DISCRETE MATHS EXAM LATEST
VERSION WITH CORRECT DETAILED
ANSWERS GRADED A+
Which of the following is a linear recurrence relation? - Answer-un = 3 un - 1
Which is the correct order for the steps to find a solution of a homogeneous linear
recurrence? - Answer-(1) find the characteristic equation
(2) find the roots of the characteristic equation
(3) compute the solution coefficients
The sequence <1, 3, 5, 11, 21, 43, ... > has the recursive formula and characteristic
equation stated below. Mark the answer that has its solution. - Answer-ai = 4/2
What is the difference between homogeneous linear recurrences and non-
homogeneous linear recurrences? - Answer-The homogeneous linear recurrences
express its elements exclusively as a function of its preceding elements, while the non-
homogeneous linear recurrences may also express its elements as a function of the
position i of the element.
The Fibonacci sequence, < 0, 1, 1, 2, 3, 5, 8, ...> has the recursive formula and initial
values as stated below. Mark the correct characteristic equation for this version of the
Fibonacci sequence. - Answer-s2-s-1 = 0
A given homogeneous linear recurrence has the following characteristic equation: s + 1
= 0. Which is the order of this homogeneous linear recurrence? - Answer-Order 1
Find the solution to the recurrence relation f(n) = 8 f(n/2) + n3. - Answer-O(n3 logn)
Find the solution to the recurrence relation f(n) = 16 f(n/4) + n. - Answer-O(n2)
Given the recurrence f(n) = 4 f(n/2) + 1, how many sub-problems will a divide-and-
conquer algorithm divide the original problem into, and what will be the size of those
sub-problems? - Answer-4 sub-problems, each of size n/2
Find the solution to the recurrence relation f(n) = 4 f(n/8) + n2. - Answer-O(n2)
What does the solution to a recurrence for a divide-and-conquer algorithm estimate? -
Answer-The number of operations needed to solve the problem
,Knowing the generating function corresponding to the sequence of the natural numbers
(except 0) in the first row on the image below, which would be the generating function
for the sequence in the second row of the image below? - Answer-x2/(1-x)2
Knowing the sequences A and B and their corresponding generation functions, which is
the generation function for sequence C? - Answer-1/1-x2
Which of these sequences corresponds to the generating function below? - Answer-< 1
, 2 , 4 , 8 , 16 , ... >
Knowing the generation function for the sequence of the natural numbers except 0 as
below in the first row, how would you describe the sequence for the generation function
in the second row? - Answer-The even natural numbers, except 0.
How do we find the number of items in neither of two sets? - Answer-Find the number of
items in the union of the two sets and subtract from the number of items in the universe.
Consider an example with 100 students where 20 are taking discrete math, 30 are
taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking
discrete math and web design, 10 are taking Java and web design and 5 are taking all
three classes. How many students are taking discrete math only? - Answer-11
Consider an example with 100 students where 20 are taking discrete math, 30 are
taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking
discrete math and web design, 10 are taking Java and web design and 5 are taking all
three classes. How many students are taking neither of the three classes? - Answer-44
When finding the number of items in either of two sets, why does it not work to simply
add the number of items in each of the two individual sets? - Answer-The items that are
in common with both sets will be counted too many times.
What is the term for the collection of elements from either of two sets? - Answer-Union
Which of the following describes a loop? - Answer-This edge connects point C to point
C.
How many vertices are in this graph? 645321 - Answer-6
Which of the following will get you from point 3 to point 5? - Answer-3 to 2 to 5
How many vertices are in this graph? figure - Answer-5
Which graph in discrete mathematics has a path of edges between every pair of
vertices in the graph? - Answer-A connected graph
,Suppose that a dating service is determining couples that would make a good match. In
the graph shown, two sets of clients are represented with vertices, and the edges
between the vertices indicate that the clients make a good match. Based on the graph,
how many possible matches are there for Erin? - Answer-3
In the graph shown, the vertices represent cities, and the edges represent flights
between those cities. The weights of the edges represent the cost of the flights, in
hundreds of dollars. Based on this, what two cities are the cheapest to fly between, and
what is the cost? - Answer-Flight: City A and City E
Cost: $200
Which of the following statements is NOT true? - Answer-There are only three types of
graphs in discrete mathematics.
