FINALTERM EXAMINATION
Fall 2009
MTH202- Discrete Mathematics
Time: 120 min
Marks: 80
Question No: 1 ( Marks: 1 ) - Please choose one
Let A = {a, b, c} and
R = {(a, c), (b, b), (c, a)} be a relation on A. Is R
► Transitive
► Reflexive
► Symmetric
► Transitive and Reflexive
Question No: 2 ( Marks: 1 ) - Please choose one
Symmetric and antisymmetric are
► Negative of each other
► Both are same
► Not negative of each other
Question No: 3 ( Marks: 1 ) - Please choose one
The statement p ↔ q ≡ q ↔ p describes
► Commutative Law:
► Implication Laws:
► Exportation Law:
► Equivalence:
Question No: 4 ( Marks: 1 ) - Please choose one
The relation as a set of ordered pairs as shown in figure is
► {(a,b),(b,a),(b,d),(c,d)}
► {(a,b),(b,a),(a,c),(b,a),(c,c),(c,d)}
► {(a,b), (a,c), (b,a),(b,d), (c,c),(c,d)}
► {(a,b), (a,c), (b,a),(b,d),(c,d)}
Question No: 5 ( Marks: 1 ) - Please choose one
The statement p →q ≡ (p ∧ ~q) →c describes
► Commutative Law:
► Implication Laws:
► Exportation Law:
► Reductio ad absurdum
http://vustudents.ning.com
Question No: 6 ( Marks: 1 ) - Please choose one
A circuit with one input and one output signal is called.
► NOT-gate (or inverter)
► OR-
► AND- gate
► None of these
Question No: 7 ( Marks: 1 ) - Please choose one
If f(x)=2x+1,
g(x)=x2 -1
then fg(x)=
► x2 -1
► 2x2 -1
► 2x3 -1
ee bhai aap f(x) main x ki jagah g(x) put karen then
simplify it
like this
2(x^2 - 1) + 1 = 2x^2 -2 + 1 = 2x^2 - 1
Question No: 8 ( Marks: 1 ) - Please choose one
Let g be the functions defined by
g(x)= 3x+2 then gog(x) =
►
9x2 + 4
► 6x+4
► 9x+8
Question No: 9 ( Marks: 1 ) - Please choose one
How many integers from 1 through 1000 are neither multiple of 3 nor multiple of
5?
► 333
► 467
► 533
► 497
Question No: 10 ( Marks: 1 ) - Please choose one
What is the smallest integer N such that
9
6
⎡ N ⎤ = ⎢⎢ ⎥⎥
► 46
► 29
► 49
Question No: 11 ( Marks: 1 ) - Please choose one
What is the probability of getting a number greater than 4 when a die is thrown?
►
1
2
►
3
2
►
1
3
Question No: 12 ( Marks: 1 ) - Please choose one
If A and B are two disjoint (mutually exclusive)
events then
P(A∪B) =
► P(A) + P(B) + P(A∩B)
► P(A) + P(B) + P(AUB)
► P(A) + P(B) - P(A∩B)
► P(A) + P(B) - P(A∩B)
► P(A) + P(B)
Question No: 13 ( 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 http://vustudents.ning.com
► 1
►
1
3
►
2
3
Question No: 14 ( Marks: 1 ) - Please choose one
The probability of getting 2 heads in two successive tosses of a balanced coin is
►
1
4
►
1
2
►
2
3
Question No: 15 ( Marks: 1 ) - Please choose one
The probability of getting a 5 when a die is thrown?
►
1
6
►
5
6
►
1
3
Question No: 16 ( Marks: 1 ) - Please choose one
If a coin is tossed then what is the probability that the number is 5
►
1
2
► 0
► 1
Question No: 17 ( Marks: 1 ) - Please choose one
If A and B are two sets then The set of all elements that belong to both A
and B , is
► A ∪ B
► A ∩ B
► A--B
► None of these
Question No: 18 ( Marks: 1 ) - Please choose one
What is the expectation of the number of heads when three fair coins are
tossed? http://vustudents.ning.com
► 1
► 1.34
► 2
► 1.5
Question No: 19 ( Marks: 1 ) - Please choose one
If A, B and C are any three events, then
P(A∪B∪C) is equal to
► P(A) + P(B) + P(C)
► P(A) + P(B) + P(C)- P(A∩B) - P (A ∩C) - P(B ∩C) + P(A ∩B ∩C)
► P(A) + P(B) + P(C) - P(A∩B) - P (A ∩C) - P(B ∩C)
► P(A) + P(B) + P(C) + P(A ∩B ∩C)
Question No: 20 ( 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: 21 ( Marks: 1 ) - Please choose one
The power set of a set A is the set of all subsets of A, denoted P(A).
