DCA6108 DISCRETE MATHEMATICS & GRAPH THEORY JULY AUG 2025

Description

SESSION	July 2025
PROGRAM	Master of Computer Applications (MCA)
SEMESTER	III
Course CODE & NAME	DCA6108 DISCRETE mATHEMATICS & GRAPH THEORY

 

 

SET-I

 

 

Q1. Find the inverse of the matrix   using the adjoint method.

Ans 1.

To find the inverse of matrix  using the adjoint method, follow these steps:

Step 1: Compute the Determinant

Let

Using cofactor expansion alang the first row:

 

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Q2. Find the solution to the following system using  elimination:

Ans 2.

Solve the system by elmmation

Step 1: Eliminate  between pairs
From (2) and (1): form (2) – 3 – (1) to cancel .

 

 

Q3. Let , , and  .

If , find the composition       .

Ans 3.

Composition  with manual substitution

Given functions and their codomains:

Definition (composition)

 

SET- II

 

Q4. Show that

Ans 4.

We show the equivalence using standard propositional equivalences. Recall the definition of implication

Start with the left-hand side:

Apply distributivity of V over A :

 

Q5. The following table shows the number of books read by students in a month:

Books Read	Frequency
0 – 5	10
5 – 10	15
10 – 15	20
15 – 20	8
20 – 25	5

 

Find the median number of books read.

Ans 5.

Median of grouped data

Classes and frequencies

Class (books)	Frequency f	Cumulathv
0-5	10	10
5-10	15	25
10-15	20	45
15-20	8	53
20-25	5	58

 

Total  Compute .

 

 

Q6. Explain the difference between connected graphs and complete graphs.

Ans 6.

In graph theory, graphs are used to represent relationships among a set of objects, where vertices (or nodes) denote the objects and edges denote the connections between them. Two important types of graphs that describe different degrees of connectivity are connected graphs and complete graphs. Although both relate to how vertices are linked, they differ fundamentally in