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{3,4}' an example of? - Answer-Multi-edges
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{4,1} ' an example of? - Answer-Simple
graph
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{3,3} ' an example of? - Answer-Loop
These two graphs are because the second graph can be obtained from the first
by dividing some edges of the first with more vertices. - Answer-homomorphic
Under what condition is the graph isomorphic? - Answer-All of the above answers are
correct
How many vertices can be odd in a Eulerian graph? - Answer-0
Which of the following is a Euler path? 12034 - Answer-0, 2, 1, 0, 3, 4
At most, of the verticies will be of an odd degree in a semi-Eulerian graph. -
Answer-2
How many Euler paths are there in this graph? 1343 - Answer-4
How many Euler circuits are in this graph? 1234 - Answer-8
How many times are you allowed to pass an edge in an Euler path or circuit? - Answer-
1
A graph with an Euler path can have at most how many odd vertices? - Answer-2
How many Euler paths does this graph have? 1342 - Answer-4
, Which of the following is an Euler path for this graph? 12cross 3 4 - Answer-2, 3, 4, 2, 1,
4
A graph with an Euler circuit can have at most how many odd vertices? - Answer-0
If vertex 3 is your starting point, which vertex can you not go to? - Answer-1
What kinds of graphs does Fleury's algorithm work for? - Answer-Graphs with an Euler
path or circuit in it.
If a graph has 0 odd vertices, at which vertex must you start? - Answer-Any vertex will
work.
Which vertices can be your starting point? 45123 - Answer-1 or 3
How many odd vertices can a graph have in order to use Fleury's algorithm? - Answer-2
What can be said about this graph? 1234 - Answer-It has an Euler circuit.
What can be said about this graph?45123 - Answer-It has an Euler path.
The start and end points must be which two vertices? 1324 - Answer-1 and 3
If a graph has 15 edges, what must the degrees of the vertices add up to? - Answer-30
The total number of degrees in a graph is 20. How many edges does it have? - Answer-
10
Eulerize this graph. 12 cross 34 - Answer-Connect vertices 2 and 4 with another edge.
Which two vertices can you connect to Eulerize this graph? 45123 - Answer-1 and 3
According to Fleury's algorithm, how many odd vertices does a graph with an Euler path
in it have? - Answer-2
According to Fleury's algorithm, how many odd vertices does a graph with an Euler
circuit in it have? - Answer-0
How many odd vertices does this graph have? 12 cross 34 - Answer-2
How many times do you visit a vertex when traveling either a Hamilton circuit or path? -
Answer-1
If you started at vertex 1, at which vertex will you end up if you are traveling a Hamilton
circuit? - Answer-1
VERSION WITH CORRECT DETAILED
ANSWERS GRADED A+
Which of the following is a linear recurrence relation? - Answer-un = 3 un - 1
Which is the correct order for the steps to find a solution of a homogeneous linear
recurrence? - Answer-(1) find the characteristic equation
(2) find the roots of the characteristic equation
(3) compute the solution coefficients
The sequence <1, 3, 5, 11, 21, 43, ... > has the recursive formula and characteristic
equation stated below. Mark the answer that has its solution. - Answer-ai = 4/2
What is the difference between homogeneous linear recurrences and non-
homogeneous linear recurrences? - Answer-The homogeneous linear recurrences
express its elements exclusively as a function of its preceding elements, while the non-
homogeneous linear recurrences may also express its elements as a function of the
position i of the element.
The Fibonacci sequence, < 0, 1, 1, 2, 3, 5, 8, ...> has the recursive formula and initial
values as stated below. Mark the correct characteristic equation for this version of the
Fibonacci sequence. - Answer-s2-s-1 = 0
A given homogeneous linear recurrence has the following characteristic equation: s + 1
= 0. Which is the order of this homogeneous linear recurrence? - Answer-Order 1
Find the solution to the recurrence relation f(n) = 8 f(n/2) + n3. - Answer-O(n3 logn)
Find the solution to the recurrence relation f(n) = 16 f(n/4) + n. - Answer-O(n2)
Given the recurrence f(n) = 4 f(n/2) + 1, how many sub-problems will a divide-and-
conquer algorithm divide the original problem into, and what will be the size of those
sub-problems? - Answer-4 sub-problems, each of size n/2
Find the solution to the recurrence relation f(n) = 4 f(n/8) + n2. - Answer-O(n2)
What does the solution to a recurrence for a divide-and-conquer algorithm estimate? -
Answer-The number of operations needed to solve the problem
,Knowing the generating function corresponding to the sequence of the natural numbers
(except 0) in the first row on the image below, which would be the generating function
for the sequence in the second row of the image below? - Answer-x2/(1-x)2
Knowing the sequences A and B and their corresponding generation functions, which is
the generation function for sequence C? - Answer-1/1-x2
Which of these sequences corresponds to the generating function below? - Answer-< 1
, 2 , 4 , 8 , 16 , ... >
Knowing the generation function for the sequence of the natural numbers except 0 as
below in the first row, how would you describe the sequence for the generation function
in the second row? - Answer-The even natural numbers, except 0.
