Exam answers CS-6201 Discrete Structures 2

{(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)}
{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
{(1, 6), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 3), (4, 5), (4, 6), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} Correct

Array is a common example of a linear structure

True Correct
False

Assume that you have an ordinary deck of 52 playing cards. How many possible 7-card poker hands are there that contain at least one face card (J, Q, K)?

C(12,3) =C2=B7 C(40,4)
C(4,1) =C2=B7 C(4,1) =C2=B7 C(4,1) =C2=B7 C(40,4)
C(12,1) =C2=B7 C(40,6)
C(52,7) - C(40,7) Correct

Binary tree is similar to an array in data structure

True
False Correct

Consider the following finite automaton A over =CE=A3 = {a,b,c}:Which of the following statements about A is/are correct?

=C9=9B =E2=88=88 L(A)
bbaacbabcac =E2=88=88 L(A) Correct
bacabca =E2=88=88 L(A)
The automaton A is a Deterministic Finite Automaton (DFA)

Consider the statement "If the product of two integers is even, then their sum is also even." Which of the following assertions is correct?

The statement is true and can be proven easily using either direct proof or proof by contradiction
The statement is true and can be proven easily using either direct proof or proof by contraposition
The statement is false as you can find a counterexample Correct
The statement is true and can be proven easily using either direct proof

Consider the statement, =E2=80=9CIf n is divisible by 30 then n is divisible by 2 and by 3 and by 5.=E2=80=9D Which of the following statements is equivalent to this statement?

If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30 Correct
If n is not divisible by 30 then n is not divisible by 2 or not divisible by 3 or not divisible by 5
If n is divisible by 2 and divisible by 3 and divisible by 5 then n is divisible by 30
If n is not divisible by 30 then n is divisible by 2 or divisible by 3 or divisible by 5

Data structure is defined as a specialized format for _______ and ______ data

inserting, deleting
processing, sorting
organizing, storing Correct
searching, processing

Describe the process of removing an element at the tail of a singly linked list

Removing an element at the tail of the singly list is very fast
Removing an element at the tail of the singly list is not easy Correct
Element at the tail of the singly linked list cannot be removed or deleted
None of the choices

Doubly linked list doubles the size of a singly linked list

True
False Correct

Every tree with at least two nodes has at least two nodes of what degree?

3
No correct answer
2
0
1 Correct

Finite automata requires minimum _______ number of stacks.

No correct answer
0 Correct
1
3
2

Head and tail are the only important components of a Linked List

True
False Correct

How many bit strings of length four do not have two consecutive 1s?

8 Correct
4
16
0

How many different choices of winners can you have if the draw is limited to first year and second year students and you only have one grand prize? Note: Given that the population of the school is as follows: 1st year = 100 students, 2nd year = 98 students, 3rd year = 102 students, 4th year = 50 students.

98
100
100 x 98
198 Correct

How many different license plates are available if the license plate pattern consists of four letters that cannot be repeated and followed by three digits that can be repeated? (Assume that all letters are uppercase and the digits are 0, 1, ... 9)

C(26,3) =C2=B7 C(10,3)
26 =C2=B7 25 =C2=B7 24 =C2=B7 23 =C2=B7 103 Correct
C(26,3) =C2=B7 103
263 =C2=B7 103

How many different passwords are possible if each password consists of six characters where each character is either an uppercase letter, a lowercase letter, or a digit, and at least one digit must be included in the password? (Note: There are 26 letters and 10 digits)

62 - 266
62 - 526 Correct
62 - 366
62 - 106

How many edges does a tree with V vertices have?

V - 1 Correct
infinite
V2
V + 1
V

How many nodes in a tree have no ancestors?

2
0
3
1 Correct

How many nonisomorphic simple graphs are there with five vertices and three edges?

4 Correct
2
1
3

How many terms does the binomial expansion of (2x+3)99 has?

100 Correct
99
101
infinite

How may isomorphic graphs are there for a graph with n number vertices?

(n + 1)! number of vertices
(2n - 1)! number of vertices
(n - 1)! number of vertices
n! number of vertices Correct

If two finite states machine M and N are isomorphic, thenA. M can be transformed to N, merely re-labelling its statesB. M can be transformed to N, merely re-labelling its edgesWhich is true?

A only Correct
Neither A nor B
Both A and B
B only

If you give a proof by mathematical induction of the statement that 2n &ge= n2, for all integers n =E2=89=A5 4, the basis step requires you to prove that which of the following is true?

