Loyola College Mathematics For Computer Applications Question Papers Download
Loyola College M.Sc. Mathematics Nov 2008 Mathematics For Computer Applications Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION
FIRST SEMESTER – November 2008
MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS
Date : 111108 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Part A (Answer ALL questions) 2 x 10 = 20
 Define Lattice homomorphism between two lattices.
 With usual notations prove that (i)(ii) .
 Define context free grammar.
 What is the difference between deterministic finite automata and nondeterministic finite automata?
 Let G = (N, T, P, S), where N = {S}, T = {a}, P: {S → SS, S → a}. Check whether G is ambiguous or unambiguous.
 Give a deterministic finite automata accepting the set of all strings over {0, 1} containing 3 consecutive 0’s.
 If R and S be two relations defined by and , then find
RS, RR and R.
 Let and ,. Write the matrix of
of R and sketch its graph.
 Define ring with an example.
 State Kuratowski’s theorem.
Part B (Answer ALL questions) 5 x 8 = 40
 (a) Show that De Morgan’s laws given by and hold in a
complemented, distributive lattice.
(OR)
(b) Let be a lattice. For any prove the following distributive inequalities:
) and .
 (a) Show that L(G) = is accepted by the grammar G = (N, T, P, S) where N = {S,A} T = {a, b}, P consists of the following productions: S → aSA, S → aZA, Z → bZB, Z → bB, BA → AB, AB → Ab, bB → bb, bA→ ba.
(OR)
(b) Let the grammar G = ({S,A}, {a, b}, P, S) where P consists of S →aAS, S → a, A → SbA , A → SS, A → ba. For the string aabbaa find a
(i) leftmost derivation
(ii) rightmost derivation
(iii) derivation tree.
 (a) (i) Define deterministic finite state automata.
(ii) Draw the state diagram for the deterministic finite state automata, M = where Q =, Σ ={a, b}, F = and δ is defined as follows:
δ  a  b 
Check whether the string bbabab is accepted by M. (3+5)
(OR)
(b) Given an nondeterministic finite automaton which accepts L. Prove that there exists a deterministic finite automaton that accepts L.
 (a) (i) Write short on Hasse diagram.
(ii) Let and relation be such that if x divides y. Draw the
Hasse diagram of . (4+4)
(OR)
(b) (i) Show that n^{3}+2n is divisible by 3 using principle of mathematical induction.
(ii) If the permutations of the elements of {1,2,3,4,5} be given by
, then find
α ^{1}^{1}. (4+4)
 (a) Prove that there is a one toone correspondence between any two left cosets of H in G.
(OR)
(b) (i) If G is a graph in which the degree of every vertex is atlest two, then prove that G
contains a cycle.
(ii) Prove that the kernel of a homomorphism g from a group to is a subgroup
of . (4+4)
Part C (Answer ANY TWO questions) 2 x 20 = 40
16.(a) Let G be (p,q)graph, then prove that the following statements are equivalent:
(i) G is a tree. (ii) Every two vertices of G are joined by a unique path (iii) G is connected
and (iv) G is acyclic and p = q+1.
(b) Let H be a subgroup of G. Then prove that any two left cosets of H in G are either
identical or have no element in common. (14+6)
 (a) Let be a Boolean Algebra. Define the operations + and · on the elements of B by,
. Show that is a boolean ring with identity 1.
(b) Prove that every chain is a distributive lattice. (15+5)
 (a) If G = (N, T, P, S) where N = {S, A,B}, T = {a,b}, and P consists of the following rules:
S → aB, S → bA, A → a, A → aS, A → bAA, B →b, B → bS, B → aBB. Then prove the following:
 S w iff w consists of an equal number of a’s and b’s
 A w iff w has one more a than it has b’s.
 B w iff w has one more b than if has a’s
(b) State and prove pumping lemma. (10+10)
Loyola College M.C.A. Computer Application April 2008 Mathematics For Computer Applications Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION
FIRST SEMESTER – APRIL 2008
MT 1902 / CA 1804 – MATHEMATICS FOR COMPUTER APPLICATIONS
Date : 05/05/2008 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
SECTION A
Answer ALL the questions. (10 x 2 = 20)
 Define least upper bound of a poset.
