional

Ans: C
10. (P≥Q)≥ (⋀Q) is __________.
A. not a well formed formula
B. tautology
C. contradiction
D. well formed formula
Ans: A
11. A relation R in a set X is symmetric if ________.
A. xRy, yRz ≥xRz.
B. xRy
C. xRy≥yRx
D. xRx
Ans: C
12. If a relation is reflexive, then all the diagonal entries in the relation matrix
must be________.
A. 0
B. 1
C. 2
D. -1
Ans: B
13. If R is reflexive, symmetric and transitive then the relation is said to be
________.
A. Binary relation
B. Compatibility relation
C. Equivalence relation
D. Partial order relation
Ans: C
14. A finite non-empty set of symbols is called _________.
A. alphabet
B. letter
C. string
D. language
Ans: A
15. Surjective function is also called ________.
A. onto
B. into
C. one to one
 D. one one onto
Ans: A
16. One to one onto function is also called __________.
A. bijective
B. injective
C. surjective
D. composite function
Ans: A
17. The composition of function is associative but not _______.
A. commutative
B. associative
C. distributive
D. idempotent
Ans: A
18. A mapping x into itself is called __________.
A. reflexive
B. symmetric
C. transitive
D. equivalence
Ans: A
19. The duality law of (P∧Q)∨T is ________.
A. (P⋀Q)∧T
B. (P∨Q)∧T
C. (P∨Q)∨F
D. (P∨Q)∧F
Ans: D
20. Min-terms of two statements are formed by introducing the connective
_________.
A. Conjunction
B. disjunction
C. Conditional
D. negation
Ans: A
21. Any vertex having degree one is called _______.
A. Simple vertex
B. pendent vertex
C. regular vertex
D. complete vertex
Ans: B
22. A graph that has neither self loops nor parallel edges is called_____graph.
A. regular
B. simple
C. complete
D. null
Ans: B
23. A graph in which every vertex has same degree is called _________graph.
 A. regular
B. simple
C. complete
D. null
Ans: A
24. Kn denotes _______graph.
A. regular
B. simple
C. complete
D. null
Ans: C
25. The number of vertices of odd degree in a graph is always________.
A. odd
B. even
C. zero
D. one
Ans: B
26. A path of a graph is said to be ______ if it contains all the edges of the graph.
A. eulerian
B. hamiltonian
C. tournament
D. planar
Ans: A
27. Traveling salesman problem is example for_______graph.
A. eulerian
B. hamiltonian
C. tournament
D. planar
Ans: B
28. If a node v is reachable from node u then the path of minimum length u to v
is called _____.
A. reachability
B. node base
C. geodesic
D. accessibility
Ans: C
29. The eccentricity of a center in a tree is defined as ______ of the tree.
A. radius
B. diameter
C. length
D. path
Ans: A
30. P ≥ Q , Q ≥R then________.
A. P ≥ R
B. R ≥P
C. Q
 D. R
Ans: A
31. If a normal form contains all minterms, then it is ________.
A. a tautology
B. a contradiction
C. a contingency
D. both a and b
Ans: A
32. Max-terms of two statements are formed by introducing the connective
_________.
A. disjunction
B. conjunction
C. negation
D. conditional
Ans: A
33. The Subset relation on a set of sets is ________.
A. partial ordering
B. equivalence relation
C. reflexive and symmetric only
D. symmetric and transitive only
Ans: A
34. A relation R is defined on the set of integers as xRy if and only if (x+y) is
even. Which of the following statement is TRUE?
A. R is not an equivalence relation.
B. R is an equivalence relation having one equivalence classes
C. R is an equivalence relation having two equivalence classes
D. R is an equivalence relation having three equivalence classes
Ans: C
35. If R = {(1, y), (1, z), (3, y)} then R power (-1)= ___________.
A. {(1, a), (y, z)}
B. {(y, 1), (z, 1), (y, 3)}
C. {(y, a), (1, z), (3, y)}
D. {(y, a), (z, a), (3, y)}
Ans: B
36. Let R ={ (a,b),(c,d),(b,b)}, S = {(d,b),(c,b),(a,d)} then R composite S =
___________
A. {(a,e),(c,b),(b,e)}
B. {(d,b),(c,b),(a,d)}
C. {(a,b),(b,b)}
D. {(c,b)}
Ans: D
37. The number of relations from A = {a,b,c} to B = {1,2} are __________.
A. 6
B. 8
C. 32
D. 64
 Ans: D
38. The minimum number of edges in a connected graph with n vertices is
___________.
A. n
B. n-1
C. n+1
D. n+2
Ans: B
39. The number of distinct simple graphs with up to three nodes is _________.
A. 7
B. 9
C. 15
D. 25
Ans: A
40. A graph is planar if and only if it does not contain ________.
A. subgraphs homeomorphic to k3 & k3,3
B. subgraphs isomorphic to k5 or k3,3
C. subgraphs isomorphic to k3 & k3,3
D. sub graphs homeomorphic to k5 or k3,3
Ans: D
41. Maximum number of edges in an n-node undirected graph without self loops
is ____.
A. [n(n-a)]/2
B. n-1
C. n
D. [n(n+a)]/2
Ans: A
42. Number of distinct nodes in any elementary path of length p is ________.
A. p
B. p-1
C. p+1
D. p*1
Ans: C
43. The total number of edges in a complete graph of n vertices is _________.
A. n
B. n/2
C. [n(n-a)]/3
D. [n(n-a)]/2
Ans: D
44. A directed complete graph of n vertices contains __________.
A. one arrow between each pair of distinct vertices
B. two arrows between each pair of distinct vertices
C. n-1 arrows between each pair of distinct vertices
D. path between every two distinct vertices
Ans: A
45. A directed graph G = (V, E) is said to be finite if its ________.
 A. set V of vertices is finite
B. set V of vertices & set E of edges are finite
C. set E of edges are finite
D. no vertices & edges are repeated
Ans: A
46. A state from which a deterministic finite state automata can never come out is
called a ____________.
A. trape state
B. starting symbol
C. transition table
D. transition diagram
Ans: A
47. If a compound statement is made up of three simple statements then the
number of rows in the truth table is _______.
A. 2
B. 4
C. 6
D. 8
Ans: D
48. Let R = {(3, 3), (6, 6), (9, 9), (12,12), (3,6), (6,3), (3, 9), (9, 3), (9, 12),(12,9)} be a
relation on the set A = {3, 6, 9, 12}. The relation is _________
A. reflexive and transitive
B. reflexive and symmetric
C. symmetric and transitive
D. equivalence relation
Ans: D
49. Let R={(1,b),(3,d),(2,b)} and S={(b,4),(2,5),(d,a)} be a relation then R
composition S=____.