20 =E2=89=A5 02
22 =E2=89=A5 22
23 =E2=89=A5 32
24 =E2=89=A5 42 Correct
21 =E2=89=A5 12

Implementation of ADT requires

choosing a suitable algorithm
None of the choices
choosing a data representation Correct
choosing a suitable algorithm and data representation

In a survey of 900 likely voters, the following question was asked: Do you support using cameras to identify red-light runners? The results of the survey follow: AnswerStrongly supportSomewhatsupportSomewhat opposeStronglyOpposeDon't knowRespondents4102129516419 What is the probability that a person in the survey selected at random favors using cameras to identify red-light runners?

069 Correct
059
031
064

In examining the algorithm of multiplying two numbers and displaying the output, what is the next step after declaring the 3 integers a,b,c

define the values of a and b Correct
compute the product of a and b
check if the numbers are positive numbers
check which of the two numbers is higher

In order to get the contents of a binary search tree in ascending order, one has to traverse it in.

post-order
pre-order
in-order Correct
not possible

It is a collection of items stored at adjacent memory location

Hash Table
Queue
Array Correct
Stack

It is impossible for a valid argument to have a true premise and

a conditional conclusion
a false conclusion Correct
a true conclusion
a negated conclusion

Let p and q be propositions. does not imply is:

True Correct
False

Let P be the statement "you can make n cents postage using 3-cent and 5-cent stamps." Suppose you want to use the Principle of Mathematical Induction to show that P is true for all n =E2=89=A5 8.You begin by proving P(8), which is true because 8 cents postage can be made with one 3-cent stamp and one 5-cent stamp.Which of the following will show that the implication P(k) -> P(k+1) in the inductive step is true for all k =E2=89=A5 8?

Take all stamps that are used to make k cents postage, remove a 5-cent postage and replace with a 3-cent stamp
Take all stamps that are used to make k cents postage and add a 3-cent stamp
none of the given Correct
Take all stamps that are used to make k cents postage and add a 5-cent stamp
Take all stamps that are used to make k cents postage, remove three 3-cent postage and replace with two 5-cent stamp

Logic is a system based on __________.

truth values
statements
truth tables
propositions Correct

Nodes in a linked list are also called elements

True Correct
False

One light bulb is selected at random from a lot of 110 light bulbs, of which 2% are defective. What is the probability that the light bulb selected is defective?

The probability is 003
The probability is 001
The probability is 002 Correct
The probability is 004

Order and sorting of array and linked list follows the same process

True
False Correct

Size of a liked list can be zero

True Correct
False

Stack applies the Last In First Out process

True Correct
False

Storage capacity of an array is static

True Correct
False

Suppose inflation decreases the value of money by 3% per year? Which formula describes an = the value (in dollars) of $1000 after n years?

an = (097)n 1000 Correct
an = -003 =C2=B7 1000
an = 1000 (003)n
an = n (097)n 1000

Suppose that P is the statement "n + 1 = n + 2". What is wrong with the following proof that the statement P is true for all non negative integers n?You assume that P(k) is true for some positive integer k, that is, that k+1 = k+2. Then you add 1 to both sides of this equation to obtain k+2 = k+3; therefore P(k+1) is true. By the principle of mathematical induction P is true for all non-negative integers n.

The proof is incorrect because you cannot add one to both sides of the equation in the inductive step
There is nothing wrong with this proof
The proof is incorrect because there is no basis step Correct
The proof is incorrect because the statement used in the inductive hypothesis is incorrect

Suppose you are hired by a company at an initial salary of $30,000. At the end of each year you receive a 3% raise, plus an addition of $1000 on your base salary. Let an equal your salary at the end of n years with the company. Find a recurrence relation for an.

an = (1000 + 003)an-1
an = 103an-1 + 1000 Correct
none of the given
an = 103(an-1 + 1000)

Suppose you want to prove that every product of integers of the form k(k+1)(k+2) is divisible by 6. If you want to prove this by cases, which of the following is a set of cases you would use?

the product ends in 3, the product ends in 6, the product ends in 9
when k is divided by 3, the remainder is 0; when k is divided by 3, the remainder is 1; when k is divided by 3, the remainder is 2 Correct
k = 3n, k =E2=89=A0 3n
k is prime, k is not prime

Suppose you want to use the principle of mathematical induction to prove that 1 + 2 + 22 + 23 + 23 + ... + 2n = + 2n+1 - 1 for all non-negative integers n. Which of theses is the correct statement P(k) in the inductive step?

