**Roll No……**

**Total No. of Questions: 13**

**J-3298 [S-1154]**

**[2037]**

**MCA (Sem.-1**

^{st})**COMPUTER MATHEMATICAL FOUNDATION (MCA-104)**

**Time: 3 Hrs. Max. Marks: 75**

**Instruction to Candidates:**

1. Section-A is Compulsory.

2. Attempt any Nine questions from Section-B.

**SECTION-A**

Q1. (a) Let D

_{i }= where R is the set of real numbers and N is the set of natural numbers, find and
(b) State De-Morgan’s laws.

(c) Let A= [a,b,c], find all the partitions of
A.

(d) Express the set of points
of the rectangle and its interior in R X R ( R is the set of real numbers) with
corners at (0,1,) , (0,4), (3,1) & (3,4) as a Cartesian product.

(e) Define an equivalence
relation and give an example of equivalence relation on A= [1,2,3,4].

(f) Let A and B are matrices
s.t. 3A-2B= and
-4A+B= find A
and B.

(g) Let A=Write A as the sum of a symmetric & a skew
symmetric matrix.

(h) Define rank of a matrix.

(i) Examine whether the equations

2x+6y= -11, 6x+20y-6z=-3, 6y-18z=-1 are
consistent?

(j) Let
A= find A

^{-1}.
(k) Show by using truth table that (p-r ) ^
(q-r)= (p v q) – r.

(l) Explain the principle of mathematic
Induction.

(m) What are the types of quantifier? Give an
example of each.

(n) Define chromatic number and find chromatic
of the graph.

(o) Define Hamiltonian graph and give an
example.

**SECTION-B**

Q2. For
integers a & b, define aRb is 2a+3b=5n for some integer n. Show that R
defines the equivalence relation on Z. Also find the equivalence class of 0.

Q3. Define
the relation p & Q on [1,2,3,4] by P=(a,b: (a-b); a-b=1 that Q=(a,b) : and
represent them clearly as graphs.

Q4. Two
finite sets set have x and sy number of
elements. The total number of subjects of the first set is four times the total
no. of subjects fo second set. Find the value of x-y.

Q5. Define
the following terms:

(a) Partition of a set.

(b) Complement of a set.

(c) Symmetric relation.

(d) Partial order relation.

Q6. If
A= show
that A

^{2}-20A+8l=0; where 1, 0 are unit matrix and null matrix of order 3. Using the result find A^{-1}.
Q7. Find
the value of k such that the system of equations

X+ky+3z=0

4x+3y+kz=0

2x+y+2z=0

Has non trivial solutions.

Q8. Using
Gauss Elimination method determine for what value of y and u the following
equations have (i) no solution. (ii) a unique solution. (iii) infinite no. of
solution.

X+y+z=6

X+2y+3z=10

X+2y+yz=u

Q9. Using
matrix inversion method solve

Q10. Use
Mathematical Induction to prove that 1+2+2+…….. +2

^{n }=2^{n+1}-1 for all non negative integer n.
Q11. Determine
whether or not the following argument is valid.

If
I like biology, then I will study it.

Either
I study biology or I fail the course.

----------------------------------------------------

If
I fail the course, then I do not like biology.

Q12. Define
a bipartile graph, complete bipartile graph, complete graph, Eluerian graph,
directed graph with an example for each.

Q13. Discuss
any shortest path algorithm with a simple example.

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