 Define a Lattice.
 What are the logic operators?
 Construct a phrase structure grammar for the language .
 Define contextsensitive language.
 For a DFA ,
show that the string 011011 is in L(M)
 State the Pigeon hole principle.
 Draw the Hasse diagram for the divisors of 32.
 Define a bipartite graph with an example.
 Prove that every cyclic group is abelian.
SECTION B
Answer ALL the questions. (5 x 8 = 40)
 (a) Prove that the complement of any element ‘a’ of a Boolean algebra is uniquely determined. Prove also that the map is an anti – automorphism of period 2 and satisfies (a Ú b)¢ = a¢ Ù b¢, (a Ù b)¢ = a¢ Ú b¢, a¢¢ = a.
(or)
(b) Discuss ‘negation’ and explain a method of constructing the truth table for P Ú ùQ and (P Ú Q) Ú ùP
 (a) Write a short note on principal conjunctive normal form and construct an equivalent formula for ù.
(or)
(b) For a grammar where P consists of the following production:
Then show that.
 (a) Let L be a set accepted by a nondeterministic finite automaton. Then prove that there exists a deterministic finite automaton that accepts L
(or)
(b) (i) Construct an equivalent deterministic automaton for a given nondeterministic automatonwhere .
(ii) If R and S are equivalence relations on the set X, prove that R Ç S is also an equivalence relation on X.
 (a) Prove that the equivalence relation ~ defined on the set A decomposes the set A into mutually disjoint equivalence classes.
(or)
(b) (i) A computer password consists of a letter of the alphabet followed by 3 or 4 digits. Find the total number of passwords that can be formed and the number of passwords in which no digit repeats.
(ii) Find the minimum number of students in a class to be sure that four out of them are born in the same month.
 (a) (i) Prove that a subgroup N of a group G is a normal subgroup of G iff the product of two left cosets of N in G is again a left coset N in G.
(ii) Define ring with an example.
(or)
(b) Prove that the following statements are equivalent for a connected graph G.
 G is Eulerian
 Every point of G has even degree
 The set of edges of G can be partitioned into cycles.
SECTION C
Answer any TWO questions. (2 x 20 = 40)
 (a) Explain conditional and biconditional connectives with an example.
(b) Define a Non – Deterministic Finite automata.
(c) For the non deterministic finite automaton,
give the transition table and show that 0100110 is in L (M).
(10 + 2 + 8)
 (a) State and prove pumping lemma for regular sets.
(b) List any four applications of pumping lemma.
(c) Prove that if and be onetoone onto functions, then is also onetoone onto and .
(10 + 4 + 6)
 (a) Show that in a graph G, any u – v walk contains a u – v path.
(b) Prove that a closed walk of odd length contains a cycle.
(c) State and prove Lagrange theorem. (4 + 4 + 12)
Loyola College M.C.A. Computer Application Nov 2008 Mathematics For Computer Applications Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION
FIRST SEMESTER – November 2008
MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS
Date : 111108 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Part A (Answer ALL questions) 2 x 10 = 20
 Define Lattice homomorphism between two lattices.
 With usual notations prove that (i)(ii) .
 Define context free grammar.
 What is the difference between deterministic finite automata and nondeterministic finite automata?
 Let G = (N, T, P, S), where N = {S}, T = {a}, P: {S → SS, S → a}. Check whether G is ambiguous or unambiguous.
 Give a deterministic finite automata accepting the set of all strings over {0, 1} containing 3 consecutive 0’s.
 If R and S be two relations defined by and , then find
RS, RR and R.
 Let and ,. Write the matrix of
of R and sketch its graph.
 Define ring with an example.
 State Kuratowski’s theorem.
Part B (Answer ALL questions) 5 x 8 = 40
 (a) Show that De Morgan’s laws given by and hold in a
complemented, distributive lattice.
(OR)
(b) Let be a lattice. For any prove the following distributive inequalities:
) and .
 (a) Show that L(G) = is accepted by the grammar G = (N, T, P, S) where N = {S,A} T = {a, b}, P consists of the following productions: S → aSA, S → aZA, Z → bZB, Z → bB, BA → AB, AB → Ab, bB → bb, bA→ ba.
