Roll No……
Total No. of Questions: 13
J-3298 [S-1154]
[2037]
MCA
(Sem.-1st)
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 Di =
where R
is the set of real numbers and N is the set of natural numbers, find
and ![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif)
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif)
(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.
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif)
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
(g) Let A=
Write A as the sum of a symmetric & a skew
symmetric matrix.
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif)
(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.
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif)
(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.
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image016.jpg)
(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 A2-20A+8l=0; where 1, 0 are unit matrix and null matrix of
order 3. Using the result find A-1.
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif)
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 ![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
![](file:///C:/Users/com16/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
Q10. Use
Mathematical Induction to prove that 1+2+2+…….. +2n =2n+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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