A. {(1,b),(3,d),(2,b)}
B. {(1,4),(3,a),(2,4)}
C. {(4,b),(2,5),(3,a)}
D. {(1,d),(3,b),(2,c)}
Ans: B
50. If R= {(x, 2x)} and S= {(x, 4x)} then R composition S=____.
A. {(x, 4x)}
B. {(x, 2x)}
C. {(x, 8x)}
D. {(x, 10x)}
Ans: C
51. If R= {(x, 2x)} and S= {(x, 5x)} then R composition S=____.
A. {(x, 4x)}
B. {(x, 2x)}
C. {(x, 8x)}
D. {(x, 10x)}
Ans: D
52. Let R={(1, 3), (4, 2), (2, 2), (3, 3), (1, 1),(4,4)} be a relation on the set A={1, 2,
 3, 4}. The relation R is ____.
A. transitive
B. reflexive
C. not symmetric
D. function
Ans: C
53. The NOR statement is a combination of ________.
A. NOT and AND
B. NOT and OR
C. AND and OR
D. NOT or OR
Ans: B
54. If a relation is reflexive then in the graph of a relation there must be a loop at
_____.
A. each node
B. only first node
C. any two nodes
D. only first and last nodes
Ans: A
55. Which of the following traversal techniques lists the nodes of binary search in
ascending order?
A. pre order
B. post order
C. in order
D. root order
Ans: C
56. The rank of the incidence matrix of any connected graph G with n vertices is
______.
A. n
B. n+1
C. n-1
D. n-2
Ans: C
57. The number of 1's in each row of an incidence matrix of a graph G is equal to
_____.
A. the degree of the corresponding vertices
B. the sum of degrees of all vertices
C. the degree of the initial vertex
D. the degree of the terminal vertex
Ans: A
58. Each column of an incidence matrix of a graph G has exactly _______.
A. one 1's
B. two 1's
C. one 2's
D. two 2's
Ans: B
59. An undirected graph is tripartite if and only if it has no circuits of _______
lengths
A. odd
B. even
C. distinct
D. equal
Ans: A
60. A graph is bipartite if and only if its chromatic number is ________.
A. 1
B. 2
C. odd
D. even
Ans: B
61. The number of pendant vertices in a full binary tree with n vertices is
________.
A. (n-a)/2
B. (n-1)/2
C. (n+a)/2
D. n/2
Ans: C
62. The number of vertices in a full binary tree is _______.
A. odd
B. even
C. equal
D. 0
Ans: A
63. A binary tree with 2k vertices of level k has at least _______ vertices.
A. 2 power k
B. 2 power (k-1)
C. 2 power (k-1)-1)
D. 2 power (k+1)-1
Ans: D
64. For a symmetric digraph, the adjacency matrix is _________.
A. symmetric
B. antisymmetric
C. asymmetric
D. symmetric and asymmetric
Ans: A
65. The total number of degrees of an isolated node is _______.
A. 0
B. 1
C. 2
D. 3
Ans: A
66. If G is a connected planar graph then it has a vertex of degree _______.
A. 3 or less
 B. 4 or less
C. 5 or less
D. 6 or less
Ans: C
67. A product of the variable and their negation in a formula is called ________.
A. an elementary sum
B. an elementary product
C. a well-formed formula
D. an equivalence of relation formula
Ans: B
68. A formula consisting of disjunctions of min-terms is called _________.
A. DNF
B. CNF
C. PDNF
D. PCNF
Ans: C
69. The less than relation < on real is __________.
A. a partial ordering since it is asymmetric and reflexive
B. a partial ordering since it is anti-symmetric and reflexive
C. not a partial ordering since it is not asymmetric and not reflexive
D. not a partial ordering since it is not anti-symmetric and not reflexive
Ans: D
70. A relation R in X is said to be a ________, if it is reflexive and symmetric.
A. void relation
B. circular
C. partial order relation
D. compatibility relation
Ans: D
71. The set X*X itself defines a relation in X is called a _____relation.
A. void
B. universal
C. partial
D. equivalence
Ans: B
72. A self complemented distributive lattice is called _______.
A. boolean algebra
B. modular lattice
C. complete lattice
D. self dual lattice
Ans: A
73. Every finite subset of a lattice has ____________.
A. a Least Upper Bound and Greatest Lower Bound
B. many Least Upper Bounds and a Greatest Lower Bound
C. many Least Upper Bounds and many Greatest Lower Bounds
D. either some Least Upper Bounds or some Greatest Lower Bounds
Ans: A
 A formula consisting of conjunctions of max-terms is called _________.
A. DNF
B. CNF
C. PCNF
D. PDNF
Ans: C
74. The set of all divisors of 24 are ___________.
A. {1, 2, 3, 4, 6, 8, 12, 24}
B. {2, 3, 4, 6, 8, 12}
C. {1, 3, 6, 12,}
D. {2, 4, 6, 8}
Ans: A
75. In a bounded lattice, an element b belongs to L is called a complement of an
element a belongs to L if ______.
A. a*b=0
B. a+b=1
C. both a and b
D. none
Ans: C
76. If each non-empty subset of a lattice has a least upper bound and greatest
lower bound then the lattice
is called ________.
A. complete
B. associative
C. absorption
D. commutative
Ans: A
77. A __________ is a complemented distributive lattice.
A. boolean homomorphism
B. boolean algebra
C. boolean isomorphism
D. boolean function
Ans: D
78. Every connected graph contains a ________.
A. tree
B. sub tree
C. spanning tree
D. spanning subtree
Ans: C
79. A minimal non-empty edge cut of G is called a _________.
A. bond
B. cycle
C. path
D. tour
Ans: A
80. A connected graph that has no cut vertices is called a ________.
 A. block
B. bond
C. cycle
D. tour
Ans: A
81. A graph is Eulerian if it contains __________.
A. Euler tour
B. Euler trail
C. Hamiltonian path
D. Euler path
Ans: A
82. Hamilton cycle is a cycle that contains every ________of G.
A. path
B. cycle
C. vertex
D. edge
Ans: C
83. A set containing no element is called ____________.
A. null set
B. finite set
C. infinite
D. equal set
Ans: A
84. A = {1,3,5,7,9} is a __________.
A. null set
B. finite set
C. singleton set
D. infinite set
Ans: B
85. The number of Indians in the world is _________.
A. finite set
B. universal set
C. infinite set
D. equal set
Ans: A
86. If in the truth table the answer column has the truth values both TRUE and
FALSE then it is said to be________.
