FINALTERM EXAMINATION
Spring 2009
MTH202- Discrete Mathematics (Session - 2)
Time: 120 min
Marks: 80
Question No: 1 ( Marks: 1 ) - Please choose one
The negation of “Today is Friday” is
► Today is Saturday
► Today is not Friday
► Today is Thursday
Question No: 2 ( Marks: 1 ) - Please choose one
An arrangement of rows and columns that specifies the truth
value of a compound proposition for all possible truth values of its
constituent propositions is called
► Truth Table
► Venn diagram
► False Table
► None of these
Question No: 3 ( Marks: 1 ) - Please choose one
The converse of the conditional statement p R q is
► q Rp
► ~q R~p
► ~p R~q
► None of these
Question No: 4 ( Marks: 1 ) - Please choose one
Contrapositive of given statement “If it is raining, I will take
an umbrella” is
► I will not take an umbrella if it is not raining.
► I will take an umbrella if it is raining.
► It is not raining or I will take an umbrella.
► None of these.
Question No: 5 ( Marks: 1 ) - Please choose one
Let A= {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3),(4,4)} then
► R is symmetric.
► R is anti symmetric.
► R is transitive.
► R is reflexive.
► All given options are true
Question No: 6 ( Marks: 1 ) - Please choose one
A binary relation R is called Partial order relation if
► It is Reflexive and transitive
► It is symmetric and transitive
► It is reflexive, symmetric and transitive
► It is reflexive, antisymmetric and transitive
Question No: 7 ( Marks: 1 ) - Please choose one
How many functions are there from a set with three
elements to a set with two elements?
► 6
► 8
► 12
Question No: 8 ( Marks: 1 ) - Please choose one
1,10,102 ,103 ,104 ,105 ,106 ,107 ,................ is
► Arithmetic series
► Geometric series
► Arithmetic sequence
► Geometric sequence
Question No: 9 ( Marks: 1 ) - Please choose one
éêxùú for x = -2.01 is
► -2.01
► -3
► -2
► -1.99
Question No: 10 ( Marks: 1 ) - Please choose one
If A and B are two disjoint (mutually exclusive)
events then
P(AUB) =
► P(A) + P(B) + P(ACB)
► P(A) + P(B) + P(AUB)
► P(A) + P(B) - P(ACB)
► P(A) + P(B) - P(ACB)
► P(A) + P(B)
Question No: 11 ( Marks: 1 ) - Please choose one
If a die is thrown then the probability that the dots on the top
are prime numbers or odd numbers is
► 1
► 1
3
► 2
3
Question No: 12 ( Marks: 1 ) - Please choose one
If P(AÇB) ¹ P(A)P(B) then the events A and B are called
► Independent
► Dependent page 270
► Exhaustive
Question No: 13 ( Marks: 1 ) - Please choose one
A rule that assigns a numerical value to each outcome in a
sample space is called
► One to one function
► Conditional probability
► Random variable
Question No: 14 ( Marks: 1 ) - Please choose one
The expectation of x is equal to
► Sum of all terms
► Sum of all terms divided by number of terms
► åxf (x)
Question No: 15 ( Marks: 1 ) - Please choose one
The degree sequence {a, b, c, d, e} of the given graph is
► 2, 2, 3, 1, 1
► 2, 3, 1, 0,1
► 0, 1, 2, 2, 0
► 2, 3,1,2,0 page305
Question No: 16 ( Marks: 1 ) - Please choose one
Which of the following graph is not possible?
► Graph with four vertices of degrees 1, 2, 3 and 4.
► Graph with four vertices of degrees 1, 2, 3 and 5.
► Graph with three vertices of degrees 1, 2 and 3.
► Graph with three vertices of degrees 1, 2 and 5.
