**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 1 ^{st} matrix is equal to number of rows in 2^{nd} matrix **

► Number of rows of 1^{st} matrix is equal to number of columns in 2^{nd} 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

** ► **^{n}**C _{k}**

► ^{k}C_{n}

► ^{n}P_{k}

► ^{k}P_{k}

**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 B^{c}) È C

► (AÇ B^{c}) È C^{c}

**► ****(A Ç B) Ç C ^{c}**

**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 ! >2^{n} 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,

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.