A. tautology
B. contradiction
C. contingency
D. equivalence relation
Ans: C
87. To prove the statement P tautologically implies the statement Q, it is enough
to prove that _________.
A. P conditional Q is a contradiction
B. P conditional Q is a tautology
 C. P biconditional is a contradiction
D. P biconditional Q is a tautology
Ans: B
88. To prove the statement P is tautologically equivalent to the statement Q, it is
enough to prove that _______.
A. P conditional Q is a contradiction
B. P conditional Q is a tautology
C. P biconditional Q is a contradiction
D. P biconditional Q is a tautology
Ans: D
89. Let R={(1,2),(3,4),(2,6.} and S={(4,3),(2,5),(6,6)} be a relation then R
composite S=____.
A. {(1,5),(3,3),(2,6)}
B. {(1,5),(3,6),(2,5)}
C. {(4,4),(2,5),(3,3)}
D. {(1,1),(3,3),(2,2)}
Ans: A
90. The binary relation R = {(0, 0), (1, a)} on A = {0, 1, 2, 3, } is _______.
A. reflexive, not symmetric, transitive
B. not reflexive, symmetric, transitive
C. reflexive, symmetric, not transitive
D. reflexive, not symmetric, not transitive
Ans: B
91. There are only five distinct Hasse diagrams for partially ordered sets that
contain _______elements.
A. 2
B. 3
C. 4
D. 6
Ans: B
92. If an edge e is said to join the vertices u and v then the vertices u and v are
called __.
A. initial vertices
B. terminal vertices
C. ends of e
D. all the above
Ans: B
93. Edges intersect only at their ends are called ________.
A. planar
B. loop
C. link
D. non plannar
Ans: B
94. Two vertices which are incident with the common edge are called
______________vertices.
A. distinct
 B. directed
C. adjacent
D. loops
Ans: C
95. An edge with identical ends is called _________.
A. complete graph
B. bipartite graph
C. loops
D. link
Ans: C
96. An edge with same ends is called ___________.
A. complete graph
B. bipartite graph
C. loops
D. link
Ans: D
97. In a graph if few edges have directions and few do not have directions then
the graph is called_________.
A. multi graph
B. directed graph
C. undirected graph
D. mixed graph
Ans: D
98. If two edges have same vertices as its terminal vertices those edges are called
____.
A. parallel
B. adjacent
C. incident
D. distinct
Ans: A
99. The graph defined by the vertices and edges of a __________ is bipartite.
A. square
B. cube
C. single
D. both square and cube
Ans: B
100. To any graph G there corresponds a vertex in a matrix called
________matrix.
A. incidence
B. adjacency
C. square
D. null
Ans: A
101. If H is a sub graph of G then G is a ______ of H.
A. proper sub grapth
B. inducted sub graph
 C. spanning subgraph
D. super graph
Ans: D
102. If the graph G1 and G2 has no vertex in common then it is said to be ______.
A. disjoint
B. edge disjoint
C. union
D. intersection
Ans: A
103. The degree of vertex v in G is __________.
A. number of edges of G incident with v
B. number of loops in G
C. number of links in G
D. number of sub graph in G
Ans: A
104. Each loop counting has _________ edges.
A. 1
B. 2
C. 3
D. 4
Ans: B
105. The negation of the statement is formed by introducing ___________.
A. not
B. and
C. or
D. if
Ans: A
106. A __________ is an ordered collection of objects.
a) Relation
b) Function
c) Set
d) Proposition
Ans: C
107.  Power set of empty set has exactly _________ subset.
a) One
b) Two
c) Zero
d) Three
Ans: A
108.  What is the Cartesian product of A = {1, 2} and B = {a, b}?
a) {(1, a), (1, b), (2, a), (b, b)}
b) {(1, 1), (2, 2), (a, a), (b, b)}
c) {(1, a), (2, a), (1, b), (2, b)}
d) {(1, 1), (a, a), (2, a), (1, b)}
Ans: C
109. The bit string for the set {2, 4, 6, 8, 10} (with universal set of natural numbers
 less than or equal to 10) is ____________________
a) 0101010101
b) 1010101010
c) 1010010101
d) 0010010101
Ans: A
110. The set difference of the set A with null set is __________
a) A
b) null
c) U
d) B
Ans: A
111. The shaded area of figure is best described by

a) A‘ (Complement of A)
b) B – (A ∩ B) – (C ∩ B)
c) A ∩ C ∩ B
d) B’ (Complement of B)
Ans: B
112. The relation between sets A,B,C as shown by venn diagram is
a) A is subset of B and B is subset of C
b) C is not a subset of A and A is subset of B
c) C is subset of B and B is subset of A
d) None of the mentioned
Ans: C
113. Let A : All badminton player are good sportsperson.
B: All person who plays cricket are good sportsperson.
Let X denotes set of all badminton players, Y of all cricket players, Z of all
good sportsperson. Then which of the following statements is correct?
a) Z contains both X and Y
b) Z contains X and Y is outside
c) X contains Y and Z
d) None of the mentioned
Ans: A
114. In the given figure the if n(A)=20,n(U)=50,n(C)=10 and n(A∩B)=5 then
n(B)=? .

a) 35
b) 20
 c) 30
d)10
Ans: A Explanation: Here n(B)= n(U) – n(A) + n(A∩B
115.  The shaded area of figure is best described by

a)A‘ (Complement of A)
b) A U B – (A ∩ B)
c) A – B
d) B
Ans: B
116.  If set A has 3 elements then number of elements in A X A X A are
a) 9
b) 27
c) 6
d)19
Ans: B
117.  The set containing all the collection of subsets is known as
a) Subset
b) Power set
c) Union set
d) None of the mentioned
Ans: B
118. State whether the given statement is true or false
The number of subsets of a set can be odd or even.
a) True
b) False
Ans: A
119. A drawer contains 12 red and 12 blue socks, all unmatched. A person takes
socks out at random in the dark. How many socks must he take out to be sure
that he has at least two blue socks?
a) 18
b) 35
c) 28
d) 14
Ans: D Explanation: Given 12 red and 12 blue socks so, in order to take out at least 2
blue socks, first we need to take out 12 shocks (which might end up red in worst
case) and then take out 2 socks (which would be definitely blue). Thus we need to
take out total 14 socks.
120. The least number of computers required to connect 10 computers to 5 routers
to guarantee 5 computers can directly access 5 routers is ______
a) 74
 b) 104
c) 30
d) 67
Ans: C Explanation: Given 12 red and 12 blue socks so, in order to take out at least 2
blue socks, first we need to take out 12 shocks (which might end up red in worst
case) and then take out 2 socks (which would be definitely blue). Thus we need to
take out total 14 socks.
