**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

^{-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<x

^{n}<y^{n }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 (CH

_{4})
(ii)
Propane (CH

_{3}H_{8}).
(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.

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