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

• Thursday, November 26, 2015
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Roll No………
Total No. of Questions: 9

Paper ID [B0104]
MCA (Sem.-1st)
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-1 I^.
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=A B.
(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.