How do we find the number of items in neither of two sets? - Answer-Find the number of
items in the union of the two sets and subtract from the number of items in the universe.
Consider an example with 100 students where 20 are taking discrete math, 30 are
taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking
discrete math and web design, 10 are taking Java and web design and 5 are taking all
three classes. How many students are taking discrete math only? - Answer-11
Consider an example with 100 students where 20 are taking discrete math, 30 are
taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking
discrete math and web design, 10 are taking Java and web design and 5 are taking all
three classes. How many students are taking neither of the three classes? - Answer-44
When finding the number of items in either of two sets, why does it not work to simply
add the number of items in each of the two individual sets? - Answer-The items that are
in common with both sets will be counted too many times.
What is the term for the collection of elements from either of two sets? - Answer-Union
Which of the following describes a loop? - Answer-This edge connects point C to point
C.
How many vertices are in this graph? 645321 - Answer-6
Which of the following will get you from point 3 to point 5? - Answer-3 to 2 to 5
How many vertices are in this graph? figure - Answer-5
Which graph in discrete mathematics has a path of edges between every pair of
vertices in the graph? - Answer-A connected graph
,Suppose that a dating service is determining couples that would make a good match. In
the graph shown, two sets of clients are represented with vertices, and the edges
between the vertices indicate that the clients make a good match. Based on the graph,
how many possible matches are there for Erin? - Answer-3
In the graph shown, the vertices represent cities, and the edges represent flights
between those cities. The weights of the edges represent the cost of the flights, in
hundreds of dollars. Based on this, what two cities are the cheapest to fly between, and
what is the cost? - Answer-Flight: City A and City E
Cost: $200
Which of the following statements is NOT true? - Answer-There are only three types of
graphs in discrete mathematics.
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{3,4}' an example of? - Answer-Multi-edges
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{4,1} ' an example of? - Answer-Simple
graph
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{3,3} ' an example of? - Answer-Loop
These two graphs are because the second graph can be obtained from the first
by dividing some edges of the first with more vertices. - Answer-homomorphic
Under what condition is the graph isomorphic? - Answer-All of the above answers are
correct
How many vertices can be odd in a Eulerian graph? - Answer-0
Which of the following is a Euler path? 12034 - Answer-0, 2, 1, 0, 3, 4
At most, of the verticies will be of an odd degree in a semi-Eulerian graph. -
Answer-2
How many Euler paths are there in this graph? 1343 - Answer-4
How many Euler circuits are in this graph? 1234 - Answer-8
How many times are you allowed to pass an edge in an Euler path or circuit? - Answer-
1
A graph with an Euler path can have at most how many odd vertices? - Answer-2
How many Euler paths does this graph have? 1342 - Answer-4
, Which of the following is an Euler path for this graph? 12cross 3 4 - Answer-2, 3, 4, 2, 1,
4
A graph with an Euler circuit can have at most how many odd vertices? - Answer-0
If vertex 3 is your starting point, which vertex can you not go to? - Answer-1
What kinds of graphs does Fleury's algorithm work for? - Answer-Graphs with an Euler
path or circuit in it.
If a graph has 0 odd vertices, at which vertex must you start? - Answer-Any vertex will
work.
Which vertices can be your starting point? 45123 - Answer-1 or 3
How many odd vertices can a graph have in order to use Fleury's algorithm? - Answer-2
What can be said about this graph? 1234 - Answer-It has an Euler circuit.
What can be said about this graph?45123 - Answer-It has an Euler path.
The start and end points must be which two vertices? 1324 - Answer-1 and 3
If a graph has 15 edges, what must the degrees of the vertices add up to? - Answer-30
The total number of degrees in a graph is 20. How many edges does it have? - Answer-
10
Eulerize this graph. 12 cross 34 - Answer-Connect vertices 2 and 4 with another edge.
Which two vertices can you connect to Eulerize this graph? 45123 - Answer-1 and 3
According to Fleury's algorithm, how many odd vertices does a graph with an Euler path
in it have? - Answer-2
According to Fleury's algorithm, how many odd vertices does a graph with an Euler
circuit in it have? - Answer-0
How many odd vertices does this graph have? 12 cross 34 - Answer-2
How many times do you visit a vertex when traveling either a Hamilton circuit or path? -
Answer-1
If you started at vertex 1, at which vertex will you end up if you are traveling a Hamilton
circuit? - Answer-1
