Engineering Mathematics, Question Paper of MCA Semester 1, Download Question Paper 3

Roll No………

Total No. of Questions: 9

Paper ID [B0104]

MCA (Sem.-1^st)

COMPUTER MATHEMATICAL FOUNDATION

 SUBJECT CODE: (MCA-104)(N2)

Time: 3 Hrs.                                                                           Max. Marks: 60

Instruction to Candidates:

1. Attempt any One question from each Section-A, B, C, & D.

2. Section-E is Compulsory.

3. Use of non-programmable Scientific Calculator is allowed.

SECTION-A

Q1.    Show that set of real numbers in [0,1] is uncountable set.

Q2     . Let R be a relation on A. Prove that

(a) If R is reflexive, so is R^-1.

(b) R is symmetric if and only if R=R^-1

(c) R is antisymmetric if and only if R  R^-1I^.

SECTION-B

Q3.    If x and y denote any pair of real numbers for which 0<x<y, prove by mathematical induction 0<xⁿ<yⁿ for all natural numbers n.

Q4.    (a) Obtain disjunctive normal forms for the following.

(i) P ^ (P=q).

(ii) P=(p=q) [v-(-q v –p)].

Q5.    Find the ranks of  A, B and A+B, where

A=

Q6.    Solve the following equations by Gauss-Jordan method. 2x-y+3z=9,  x+y+z=6, x-y+z=2.

SECTION-D

Q7.    (a) Show that the degree of a vertex of  a simple graph G on ‘n’ vertices cannot exceed n-1.

(b)     A simple graph with ‘n’ vertices and k components cannot have more than  edges.

Q8.    Define breadth first search algorithm (BFS) and back tracking algorithm for shortest path with example.

SECTION-E

Q9.    (a) Draw the truth table for ~ (p v q) v (-p ^ -q).

(b) Define principle of mathematical induction.

(c) Prove that A-B=AB.

(d) Using Venn diagram show that A ∆ (B∆C)= (A∆B)∆C.

(e) If A and B are two m x n matrices and 0 is the null matrix of the type m x n, show that A+B=0 implies A= -B and B=-A.

(f) If A and B are two equivalent matrices, then show that rank A= rank B.

(g) Prove that every invertible matrix possesses a unique inverse.

(h) Draw the graphs of the chemical molecules of

(i) Methane (CH₄)

(ii) Propane (CH₃H₈).

(i) Draw the diagraph G corresponding to adjacency matrix A=

(j) Give an example of a graph that has an Eluerian circuit and also Hamiltonian circuit.