121.  A bag contains 25 balls such as 10 balls are red, 7 are white and 8 are blue.
What is the minimum number of balls that must be picked up from the bag
blindfolded (without replacing any of it) to be assured of picking at least one
ball of each colour?
a) 10
b) 18
c) 63
d) 35
Ans: B
Explanation: Consider three buckets red, white and blue and we want the total
number of balls such that each bucket contain at least one ball. Now consider the
state of picking up a ball without replacement : (normally you consider the worst
case scenario in these cases) Starting 10 balls all are red and thus goes to bucket
name Red. Now again picking up the ball gives 7 balls which are of same colour
and put all of them in bucket named White. The next pick will definitely be of
different colour thus: we picked 10 + 7 + 1 = 18.
122. The number of binary strings of 17 zeros and 8 ones in which no two ones are
adjacent is ___________
a) 43758
b) 24310
c) 32654
d) 29803
Ans: A
Explanation: First place 17 zeroes side by side _ 0 _ 0 _ 0 … 0 _ and 8 1’s can be
placed in any of the (17+1) available gaps hence the number of ways = n+1Ck =
43758.
123. How many words that can be formed with the letters of the word
‘SWIMMING’ such that the vowels do not come together? Assume that
words are of with or without meaning.
a) 430
b) 623
c) 729
d) 1239
Ans: C
Explanation: The word ‘SWIMMING contains 8 letters. Of which, I occurs twice
and M occurs twice. Therefore, the number of words formed by this word = 8!/2!
∗2! = 10080. In order to find the number of permutations that can be formed
where the two vowels I and I come together, we group the letters that should come
together and consider that group as one letter. So, the letters are S,W,M,M,N,G,
 (I,I). So, the number of letters are 7 the number of ways in which 7 letters can be
arranged is 7! = 5040. In I and I, the number of ways in which I and I can be
arranged is 2!. Hence, the total number of ways in which the letters of the
‘SWIMMING’ can be arranged such that vowels are always together are 7!/2!
∗2! = 5040 ways. The number of words in which the vowels do not come together
is =(10080 – 5040) = 5040.
124. In how many ways can 10 boys be seated in a row having 28 seats such that
no two friends occupy adjacent seats?
a) 13P5
b) 9P29
c) 19P10
d) 15P7
Ans: C
Explanation: First let us take the 18 unoccupied seats. They create 19 slots i.e.,
one on the left of each seat and one on the right of the last one. So we can place
the 10 boys in any of these 19 slots that are, 19P10 ways.
125. How many ways can 8 prizes be given away to 7 students, if each student is
eligible for all the prizes?
a) 40325
b) 40320
c) 40520
d) 40720
Ans: B
Explanation: Now the first student is eligible to receive any of the 8 available
prizes (so 8 ways), the second student will receive a prize from rest 7 available
prizes (so 7 ways), the third student will receive his prize from the rest 6 prizes
available(so 6 ways) and so on. So total ways would be 8! = 8*7*6*5*4*3*2*1 =
40320. Hence, the 7 prizes can be distributed in 40320 ways.
126. How many numbers of three digits can be formed with digits 1, 3, 5, 7 and 9?
a) 983
b) 120
c) 345
d) 5430
Ans: B
Explanation: Here number of digits, n = 5 and number of places to be filled-up r
= 3. Hence, the required number is 5P3 = 5!/2!*3! = 120
127. How many words can be formed with the letters of the word ‘CASTLE’ when
‘O’ and ‘A’ occupying end places.
a) 217
b) 48
c) 75
d) 186
Ans: B
Explanation: When ‘O’ and ‘A’ are occupying end-places => A.S.T.L. (CE). We
can see that (CE) are fixed, hence A, S, T, L can be arranged in 4! Ways and (C,
E) can be arranged themselves is 2! ways. So, the number of words formed = 4! x
 2! = 48 ways.
128. Determine the number of ways of choosing a cricket team (consists of 11
players) out of 18 players if a particular player is never chosen.
a) 12798
b) 22800
c) 31824
d) 43290
Ans: C
Explanation: When ‘O’ and ‘A’ are occupying end-places => A.S.T.L. (CE). We
can see that (CE) are fixed, hence A, S, T, L can be arranged in 4! Ways and (C,
E) can be arranged themselves is 2! ways. So, the number of words formed = 4! x
2! = 48 ways.
129. How many different choices can be made from 5 roses, 4 marigold and 8
sunflowers if at least one flower is to be chosen for making of garland?
a) 269
b) 270
c) 281
d) 320
Ans: A
Explanation: Number of ways of selecting roses = (5+1) = 6 ways, number of
ways of selecting marigold = (4+1) = 5 ways, and the number of ways of selecting
sunflowers = (8+1) = 9 ways. Total number of ways of selecting flowers= 6 * 5 *
9 = 270. But this includes when no flowers or zero flowers is selected (There is no
flowers of a different type, hence n=0 => 2n = 20 = 1). Hence, the number of ways
of selecting at least one fruit = 270 – 1 = 269.
130. In how many ways 6 pens can be selected from 15 identical black pens?
a) 9*3!
b) 21
c) 14!
d) 1
Ans: D
Explanation: Here the pens are identical, the total number of ways of selecting 6
pens is 1
131. Determine the number of ways of selecting one or more letters from the
letters BBBBBB?
a) 6
b) 73
c) 23
d) 56
Ans: A
Explanation: The number of ways of selecting one ‘B’s = 1, selecting two ‘B’s =
1, selecting three ‘B’s = 1, selecting four ‘B’s = 1, selecting five ‘B’s = 1 and
selecting six ‘B’s = 1. Hence, the required number of ways = 6.
132. Determine the number of ways such that 5 men and 5 women be seated at a
round table if no two women are seated together.
a) 654870
 b) 144521
c) 362160
d) 5634
Ans: C
Explanation: The men and women can be seated alternately so that no two
women will sit together. Hence, 4 women can be seated on alternate seats at a
round table in (4 – 1)! or 6
ways. Now, the 5 men can be seated in the remaining seats in 5! or 120 ways.
Therefore the total number of ways in this case will be (10-1)! – (120 * 6) =
362160.
133. Find the number of ways in which 4 people E, F, G, H, A, C can be seated at a
round table, such that E and F must always sit together.
a) 32
b) 290
c) 124
d) 48
Ans: D
Explanation: E and F can sit together in all arrangements in 2! Ways. Now, the
arrangement of the 5 people in a circle can be done in (5 – 1)! or 24 ways.