► False
► True
Question No: 22 ( Marks: 1 ) - Please choose one
A walk that starts and ends at the same vertex is called
► Simple walk
► Circuit
► Closed walk
Question No: 23 ( Marks: 1 ) - Please choose one
If a graph has any vertex of degree 3 then
► It must have Euler circuit
► It must have Hamiltonian circuit
► It does not have Euler circuit
Question No: 24 ( Marks: 1 ) - Please choose one
The square root of every prime number is irrational
► True
► False
► Depends on the prime number given
Question No: 25 ( Marks: 1 ) - Please choose one
A predicate is a sentence that contains a finite number of variables
and becomes a statement when specific values are substituted for the
variables
► True
► False
► None of these
Question No: 26 ( Marks: 1 ) - Please choose one
If r is a positive integer then gcd(r,0)=
► r
► 0
► 1
► None of these
Question No: 27 ( Marks: 1 ) - Please choose one
Combinatorics is the mathematics of counting and arranging objects
► True
► False
► Cannot be determined
Question No: 28 ( Marks: 1 ) - Please choose one
A circuit that consist of a single vertex is called
► Trivial
► Tree
► Empty
Question No: 29 ( Marks: 1 ) - Please choose one
In the planar graph, the graph crossing number is
► 0
► 1
► 2
► 3
Question No: 30 ( Marks: 1 ) - Please choose one
How many ways are there to select five players from a 10 member
tennis team to make a trip to a match to another school?
► C(10,5)
► C(5,10)
► P(10,5)
► None of these
Question No: 31 ( Marks: 1 ) - Please choose one
The value of 0! Is
► 0
► 1
► Cannot be determined
Question No: 32 ( Marks: 1 ) - Please choose one
If the transpose of any square matrix and that matrix are same then matrix is
called
► Additive Inverse
► Hermition Matrix
► Symmetric Matrix
Question No: 33 ( Marks: 1 ) - Please choose one
The value of
( )
( 1)!
1 !
n
n
−
+
is
► 0
► n(n-1)
►
( 2 )
1
n + n
► Cannot be determined
Question No: 34 ( Marks: 1 ) - Please choose one
If A and B are two disjoint sets then which of the following must be true
► n(A∪B) = n(A) + n(B)
► n(A∪B) = n(A) + n(B) - n(A∩B)
► n(A∪B)= ø
► None of these
Question No: 35 ( Marks: 1 ) - Please choose one
Any two spanning trees for a graph
► Does not contain same number of edges
► Have the same degree of corresponding edges
► contain same number of edges
► May or may not contain same number of edges
Question No: 36 ( Marks: 1 ) - Please choose one
When P(k) and P(k+1) are true for any positive integer k, then P(n) is not true for all +ve
Integers.
► True
► False
Question No: 37 ( Marks: 1 ) - Please choose one
n2 > n+3 for all integers n ≥3.
► True
► False
Question No: 38 ( Marks: 1 ) - Please choose one
Quotient –Remainder Theorem states that for any positive integer d, there exist unique
integer q and r such that _______________ and 0≤r<d.
► n=d.q+ r
► n=d.r+ q
► n=q.r+ d
► None of these
Question No: 39 ( Marks: 1 ) - Please choose one
Euler formula for graphs is
► f = e-v
► f = e+v +2
► f = e-v-2
► f = e-v+2
Question No: 40 ( Marks: 1 ) - Please choose one
The degrees of {a, b, c, d, e} in the given graph is
a b
d c
e
► 2, 2, 3, 1, 1
► 2, 3, 1, 0, 1
► 0, 1, 2, 2, 0
► 2,3,1,2,0
Question No: 41 ( Marks: 2 )
Let
1 3 7
5 2 9
A
⎡ ⎤
=⎢ ⎥
⎣ ⎦
then find At
Question No: 42 ( Marks: 2 )
Write the contra positive of the following statements:
1. For all integers n, if n2 is odd then n is odd.
2. If m and n are odd integers, then m+n is even integer.
Question No: 43 ( Marks: 2 )
How many distinguishable ways can the letter of the word
HULLABALOO be arranged.
Question No: 44 ( Marks: 3 )
Find the variance σ2 of the distribution given in the following table.
xi 1 3 4 5
f(xi) 0.4 0.1 0.2 0.3
Question No: 45 ( Marks: 3 )
Prove that every integer is a rational number.
Question No: 46 ( Marks: 3 )
a. Evaluate P(5,2)
b. How many 5-permutations are there of a set of five objects?
Question No: 47 ( Marks: 5 )
Is it possible to have a simple graph with four vertices of degree 1, 1, 3, and 3.If
no then give reason?(Justify your answer)
Question No: 48 ( Marks: 5 )
Find the GCD of 500008, 78 using Division Algorithm.
Question No: 49 ( Marks: 5 )
Find the M number of ways that ten chocolates can be divided among
three children if the youngest child is to receive four chocolates and each
of the others three chocolates.
Question No: 50 ( Marks: 10 )
In the graph below, determine whether the following walks are paths, simple
paths, closed walks, circuits,
simple circuits, or are just walk?
i) v0 e1 v1 e10 v5 e9 v2 e2 v1
ii) v4 e7 v2 e9 v5 e10 v1 e3 v2 e9 v5
iii) v2
iv) v5 v2 v3 v4 v4v5
v) v2 v3 v4 v5 v2v4 v3 v2
By ADEEL ABBAS, Bhakkar. AdeelAbbasbk@gmail.com
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