Question No: 17 ( Marks: 1 ) - Please choose one
The graph given below
► Has Euler circuit
► Has Hamiltonian circuit
► Does not have Hamiltonian circuit
Question No: 18 ( Marks: 1 ) - Please choose one
Let n and d be integers and d 1 0. Then n is divisible by d or d
divides n
If and only if
► n= k.d for some integer k
► n=d
► n.d=1
► none of these
Question No: 19 ( Marks: 1 ) - Please choose one
The contradiction proof of a statement paq involves
► Considering p and then try to reach q
► Considering ~q and then try to reach ~p
► Considering p and ~q and try to reach contradiction
► None of these
Question No: 20 ( Marks: 1 ) - Please choose one
An integer n is prime if, and only if, n > 1 and for all positive
integers r and s, if
n = r·s, then
► r = 1 or s = 1.
► r = 1 or s = 0.
► r = 2 or s = 3.
► None of these
Question No: 21 ( Marks: 1 ) - Please choose one
The method of loop invariants is used to prove correctness of
a loop with respect to certain pre and post-conditions.
► True
► False
► None of these
Question No: 22 ( Marks: 1 ) - Please choose one
The greatest common divisor of 27 and 72 is
► 27
► 9
► 1
► None of these
Question No: 23 ( Marks: 1 ) - Please choose one
If a tree has 8 vertices then it has
► 6 edges
► 7 edges
► 9 edges
Question No: 24 ( Marks: 1 ) - Please choose one
Complete graph is planar if
► n = 4
► n>4
► n £ 4
Question No: 25 ( Marks: 1 ) - Please choose one
The given graph is
► Simple graph
► Complete graph
► Bipartite graph
► Both (i) and (ii)
► Both (i) and (iii)
Question No: 26 ( Marks: 1 ) - Please choose one
The value of 0! Is
► 0
► 1 pg160
► Cannot be determined
Question No: 27 ( Marks: 1 ) - Please choose one
Two matrices are said to confirmable for multiplication if
► Both have same order
► Number of columns of 1st matrix is equal to number
of rows in 2nd matrix
► Number of rows of 1st matrix is equal to number of
columns in 2nd matrix
Question No: 28 ( Marks: 1 ) - Please choose one
The value of (-2)! Is
►0
1
►Cannot be determined
Question No: 29 ( Marks: 1 ) - Please choose one
The value of ( )
( 1)!
1 !
n
n
+
- is
► 0
► n(n-1)
► n2 + n
► Cannot be determined
Question No: 30 ( Marks: 1 ) - Please choose one
The number of k-combinations that can be chosen from a set of n
elements can be written as
► nCk pg223
► kCn
► nPk
► kPk
Question No: 31 ( Marks: 1 ) - Please choose one
If the order does not matter and repetition is allowed then
total number of ways for selecting k sample from n. is
► nk
► C(n+k-1,k) page 228
► P(n,k)
► C(n,k)
Question No: 32 ( Marks: 1 ) - Please choose one
If the order matters and repetition is not allowed then total
number of ways for selecting k sample from n. is
► nk
► C(n+k-1,k)
► P(n,k) page 228
► C(n,k)
Question No: 33 ( Marks: 1 ) - Please choose one
To find the number of unordered partitions, we have to count the
ordered partitions and then divide it by suitable number to erase
the order in partitions
► True pg231
► False
► None of these
Question No: 34 ( Marks: 1 ) - Please choose one
A tree diagram is a useful tool to list all the logical
possibilities of a sequence of events where each event can
occur in a finite number of ways.
► True
► False
Question No: 35 ( Marks: 1 ) - Please choose one
If A and B are finite (overlapping) sets, then which of the
following must be true
► n(AEB) = n(A) + n(B)
► n(AEB) = n(A) + n(B) - n(ACB)
► n(AEB)= o
► None of these
Question No: 36 ( Marks: 1 ) - Please choose one
What is the output state of an OR gate if the inputs are 0 and 1?
► 0
► 1
► 2
► 3
Question No: 37 ( Marks: 1 ) - Please choose one
In the given Venn diagram shaded area represents:
► (A C B) E C
► (A E Bc ) E C
► (AC Bc) E Cc page 53
► (A C B) C Cc
Question No: 38 ( Marks: 1 ) - Please choose one
Let A,B,C be the subsets of a universal set U.