Therefore, the total number of ways will be 24 x 2 = 48.
134. There are 6 equally spaced points A, B, C, D, E and F marked on a circle with
radius R. How many convex heptagons of distinctly different areas can be
drawn using these points as vertices?
a) 7! * 6
b) 7C5
c) 7!
d) same area
Ans: D
Explanation: Since all the points are equally spaced; hence the area of all the
convex heptagons will be same.
135. There are 2 twin sisters among a group of 15 persons. In how many ways can
the group be arranged around a circle so that there is exactly one person
between the two sisters?
a) 15 *12! * 2!
b) 15! * 2!
c) 14C2
d) 16 * 15!
Ans: A
Explanation: We know that n objects can be arranged around a circle
in (n−1)!/2. If we consider the two sisters and the person in between the brothers
as a block, then there will 12 others and this block of three people to be arranged
around a circle. The number of ways of arranging 13 objects around a circle is in
12! ways. Now the sisters can be arranged on either side of the person who is in
between the sisters in 2! ways. The person who sits in between the two sisters can
be any of the 15 in the group and can be selected in 15 ways. Therefore, the total
number of ways 15 *12! * 2!.
136. The number of words of 4 consonants and 3 vowels can be made from 15
consonants and 5 vowels, if all the letters are different is ________
a) 3! * 12C5
b) 16C4 * 4C4
c) 15! * 4
d) 15C4 * 5C3 * 7!
Ans: D
Explanation: There are 4 consonants out of 15 can be selected in 15C4 ways and 3
vowels can be selected in 5C3 ways. Therefore, the total number of groups each
containing 4 consonants and 3 vowels = 15C4 * 4C3. Each group contains 7 letters
which can be arranged in 7! ways. Hence, required number of words = 15C4 * 5C3 *
7!.
137. How many ways are there to arrange 7 chocolate biscuits and 12 cheesecake
biscuits into a row of 19 biscuits?
a) 52347
b) 50388
c) 87658
d) 24976
Ans: B
Explanation: Consider the situation as having 19 spots and filling them with 7
chocolate biscuits and 19 cheesecake biscuits. Then we just choose 7 spots for the
chocolate biscuits and let the other 10 spots have cheesecake biscuits. The number
of ways to do this job is 19C7 = 50388.
138. If a, b, c, d and e are five natural numbers, then find the number of ordered
sets(a, b, c, d, e) possible such that a+b+c+d+e=75.
a) 65C5
b) 58C6
c) 72C7
d) 74C4
Ans: D
Explanation: Let assumes that there are 75 identical balls which are to be
arranged in 5 different compartments (Since a, b, c, d, e are distinguishable). If the
balls are arranged in the row. We have 74 gaps where we can place a ball in each
gap since we need 5 compartments we need to place only 4 balls. We can do this
in 74C4 ways.
139. There are 15 people in a committee. How many ways are there to group these
15 people into 3, 5, and 4?
a) 846
b) 2468
c) 658
d) 1317
Ans: D Explanation: The number of ways to choose 3 people out of 9 is 15C3. Then,
number of ways to choose 5 people out of (15-3) = 12 is 12C5. Finally, the number
of ways to choose 4 people out of (12-4) = 8 is 8C4. Hence, by the rule of
product, 15C3 + 12C5 + 8C4 = 1317.
140. There are six movie parts numbered from 1 to 6. Find the number of ways in
 which they be arranged so that part-1 and part-3 are never together.
a) 876
b) 480
c) 654
d) 237
Ans: B
Explanation: The total number of ways in which 6 part can be arranged = 6! =
720. The total number of ways in which part-1 and part-3 are always together: = 5!
*2! = 240. Therefore, the total number of arrangements, in which they are not
together is = 720 − 240 = 480.
141.  How many ways are there to divide 4 Indian countries and 4 China countries
into 4 groups of 2 each such that at least one group must have only Indian
countries?
a) 6
b) 45
c) 12
d) 76
Ans: A
Explanation: The number of ways to divide 4+4=8 countries into 4 groups of 2
each is as follows: (10C2 * 10C2 * 10C2 * 10C2)/4! = 30. Since it is required that at
least one group must have only Indian countries, we need to subtract 30 from the
number of possible groupings where all 4 groups have 1 Indian country and 1
China country each. This is equivalent to the number of ways to match each of the
4 Indian countries with one China country: 4! = 24. Therefore, the answer is 30 –
24 = 6.
142. From a group of 8 men and 6 women, five persons are to be selected to form a
committee so that at least 3 women are there on the committee. In how many
ways can it be done?
a) 686
b) 438
c) 732
d) 549
Ans: A
Explanation: We may have (2 men and 3 women) or (1 men and 4 woman) or (5
women only). The Required number of ways = (8C2 × 6C3) + (8C1 × 6C4) + (6C5) =
686.
143.  Evaluate the expression (y+1)4 – (y-1)4.
a) 3y2 + 2y5
b) 7(y4 + y2 + y)
c) 8(y3 + y1)
d) y + y2 + y3
Ans: C
Explanation: By using Binomial theorem,the expression (y+1)4 – (y-1)4 can be
expanded as = (y+1)4 = 4C0y4 + 4C1y3 + 4C2y2 + 4C3y1 + 4C4y0 and (y-
1)4 = 4C0y4 – 4C1y3 + 4C2y2 – 4C3y1 + 4C4y0. Now, (y+1)4 – (y-1)4 =
(4C0y4 + 4C1y3 + 4C2y2 + 4C3y1 + 4C4y0) – (4C0y4 – 4C1y3 + 4C2y2 – 4C3y1 + 4C4y0) =
 2(4C1y3 + 4C3y1) = 8(y3 + y1).
144.  Find the coefficient of x7 in (x+4)9.
a) 523001
b) 428700
c) 327640
d) 129024
Ans: D
Explanation: It is known that (r+1)th term, in the binomial expansion of (a+b)n is
given by, Tr+1 = nCran-rbr. Assuming that x7 occurs in the (r+1)th term of the
expansion (x+4)9, we obtain Tr+1 = 129024x4.
145. Determine the 7th term in the expansion of (x-2y)12.
a) 6128y7
b) 59136y6
c) 52632x6
d) 39861y5
Ans: B
Explanation: By assuming that x7 occurs in the (r+1)th term of the expansion (x-
2y)12, we obtain Tr+1 = nCran-rbr = 12C6 x6 (2y)6 = 59136y6.
146. Calculate the value of 8C5.
a) 79
b) 43
c) 120
d) 56
Ans: D
Explanation: We can use the formula nCk = n!k!(n−k)! to calculate the value
of 8C5 = 8!/5!(8−5)! = 56.
