Description

 

SESSION	NOV 2024
PROGRAM	Master of CoMPUTER APPLICATIONS (MCA)
SEMESTER	I
course CODE & NAME	DCA6108 DISCRETE mATHEMATICS & GRAPH THEORY

 

 

 

Set-I

 

 

1. Express the System of Equations in Matrix Form and Solve Using the Elimination Method

The system of equations given is:

Ans 1.

We represent this system in matrix form as , where:

To solve using the elimination method:

1. Step 1: Eliminate  from all equations except the first.Divide Row 1 (R1) by 2 to make the pivot element 1. Then subtract appropriate multiples of R1 from the other rows. This results in the updated matrix:

 

Its Half solved only

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2. Two APs have the same common difference. The difference between their 100^th terms is 100,

what is the difference between their 1000^th terms?

Ans 2.

Difference Between 1000th Terms of Two APs

We are given that two arithmetic progressions (APs) have the same common difference ( ). The difference between their 100th terms is 100. We need to find the difference between their 1000th terms.

General Form of AP:

Let the first AP be:
.

Let the second AP be:
.

 

 

3. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:

• The number of people who read at least one of the newspapers.
• The number of people who read exactly one of the newspapers.

Ans 3.

3. Newspaper Survey Problem

Given Data:

• Total people surveyed ( ): 60
• , ,
• , , ,

We use the principle of inclusion-exclusion to answer both parts.

(a) Number of people who read at least one newspaper:

Using the formula for the union of three sets:

 

 

4. Logical Proof:

Ans 4.

Proof:

We prove the equivalence using truth tables and logical equivalences.

Step 1: Recall definitions of logical operators

1. Implication:
2. Conjunction: is true only if both  and  are true.

 

Step 2: Expand :

Using the definition of implication:

Substitute :

 

5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying numbers of mangoes. The following was the distribution of mangoes according to the number of boxes:

Number of mangoes	50 – 52	53 – 55	56 – 58	59 – 61	62 – 64
Number of boxes	15	110	135	115	25

 

Ans 5.

 

Find the mean number of mangoes kept in a packing box.

 

5. Mean Number of Mangoes in a Packing Box

Given Data:

The data represents the number of mangoes in packing boxes and their corresponding frequencies:

Number of Mangoes (Class Interval)	Midpoint ( )	Number of Boxes ( )
50 – 52	51	15
53 – 55	54	110
56 – 58	57	135
59 – 61	60	115
62 – 64	63	25

Step 1: Formula for Mean

The mean is calculated using the formula:

 

 

 

6. Explain about walk, Path & Circuit and also define Regular & complete connected graphs.

Ans 6.

Walk, Path, Circuit, Regular & Complete Connected Graphs

Walk

A walk in a graph is a sequence of vertices and edges where:

1. Each edge is incident to its preceding and succeeding vertices.
2. Vertices and edges can repeat.

Example: In a graph with vertices  and edges , the sequence  is a walk.

Path

A path is a specific type of walk where no vertex or edge is repeated. Paths are finite and simple.

Example: If a graph has vertices  with edges ,  is a path.

Circuit