MTH202 (FINALTERM EXAM Spring 2009))


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 ® q is

q ®p

► ~q ®~p

► ~p ®~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

  is

       ► Arithmetic series

Arithmetic series

► Geometric series

► Arithmetic sequence

► Geometric sequence 

 

Question No: 9    ( Marks: 1 )    – Please choose one

  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(AÈB) =

► P(A) + P(B) + P(AÇB)

► P(A) + P(B) + P(AUB)

► P(A) + P(B) – P(AÇB)

► P(A) + P(B) – P(AÇB)

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

 

Question No: 12    ( Marks: 1 )    – Please choose one

 If  then the events A and B are called

       ► Independent

► Dependen

► 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

 

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

 

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 pàq 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

 

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

► 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   is

► 0

► n(n-1)

       ►

► 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

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)

► 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)

► 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

       ► 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(AÈB) = n(A) + n(B)

► n(AÈB) = n(A) + n(B) – n(AÇB)

► n(AÈB)= ø

► 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 Ç B) È C

► (A È Bc) È C

► (AÇ Bc) È  Cc

(A Ç B) Ç  Cc

 

Question No: 38    ( Marks: 1 )    – Please choose one

 Let A,B,C be the subsets of auniversal setU.

Then  is equal to:

?

?

?

?

 

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.

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

 

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

…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 aHamilton circuit? If  it 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 = 450×25 + 175

 450 = 175×2 + 100

 175 = 100×1 + 75

 100 = 75×1 + 25

 75 = 25×3 + 0

 

Linear combination= 25 = 127×450 + -5×11425

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.

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