147. In how many ways can you select 9 cupcakes from a box containing 17
cupcakes?
a) 42769
b) 45398
c) 24310
d) 36214
Ans: C
Explanation: The number of ways to choose 9 cupcakes out of a set of 17
is 17C9 = 17!/9!(17−9)! = 24,310
148. How many 4-digit numbers can be formed by using 2, 4, 6, 8, 10, 12 without
repetition of digits?
a) 15
b) 42
c) 70
d) 127
Ans: A
Explanation: Here making a 4-digit number is equivalent to filling 4 places with 6
numbers. So, the number of ways of filling all the four places is 6C4 = 15. Hence,
the total possible 4-digit numbers from the above 6 numbers are 15.
149. What is the coefficient of x9 in the expansion of (x+5)14?
 a) 5! * 14C6
b) 14C5
c) 54 * 14C5
d) 34 * 11C5
Ans: C
Explanation: the binomial theorem is (x+y)a = Σ aCi xa-i yi. In order to get the
coefficient of x9, we need to have a-i=9. Since a=14, i=5. Thus, the answer is aC5 *
y4 = 54 * 14C5.
150. Determine the independent term of x7 in the expansion of (3x2 + 4)12.
a) 220 * 46
b) 230
c) 548* 3!
d) 220 * 36 * 46
Ans: D
Explanation: By using Binomial theorem = n∑k=0 (nk) xkyn-k = n0x0yn + n1x1yn-1 +
n2x2yn-2 + … + nnxny0, where (nk) = n!/k!(n−k)!. Now, Tr+1 = nCran-rbr, T9+1 = 12C6a12-
6 6
b  = 220 * (3x2)6 * (4)6 = 220 * 36 * 46. Hence the coefficient is 220 * 36 * 46.
151.  Find the coefficient of x8 in the expansion of (x+2)11.
a) 640
b) 326
c) 1320
d) 456
Ans: C
Explanation: The coefficient of the 8th term is 11C8 = 165. Hence, the 8th term of
the expansion is 165 * 23 * x8 = 1320x8, where the coefficient is 1320.
152. A Poset in which every pair of elements has both a least upper bound and a
greatest lower bound is termed as _______
a) sublattice
b) lattice
c) trail
d) walk
Ans: B
153. In the poset (Z+, |) (where Z+ is the set of all positive integers and | is the
divides relation) are the integers 9 and 351 comparable?
a) comparable
b) not comparable
c) comparable but not determined
d) determined but not comparable
Ans: A
154.  If every two elements of a poset are comparable then the poset is called
________
a) sub ordered poset
b) totally ordered poset
c) sub lattice
d) semigroup
Ans: B
155. ______ and _______ are the two binary operations defined for lattices.
a) Join, meet
b) Addition, subtraction
c) Union, intersection
d) Multiplication, modulo division
Ans: A
156. The graph given below is an example of _________

a) non-lattice poset
b) semilattice
c) partial lattice
d) bounded lattice
Ans: A
Explanation: The graph is an example of non-lattice poset where b and c have
common upper bounds d, e and f but none of them is the least upper bound.
157. The maximum number of edges in a bipartite graph on 14 vertices is
___________
a) 56
b) 14
c) 49
d) 87
Ans: C Explanation: Maximum number of edges occur in a complete bipartite graph
when every vertex has an edge to every opposite vertex in the graph. Number of
edges in a complete bipartite graph is a*b, where a and b are no. of vertices on
each side. This quantity is maximum when a = b i.e. when there are 7 vertices on
each side. So answer is 7 * 7 = 49.
158. In a ______ the degree of each and every vertex is equal.
a) regular graph
b) point graph
c) star graph
d) euler graph
Ans: C
Explanation: A regular graph has the same degree in each of its vertices. In a
regular bipartite graph, if the common degree of each vertices is 1, the two parts
are of the same size.
159. The time complexity to test whether a graph is bipartite or not is said to be
_______ using depth first search.
a) O(n3)
b) linear time
c) O(1)
 d) O(nlogn)
Ans: B
Explanation: It is possible to test whether a graph is bipartite, and to return either
a two-coloring (if it is bipartite) or an odd cycle (if it is not) in linear time i.e, O(n)
using depth first search. In case of the intersection of n line segments or other
simple shapes in the Euclidean graph, it is possible to test whether the graph is
bipartite and it will return either a two-coloring or an odd cycle in time O(nlogn),
even though the graph itself has up to O(n2) edges.
160. What is the maximum number of edges in a bipartite graph on 14 vertices?
a) 78
b) 15
c) 214
d) 49
Ans: D
Explanation: By definition, the maximum possible number of edges in a bipartite
graph on ‘n’ vertices = (1/4) x n2.
Substituting n = 14, we get maximum number of edges in a bipartite graph on 14
vertices,= (1/4) x (14)2
= (1/4) x 14 x 14
= 49
∴ Maximum number of edges in a bipartite graph on 14 vertices = 49.
161.  In a complete bipartite graph, the intersection of two sub graphs is ______
a) 1
b) null
c) 210
d) 412
Ans: B Explanation: In a complete Bipartite graph, there must exist a partition say,
V(G)=X∪Y and X∩Y=∅, that means all edges share a vertex from both set X and
Y.
162. Bipartite graphs are used in ________
a) modern coding theory
b) colouring graphs
c) neural networks
d) chemical bonds
Ans: A
Explanation: All types of cyclic graphs are examples of cyclic graphs. A cyclic
graph is considered bipartite if all the cycles involved are of even length. Bipartite
graphs are widely used in modern coding theory apart from being used in
modeling relationships.
163. Every complete bipartite graph must not be _______
a) planar graph
b) line graph
c) complete graph
d) subgraph
Ans: C
Explanation: The below bipartite graph is not a complete graph as there is no
 edge between A-B, B-C, C-D, C-Q, P-Q, Q-R, Q-D and so it is not a complete
graph.

164. G is a simple undirected graph and some vertices of G are of odd degree. Add
a node n to G and make it adjacent to each odd degree vertex of G. The
resultant graph is ______
a) Complete bipartite graph
b) Hamiltonian cycle
c) Regular graph
d) Euler graph
Ans: D
Explanation: In any simple undirected graph, total degree of all vertices is even
(since each edge contributes 2 degrees). So number of vertices having odd degrees
must be even, otherwise, their sum would have been odd, making total degree also
odd. Now single vertex n is connected to all these even number of vertices (which
have odd degrees). So, degree of n is also even. Moreover, now degree of all
vertices which are connected to v is increased by 1, hence earlier vertices which
had odd degree now have even degree. So now, all vertices in the graph have even
degree, which is necessary and sufficient condition for euler graph.
165. What is the number of vertices in an undirected connected graph with 39
edges, 7 vertices of degree 2, 2 vertices of degree 5 and remaining of degree 6?
a) 11
b) 14
c) 18
d) 19
Ans: C
Explanation: We know that, sum of degree of all the vertices = 2 * number of
edges
2*7 + 5*2 + 6*x = 39*2
x=9
Number of vertices = 7 + 2 + 9 = 18.
166. ______ is the maximum number of edges in an acyclic undirected graph with
k vertices.
a) k-1
b) k2
c) 2k+3
d) k3+4
Ans: A
Explanation: This is possible with spanning trees since, a spanning tree with k
nodes has k – 1 edges.
167. The minimum number of edges in a connected cyclic graph on n vertices is
 _____________
a) n – 1
b) n
c) 2n+3
d) n+1
Ans: B
Explanation: For making a cyclic graph, the minimum number of edges have to
be equal to the number of vertices. SO, the answer should be n minimum edges.
168. The maximum number of edges in a 8-node undirected graph without self
loops is ____________
a) 45
b) 61
c) 28
d) 17
Ans: C Explanation: In a graph of n vertices we can draw an edge from a vertex to n-1
vertex we will do it for n vertices and so total number of edges is n*(n-1). Now
each edge is counted twice so the required maximum number of edges is n*(n-
1)/2. Hence, 8*(8-1)/2 = 28 edges.
169. Every Isomorphic graph must have ________ representation.
a) cyclic
b) adjacency list
c) tree
d) adjacency matrix
Ans: D
Explanation: A graph can exist in different forms having the same number of
vertices, edges and also the same edge connectivity, such graphs are called
isomorphic graphs. Two graphs G1 and G2 are said to be isomorphic if −> 1) their
number of components (vertices and edges) are same and 2) their edge
connectivity is retained. Isomorphic graphs must have adjacency matrix
representation.
170. A cycle on n vertices is isomorphic to its complement. What is the value of n?
a) 5
b) 32
c) 17
d) 8
Ans: A
Explanation: A cycle with n vertices has n edges. Number of edges in cycle = n
and number of edges in its complement = (n*(n−1)/2) – n. To be isomorphism,
both graphs should have equal number of edges. This gives, (n*(n-1)/2) – n = n
⇒n=5
171. How many perfect matchings are there in a complete graph of 10 vertices?
a) 60
b) 945
c) 756
d) 127
Ans: B
 Explanation: Perfect matching is a set of edges such that each vertex appears only
once and all vertices appear at least once (exactly one appearance). So for n
vertices perfect matching will have n/2 edges and there won’t be any perfect
matching if n is odd. For n=10, we can choose the first edge in 10C2 = 45 ways,
second in 8C2=28 ways, third in 6C2=15 ways and so on. So, total number of ways
45*28*15*6*1=113400. But perfect matching being a set, order of elements is not
important and the permutations 5! of the 5 edges are same only. So, total number
of perfect matching is 113400/5! = 945.
172. A complete n-node graph Kn is planar if and only if _____________
a) n ≥ 6
b) n2 = n + 1
c) n ≤ 4
d) n + 3
Ans: C
Explanation: Any graph with 4 or less vertices is planar, any graph with 8 or less
edges is planar and a complete n-node graph Kn is planar if and only if n ≤ 4.
173. An isomorphism of graphs G and H is a bijection f the vertex sets of G and H.
Such that any two vertices u and v of G are adjacent in G if and only if
____________
a) f(u) and f(v) are contained in G but not contained in H
b) f(u) and f(v) are adjacent in H
c) f(u * v) = f(u) + f(v)
d) f(u) = f(u)2 + f(v)2
Ans: B
174. What is the grade of a planar graph consisting of 8 vertices and 15 edges?
a) 30
b) 15
c) 45
d) 106
Ans: A
Explanation: If G is a planar graph with n vertices and m edges then r(G) = 2m
i.e. the grade or rank of G is equal to the twofold of the number of edges in G. So,
the rank of the graph is 2*15=30 having 8 vertices and 15 edges
175. A _______ is a graph with no homomorphism to any proper subgraph.
a) poset
b) core
c) walk
d) trail
Ans: B
176. The chromatic number of a graph is the property of ____________
a) graph coloring
b) graph ordering
c) group ordering
d) group coloring
Ans: B
177.  Determine the density of a planar graph with 34 edges and 13 nodes.
 a) 22/21
b) 12/23
c) 328
d) 576
Ans: A
Explanation: The density of a planar graph or network is described as the ratio of
the number of edges(E) to the number of possible edges in a network with(N)
nodes. So, D = E − N + 1/ 2 N − 5. Hence, the required answer is:
D=(34-13+1)/(2*13-5) = 22/21. A completely sparse planar graph has density 0
and a completely dense planar graph has degree 1.
178. A non-planar graph can have ____________
a) complete graph
b) subgraph
c) line graph
d) bar graph
Ans: B
179. An undirected graph G which is connected and acyclic is called ____________
a) bipartite graph
b) cyclic graph
c) tree
d) forest
Ans: C
180. An n-vertex graph has ______ edges.
a) n2
b) n-1
c) n*n
d) n*(n+1)/2
Ans: B
181. A polytree is called _______________
a) directed acyclic graph
b) directed cyclic graph
c) bipartite graph
d) connected graph
Ans: A
182. The tree elements are called __________
a) vertices
b) nodes
c) points
d) edges
Ans: B
183. In an n-ary tree, each vertex has at most ______ children.
a) n
b) n4
c) n*n
d) n-1
Ans: A
184. A linear graph consists of vertices arranged in a line.
a) false
b) true
c) either true or false
d) cannot determined
Ans: B
185. Two labeled trees are isomorphic if ____________
a) graphs of the two trees are isomorphic
b) the two trees have same label
c) graphs of the two trees are isomorphic and the two trees have the same label
d) graphs of the two trees are cyclic
Ans: C
186. A graph which consists of disjoint union of trees is called ______
a) bipartite graph
b) forest
c) caterpillar tree
d) labeled tree
Ans: B
187.  What is a bipartite graph?
a) a graph which contains only one cycle
b) a graph which consists of more than 3 number of vertices
c) a graph which has odd number of vertices and even number of edges
d) a graph which contains no cycles of odd length
Ans: D
188. How many edges are there in a complete graph of order 9?
a) 35
b) 36
c) 45
d) 19
Ans: B
Explanation: In a complete graph of order n, there are n*(n-1) number of edges
and degree of each vertex is (n-1). Hence, for a graph of order 9 there should be 36
edges in total.
