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Discrete Math Quiz-2 Solutions

The document is a quiz from a discrete mathematics class containing 5 multiple choice questions about graph theory, probability, and proofs. It provides the questions, answers, and instructions for students to write their information and answers clearly. It also contains a tutorial evaluation with 2 questions, one involving solving a recurrence relation and finding the unique solution, and another applying Floyd Warshall's algorithm to find shortest paths on a graph.

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mehra.harshal25
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72 views4 pages

Discrete Math Quiz-2 Solutions

The document is a quiz from a discrete mathematics class containing 5 multiple choice questions about graph theory, probability, and proofs. It provides the questions, answers, and instructions for students to write their information and answers clearly. It also contains a tutorial evaluation with 2 questions, one involving solving a recurrence relation and finding the unique solution, and another applying Floyd Warshall's algorithm to find shortest paths on a graph.

Uploaded by

mehra.harshal25
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Roll No.

______________ Name__________________________ Group__________

Computer Science & Engineering Department


Thapar Institute of Engineering & Technology, Patiala
Discrete Mathematical Structures (UCS405)
Quiz-2 (Solution)
Total Time: 25 min Max Marks: 5

Instructions for students:


 Missing roll no. or name will be considered as an absent.
 Any cutting or overwriting will be considered as a wrong answer.
 Write the answer only in the space provided otherwise no marks will be given.
________________________________________________________________________

1. What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of
degree 2, 3 vertices of degree 4 and remaining of degree 3?
Ans: 19

2. A bag contains 10 red marbles, 10 white marbles, and 10 blue marbles. What is the minimum no.
of marbles you have to choose randomly from the bag to ensure that we get 4 marbles of same
color?
Ans: 10

3. What is wrong with the following proofs? Clearly mention the step number which you think is
wrong.
Consider the “proof” that 2=1.
“Proof”:
Step 1. -2 = -2
Step 2. 4-6 = 1-3
Step 3. 4-6 + 9/4 = 1-3 + 9/4
Step 4. (2-3/2) ^ 2 = (1-3/2) ^ 2
Step 5. 2-3/2= 1-3/2
Step 6. 2 = 1
Ans: Step 5

4. Let G be an n-vertices undirected connected graph. Then the spanning tree of G will contain
exactly _____________________ number of edges.
Ans: n-1

5. Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}.
∑* with the concatenation operation for strings forms a group ____________ (True / False).

Ans: False

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Roll No. ______________ Name__________________________ Group__________

Tutorial Evaluation Max Marks: 10

Instructions for students:


 Missing roll no. or name will be considered as an absent.
 This sheet contains 2 questions. Attempt both.
Q 1: Solve the recurrence relation and find its unique solution.

Fn  3Fn1  10 Fn2  7.5 n Where F0  4 and F1  3

Ans: Fn  3Fn1  10 Fn 2 and f (n)  7.5 n

x2 - 3x – 10 = 0
(x - 5) (x + 2) = 0
x1 = 5 x2 = -2

ah = a. 5n + b (-2) n where a, b are constants.

Since f (n)  7.5n


x=5

Its Solution is A n x n

at = A n x n = A n 5n

After putting the solution in the Recurrence Relation we get:

A n 5n = 3 A (n-1) 5n-1 + 10 A (n-2) 5n-2 + 7. 5n

Dividing both sides by 5n-2 we get:

A n 52 = 3 A (n-1) 5 + 10 A (n-2) 50 + 7. 52

25 A n = 15 A n – 15 A + 10 A n – 20 A +175

35 A = 175

A=5

Fn = A n 5n = 5 n 5n = n 5n+1
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Solution of the Recurrence Relation can be written as:

Fn = ah + at

= a. 5n + b (-2)n + n 5n+1

Putting values of F0 = 4 and F1 = 3

We get a = -2 and b = 6

Solution is:

Fn = n 5n+1 + 6 (-2)n – 2. 5n

Q 2: Apply Floyd Warshall’s Algorithm on the given graph and find values of D0, D1, D2, D3, D4.

Ans:

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