(OR)
(b) Let the grammar G = ({S,A}, {a, b}, P, S) where P consists of S →aAS, S → a, A → SbA , A → SS, A → ba. For the string aabbaa find a
(i) leftmost derivation
(ii) rightmost derivation
(iii) derivation tree.
 (a) (i) Define deterministic finite state automata.
(ii) Draw the state diagram for the deterministic finite state automata, M = where Q =, Σ ={a, b}, F = and δ is defined as follows:
δ  a  b 
Check whether the string bbabab is accepted by M. (3+5)
(OR)
(b) Given an nondeterministic finite automaton which accepts L. Prove that there exists a deterministic finite automaton that accepts L.
 (a) (i) Write short on Hasse diagram.
(ii) Let and relation be such that if x divides y. Draw the
Hasse diagram of . (4+4)
(OR)
(b) (i) Show that n^{3}+2n is divisible by 3 using principle of mathematical induction.
(ii) If the permutations of the elements of {1,2,3,4,5} be given by
, then find
α ^{1}^{1}. (4+4)
 (a) Prove that there is a one toone correspondence between any two left cosets of H in G.
(OR)
(b) (i) If G is a graph in which the degree of every vertex is atlest two, then prove that G
contains a cycle.
(ii) Prove that the kernel of a homomorphism g from a group to is a subgroup
of . (4+4)
Part C (Answer ANY TWO questions) 2 x 20 = 40
16.(a) Let G be (p,q)graph, then prove that the following statements are equivalent:
(i) G is a tree. (ii) Every two vertices of G are joined by a unique path (iii) G is connected
and (iv) G is acyclic and p = q+1.
(b) Let H be a subgroup of G. Then prove that any two left cosets of H in G are either
identical or have no element in common. (14+6)
 (a) Let be a Boolean Algebra. Define the operations + and · on the elements of B by,
. Show that is a boolean ring with identity 1.
(b) Prove that every chain is a distributive lattice. (15+5)
 (a) If G = (N, T, P, S) where N = {S, A,B}, T = {a,b}, and P consists of the following rules:
S → aB, S → bA, A → a, A → aS, A → bAA, B →b, B → bS, B → aBB. Then prove the following:
 S w iff w consists of an equal number of a’s and b’s
 A w iff w has one more a than it has b’s.
 B w iff w has one more b than if has a’s
(b) State and prove pumping lemma. (10+10)
Loyola College B.C.A. Computer Application April 2012 Mathematics For Computer Applications Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.C.A. DEGREE EXAMINATION – COMPUTER APPL.
SECOND SEMESTER – APRIL 2012
MT 2101 – MATHEMATICS FOR COMPUTER APPLICATIONS
Date : 23042012 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
Part A
Answer ALL questions: (10 x 2 = 20)
 Give an example of skew symmetric matrix.
 Prove that .
 If α and β are the roots of the equation, find α+β, αβ.
 Find the first order partial derivatives for .
 Evaluate .
 Write down the Bernoulli’s formula for integration.
 Find the complementary function for .
 Form partial differential equation by eliminating arbitrary constants from .
 Write the approximation formula to find the root using Regula Falsi method.
 How many types in Simpson’s rule.
Part B
Answer any FIVE questions: (5 x 8 = 40)
 Find the rank of the matrix .
 Prove that .
 Solve the equation whose roots are in A.P.
 If where ,then prove that .
 Evaluate .
 Solve the equation .
 Solve .
 Apply Simpson’s rule to evaluate correct to 2 decimal places by dividing the range into 8 equal parts.
Part C
Answer any TWO questions: (2 x 20 = 40)
 (a)Find the Eigen values and Eigen vectors of the matrix . (12)
(b)Prove that . (8)
 (a)Solve . (12)
(b)Find the radius of curvature for the curve at . (8)
 (a)Prove that . (8)
(b)Solve the equation . (12)
 (a)Using NewtonRaphson method find the root of the equation (15)
(b)The velocity of a particle at distance S from a point on it’s path is given by the following table
S(ft)  0  10  20  30  40  50  60 
V(ft/s)  47  58  64  65  61  52  38 
Estimate the time taken to travel 60 ft using Trapezoidal rule. (5)