189. How many cycles are there in a wheel graph of order 5?
a) 6
b) 10
c) 25
d) 7
Ans: D
Explanation: In a cycle of a graph G if we join all the vertices to the centre point,
then that graph is called a wheel graph. There is always a Hamiltonian cycle in a
wheel graph and there are n2-3n+3 cycles. So, for order 5 the answer should be 7.
190. In preorder traversal of a binary tree the second step is ____________
a) traverse the right subtree
b) traverse the left subtree
c) traverse right subtree and visit the root
 d) visit the root
Ans: B
191. An important application of binary tree is ______
a) Huffman coding
b) stack implementation
c) queue implementation
d) traverse a cyclic graph
Ans: A
Explanation: A binary tree is used to sort a list of elements; the inorder traversal
will do this automatically. Better tree sorting algorithm will involve balancing the
trees. The binary coding, in particular for the Huffman coding is an immediate
application of binary trees.
192.  From the following code identify the which traversal of a binary tree is this
__________
//if node has left child
order(node.left)
//if node has right child
order(node.right)
visit(node)
a) Inorder traversal
b) preorder traversal
c) postorder traversal
d) Euler tour traversal

Ans: C
Explanation: In a postorder traversal of a binary tree first is to traverse the left
subtree, second traverse the right subtree of the tree and third is to visit the node
193. An immediate application of a Depth First Search traversal is __________
a) count the number of leaf nodes
b) perform Inorder traversal in easy way
c) count number of nodes
d) implement preorder traversal
Ans: A
194. What is the postfix expression of 9+3*5/(10-4)?
a) 9 3 + * 5 / 10 4 –
b) 9 3 5 + * / 10 4 –
c) 9 3 + 5 * / 10 4 –
d) 9 3 5 * / + 10 – 4
Ans: C
Explanation: The expression, 9+3*5/(10-4)
= 9+3*5/(10 4-)
= 9+35/*(10 4-)
= 935/*+(10 4-)
So the output is:9 3 5 / * + 10 4 -.(solve by tree diagram)
195. What is the postfix expression of (A+B)-C*(D/E))+F?
 a) A B + C D E / * – F +
b) A B C D E + / * F – +
c) A B C + * D E / F + –
d) A B + C – * D E / F +
Ans: A
Explanation: The expression is (A+B)-C*(D/E))+F
= (A+B)-C*(DE/)+F
= (A+B)-C*(DE/)F+
= (A+B)-C(DE/)*F+
= (A+B)C(DE/)*-F+
= (AB+)C(DE/)*-F+
So the output is: AB+CDE/*-F+.
196. Prefix expression can be evaluated _________
a) from right to left
b) from left to right
c) from the exact middle
d) from second right element
Ans: B
197. _____ is a disjunctive normal form.
a) product-of-sums
b) product-of-subtractions
c) sum-of-products
d) sum-of-subtractions
Ans: C
198. The truth table for (p ∨ q) ∨ (p ∧ r) is the same as the truth table for:
A. p ∨ q
B. (p ∨ q) ∧ r
C. (p ∨ q) ∧ (p ∧ r)
D. (p ∨ q) ∧ (p ∨ r)
Ans: A
199.  How many have all the vowels together in word MISAPPREHENSION:
A. 15!/2!2!2!2!2!
B. 10!/2!2!2! × 6!/2!2!
C. 13!/2!2!2!2!
D. None of the above
Ans: B
200. The Boolean function [∼(∼p∧q)∧∼(∼p∧∼q)]∨(p∧r) is equal to the Boolean
function:
A. q
B. p ∧ r
C. p
D. None of the above
Ans: C
201.  Which of the following statements is FALSE:
A. (P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to ∼Q ∧ ∼P
 B. (P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to Q ∨ P
C. (P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to Q ∨ (P ∧ ∼Q)
D. (P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to [(P ∨ ∼P) ∧ Q] ∨ (P ∧ ∼Q)
Ans: A
202. Hasse diagrams are drawn
A. Partially ordered sets
B. Lattices
C. Boolean algebra
D. None of these
Ans: A
203. The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1
for a, b ∈ Z, is a group. The identity element of this group is
A.0
B. 1
C. -1
D.12
Ans: C
204.  (Z,*) is a group with a*b = a+b+1   ∀ a, b ∈Z. The inverse of a is
A0
.
B.-2
C.a-2
D -a-2
.
Ans: D
205.  Match the following
A. Groups                     I. Associativity
B. Semi groups           II. Identity
C. Monoids                  III. Commutative
D. Abelian Groups      IV Left inverse
A A  B  C  D
. IV  I   II   III
B.A  B  C  D
III  I   IV   II
C.A  B  C  D
II   III  I   IV
D A  B  C  D
. I    II  III  IV
Ans: A
206. Which of the following statements is FALSE ?

A.The set of rational numbers is an abelian group under addition

B.The set of rational integers is an abelian group under addition
C.The set of rational numbers form an abelian group under multiplication
D.None of these
Ans: D
207. The set of all nth roots of unity under multiplication of complex numbers
form a/an
A.semi group with identity
B.commutative semigroups with identity
C.group
D.abelian group
Ans: D
208.  Which of the following is TRUE ?
A.Set of all rational negative numbers forms a group under
multiplication
B.Set of all non-singular matrices forms a group under multiplication
C Set of all matrices forms a group under multipication
.
D Both (b) and (c)
.
Ans: B
209. In the group G = {2, 4, 6, 8) under multiplication modulo 10, the identity
element is
A6
.
B.8
C.4
D2
.
Ans: A
210. Let (Z, *) be an algebraic structure, where Z is the set of integers and the
operation * is defined by n * m = maximum (n, m). Which of the following
statements is TRUE for (Z, *) ?
A.(Z, *) is a monoid
B.(Z, *) is an abelian group
C.(Z, *) is a group
D.None of these
Ans: D
211. In a classroom containing 28 people, there are 18 people who speak English,
15 people who speak Hindi and 22 people who speak Telugu, 9 people speak
both English and Hindi, while 1 people speak both Hindi and Telugu. Further
13 persons speak both Telugu and English. How many people speak all three
languages?
A9
.
B.8 
C.7
D6
.
Ans: D
212. A monoid is always a
A. A group
 B. a commutative group
C. a non abelian group
D. groupoid
Ans: D
213. A monoid is always a
A. A group
B. a commutative group
C. a non abelian group
D. semi - group
Ans: D