1 + 2 + 22 + 23 + 24 + + 2k + 2k+1
2k+1 - 1
2k = + 2k+1 - 1
1 + 2 + 22 + 23 + 24 + + 2k = 2k+1 - 1 Correct

Suppose you want to use the principle of mathematical induction to prove that 1 + 2 + 22 + 23 + 23 + ... + 2n = + 2n+1 - 1 for all positive integers n. Which of these is the correct implication p(k) -> P(k+1) to be used in the inductive step?

1 + 2 + 22 + 23 + + 2k = 2k+1 - 1 -> 1 + 2 + 22 + 23 + 24 + + 2k + 2k+1 - 1
1 + 2 + 22 + 23 + + 2k = 2k+1 - 1 -> 1 + 2 + 22 + 23 + + 2k + 2k+1 = 2k+1 - 1 + 2k+1
2k -> 2k+1 - 1
1 + 2 + 22 + 23 + + 2k = 2k+1 - 1 -> 1 + 2 + 22 + 23 + + 2k + 2k+1 = 2k+2 - 1 Correct

Suppose you wish to prove this statement "If n is an integer, then n =E2=89=A4 n3." Which of the following is correct?

The given statement is true and can be proven easily using a direct proof
The given statement is true and can be proven easily using contradiction
The given statement is false because a counterexample can be found Correct
The given statement is true and can be proven easily using mathematical induction

The basic limitation of finite automata is that

All of the mentioned
It cannot remember arbitrary large amount of information Correct
It cannot process strings of length greater than 5
It sometimes fails to recognize regular grammar
It sometimes recognize grammar that are not regular

The first node of a linked list is called head

True Correct
False

The first node of a linked list is referred to as _____________

main node
base pointer
head Correct
pointer

The first step to insert an element at the tail of a linked list is to increase the size of the linked list

True
False Correct

The following breakdown of a total of 18,686 transportation fatalities that occured in 2007 was obtained from records compiled by the U.S. Department of Transportation (DOT).Mode of Transportation Car Train Bicycle PlaneNumber of Fatalities 16,525842698538 What is the probability that a victim randomly selected from this list of transportation fatalities for 2007 died in a train or a plane accident? Round answer to two decimal places.

005
008
011
007 Correct

The method of accessing the element in a linked list is _______

Traversal Correct
By specifying the index element
Random access
Direct access

The next node in the sequence of a linked list is called _____________

Link
Pointer
Node's successor Correct
Index

The number of cars entering a tunnel leading to an airport in a major city over a period of 200 peak hours was observed and the following data were obtained: Cars, xFrequency of Occurrence152040654515Find the empirical probability distribution for this experiment.

Cars, x Frequency of Occurrence 0075 0225 0325 02 01 0075
Cars, x Frequency of Occurrence 0075 01 02 0325 0225 0075 Correct
Cars, x Frequency of Occurrence 0077 0102 0198 0323 0225 0075
Cars, x Frequency of Occurrence 0077 0098 0202 0323 0227 0073

The number of different directed trees with 3 nodes is

2
4
3 Correct
5

The number of leaf nodes in a complete binary tree of depth d is

2d-1+1
2d Correct
2d+1+1
2d+1

The preorder and post order traversal of a Binary Tree generates the same output. The tree can have maximum

Two nodes
Any number of nodes
One node Correct
Three nodes

The results of a time study conducted by the production manager of Ace Novelty are shown in the accompanying table, where the number of space action-figures produced each quarter hour during an 8-hour workday has been tabulated. Find the empirical probability distribution associated with this experiment. Figures Produced (in dozens)Frequency of Occurrence304310326336348356362

Figures Frequency of Produced Occurrence(in dozens)30 0062531 0187532 02533 0187534 0187535 036 0125
Figures Frequency of Produced Occurrence(in dozens)30 0125131 032 0187633 0187434 0249935 0187536 00625
Figures Frequency of Produced Occurrence(in dozens)30 0125131 0000132 0187433 0187434 02535 0187536 00625
Figures Frequency of Produced Occurrence(in dozens)30 012531 032 0187533 0187534 02535 0187536 00625 Correct

The size need to be specific during declaration

The statement applies to both array and linked list
The statement describes a linked list
The statement describes an array Correct
The statement does not apply to array and linked list

The size of a singly linked list is always fixed but the size can grow dynamically

True
False Correct

This is a statement that is always false.

tautology
implication
conjunction
contradiction Correct

Discrete Structures 2

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