time: 2½ hrs. marks: 75 all questions are compulsory...

CC3R1

Time: 2½ Hrs. Marks:75

Note: 1. All questions are compulsory. 2. Figures to the right indicate full marks.

Q.I Attempt the following. (Any Four) (20)

(1) Define the following.

(a) Matrix of a relation (b)Symmetric relation

(c) Antisymmetric relation (d)Equivalence relation (e)Linearly ordered set

(2) Let be the relation on = { , , , } defined by is less than .

(a) Write as the set of ordered pairs.

(b) Find − .

(c) Describe − in words.

(3) Let be the following relation on = { , , , , , }. = { , , , , , , , , , , , , , , , , , }. (a) Find �2. (b) Draw the diagraph of .

(c) List all paths of length 2 starting from vertex 3.

(4) Let be the following relation on = { , , , }. = { , , , , , , , , , , , }. Determine whether the relation is reflexive, symmetric, asymmetric, antisymmetric or

transitive.

(5) Consider the partial order of divisibility on set = { , , , , , , , }. Draw Hasse

diagram of the poset and determine whether the poset is linearly ordered.

(6) Let be a relation �� on = { , , , , , , , } and Let = { , }. Find

(a) all upper bounds (b)lower bounds of for the poset .

(7) Solve the recurrence relation � = − �− − �− − �− with initial conditions = , = , = .

(8) Let be the following relation on = { , , , }. = { , , , , , , , , , }. Check whether is transitive, if not, find the

transitive closure of the relation .

Q.II Attempt the following. (Any Four) (20)

(1) Define the following.

(a) Loop (b)Path (c)Regular graph

(d) Connected graph (e)Degree of a vertex

(2) Consider the following graphs � = �, �, γ and answer the questions.

(a) Describe � formally.

(b) Find the degree of each vertex.

(c) Verify the sum of degrees and number

of edges in �.

(3) Find the connectivity matrix for digraph � using sum of powers of adjacency matrix.

How many paths are there from vertex to vertex of length 3.

(4) Prepare the table of adjacency structure and write the linked representation of the graph.

(5) For the graphs � and � , draw the graphs for

(a) � � (b) � � (c) � ⨁�

(6) Represent the expression − ( − × ( ÷ − ) ) in a binary tree. Give doubly

linked list representation of the binary tree created.

(7) For the following binary search tree

(a) search for value=

(b) insert value=

(8) Starting with vertex s, apply DFS algorithm to the following graph.

Q.III Attempt the following. (Any Four) (20)

(1) In how many ways can 5 men and 5 women stand in a row, so that no two men and no

two women are adjacent to each other?

(2) Out of 4 officers and 10 clerks in an office, a committee consisting of 2 officers and 3

clerks is to be formed. In how many ways can this be done, if

(a) any officer and any clerk can be included ?

(b) one particular clerk must be on the committee ?

(c) one particular officer cannot be on the committee ?

(3) In a survey on eating fruits of IT professionals, it was found that 22 like apple, 25 like

grapes, 39 like banana, 9 like grapes and apple, 17 like apple and banana, 20 like

grapes and banana, 6 like all three fruits and 4 like none. How many IT professionals

were surveyed?

(4) Prove Pascal’s identity using the formula for ��.

(5) A train required 15 hours to complete the journey of 972 kms from Pune to Indore. It is

known that the speed of the train was 50 kms/hr in first 3 hours and 40 kms/hr in last 3

hours. Show that the train must have travelled at least 234 kms within a certain period

of three consecutive hours.

(6) Consider the FSA defined by the state diagram in figure. Find

(a) states.

(b) input symbols.

(c) initial state and accepting states.

(d) � , .

(e) state transition table.

(7) Let input symbols � = { , }. Construct the finite state automation such that ={w│w contains number of a' s divisible by 2}. Also, define the transition state

function.

(8) Let = {� , � }, � = { , }, � = { , , }, � is initial state.

The transition function : × � ⟶ is defined as � , = � , � , = � , � , = � , � , = � . The output function : × � ⟶ � is given by

� , = , � , = , � , = , � , = .

Draw the state transition table and the state transition diagram of FSM

Q.IV Attempt the following. (Any Three) (15)

(1) Let be the following relation on = { , , , , }. = { , , , , , , , , , , , , , , , , , } . Check whether the poset , is a lattice.

(2) Solve the recurrence relation by characteristic root method. � = �− − �− , = , =

(3) Find the shortest path between the vertices of a weighted digraph � using Warshall’s algorithm.

(4) Perform preorder and postorder search for the given tree.

(5) Determine the number of integers between 1 and 250 that are divisible by any of the

integers 2, 3, 5 and 7.

(6) Consider the finite state automation in figure

Determine which of the following words are accepted by .

(a) 00

(b) 0010

(c) 10101

(d) 000011

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CC3R4

Time: 2½ hrs. Marks:75

Note: 1. All questions are compulsory with internal options. 2. Figures to the right indicate full marks.

Q. 1 Answer the following: (Attempt any four) (20)

(a) Explain the view and physical level used for abstraction.

(b) What are the limitations of data processing?

(c) What is meant by generalization in E-R model?

(d) Explain the use of cardinality. (e) Explain the relational model with suitable example. (f) Explain the term normalization.

(g) What is meant by lossless join decomposition?

(h) Write a note on three tier architecture.


(a) Explain the CHECKED constraint with suitable example.

(b) Explain the types of joins used in SQL.

(c) Explain alter and update commands used in SQL.

(d) Explain the following functions used in SQL. i> MIN() ii>COUNT()

(e) Explain the UNION operation used in SQL.

(f) Explain the joins used in relational algebra.

(g) Illustrate relational calculus.

(h) What is meant by sub query? How it is used in tables?


(a) Explain the Tree based indexing used in file organisation.

(b) How to modify and delete the trigger?

(c) How to alter the stored procedure?

(d) Explain the types of triggers.

(e) Explain cost model with example.

(f) What are the advantages and disadvantages of view?

(g) Explain the clustered files used in file organisation.

(h) How the stored procedure is used in SQL?

Q. 4 Answer the following: (Attempt any three) (15)

(a) Explain the Rename and Division operation used in relational algebra.

(b) Explain NOT NULL constraint used in SQL.

(c) Explain the data models used for basic building blocks.

(d) What are the different types of attributes?

(e) Explain referential integrity constraint.

(f) How to use data integrity through triggers?

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CC3R8




(a) What is a preprocessor directive? Explain its use. