DCA6108 DISCRETE MATHEMATICS & GRAPH THEORY APRIL 2025

Description

SESSION	APRIL 2025
PROGRAM	MASTER OF COMPUTER APPLICATIONS (MCA)
SEMESTER	I
COURSE CODE & NAME	DCA6108 DISCRETE MATHEMATICS & GRAPH THEORY

 

 

 

SET – I

 

1. Find the inverse of the matrix

using the adjoint method.

Ans 1.

Inverse of a Matrix Using Adjoint Method

We are given the matrix:

Step 1: Find the Determinant of A (|A|)

Using cofactor expansion (along the first row):

Calculate each 2×2 determinant:

 

Its Half solved only

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

Ans 2.

Solving System of Equations Using Elimination Method

Given system:

Step 1: Eliminate x from Equations (1) and (3)

From (1):

 

 

3. Let . If , find the composition , also check the bijective-ness of mappings. Marks:

Ans 3.

Function Composition and Bijective Nature

Given:

• Sets:  (Set of integers)
• Functions:

We are asked to:

 

 

4. Verify that

Ans 4.

Propositional Logic Verification

We are given a proposition:

and asked to verify whether it is always true (i.e., a tautology).

Step 1: Understand the Expressions

 

 

5. The table shows the height distribution (in cm) of students in a school:

Height (cm)	Frequency
140 – 150	5
150 – 160	12
160 – 170	17
170 – 180	8
180 – 190	5

Find the median height of the students.

Ans 5.

Median of Height Distribution

Given table:

Height (cm)	Frequency (f)
140 – 150	5
150 – 160	12
160 – 170	17
170 – 180	8

 

 

 

6. Explain the Degree in Directed Graph.

Ans 6.

Degree in Directed Graph

Definition of Degree in Directed Graphs

In graph theory, a directed graph (digraph) is a set of vertices connected by directed edges (arcs). Each edge has a direction, meaning it goes from one vertex to another.

In a directed graph, each vertex has two degrees:

1. In-degree (deg⁻): Number of edges coming into a vertex.