Then (AÈB)ÈC is equal to:
► AÇ(BÈC)
► AÈ(BÇC)
►Æ
► AÈ(BÈC)
Question No: 39 ( Marks: 1 ) - Please choose one
n ! >2n for all integers n 34.
► True
► False
Question No: 40 ( Marks: 1 ) - Please choose one
+,-,´,¸ are
► Geometric expressions
► Arithmetic expressions
► Harmonic expressions
Question No: 41 ( Marks: 2 )
Find a non-isomorphic tree with five vertices.
There are three non-isomorphic trees with five vertices as shown
(where every tree with five vertices has 5-1=4 edges).
Question No: 42 ( Marks: 2 )
Define a predicate.
A predicate is a sentence that contains a finite number of
variables and becomes a statement when specific values are
substituted for the variables.
The domain of a predicate variable is the set of all values that
may be substituted in place of the variable.
Let the declarative statement:
“x is greater than 3”.
We denote this declarative statement by P(x) where
x is the variable,
P is the predicate “is greater than 3”.
The declarative statement P(x) is said to be the value of the
propositional function P at x.
Question No: 43 ( Marks: 2 )
Write the following in the factorial form:
(n +2)(n+1) n
( 2)( 1) !
!
n n n
n
+ +
Question No: 44 ( Marks: 3 )
Determine the probability of the given event
“An odd number appears in the toss of a fair die”
Sample space will be..S={1,2.3,4,5,6}…there are 3 odd numbers so,
For odd numbers, probability will be
3
6…Ans
Question No: 45 ( Marks: 3 )
Determine whether the following graph has Hamiltonian circuit.
This graph is not a Hamiltonian circuit, because it does not
satisfy all conditions of it.
E.g. it has unequal number of vertices and edges. And its path
cannot be formed without repeating vertices.
Question No: 46 ( Marks: 3 )
Prove that If the sum of any two integers is even, then so is their
difference.
Theorem: ∀ integers m and n, if m + n is even, then so is m - n.
Proof:
Suppose m and n are integers so that m + n is even. By definition
of even, m + n = 2k for some integer k. Subtracting n from both
sides gives m = 2k - n. Thus,
m - n = (2k - n) -
n
by
substitution
= 2k - 2n
combining
common
terms
= 2(k - n)
by
factoring
out a 2
But (k - n) is an integer because it is a difference of integers.
Hence, (m - n) equals 2 times an integer, and so by definition of
even number, (m - n) is even.
This completes the proof.
Question No: 47 ( Marks: 5 )
Show that if seven colors are used to paint 50 heavy bikes, at
least 8 heavy bikes will be the same color.
N=50
K=7
C(7+50-1,7)
C(56,7)
56!/(56-7)!7!
56!/49!.7!
Question No: 48 ( Marks: 5 )
Determine whether the given graph has a Hamilton
does, find such a circuit, if it does not , given an argument to
show why no such circuit exists.
(a)
This graph does not have Hamiltonian circuit, because it does not
satisfy the conditions. Circuit may not be completed without
repeating edges. It has also unequal values of edges and vertices.
(b)
This graph is a Hamiltonian circuit ..Its path is a b c d e a
Question No: 49 ( Marks: 5 )
Find the GCD of 11425 , 450 using Division Algorithm.
LCM = 205650
11425 = 450x25 + 175
450 = 175x2 + 100
175 = 100x1 + 75
100 = 75x1 + 25
75 = 25x3 + 0
Linear combination= 25 = 127x450 + -5x11425
GCD= 25…Ans
Question No: 50 ( Marks: 10 )
Write the adjacency matrix of the given graph also find
transpose and product of adjacency matrix and its transpose
(if not possible then give reason)
Adjacency matrix= 0 1 0 0 0
1 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 0
Transpose = 0 1 0 0 0
1 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 0
Its transpose is not possible…it’s same. Because there is no loop.
It is not directed graph.
By ADEEL ABBAS, Bhakkar. AdeelAbbasbk@gmail.com
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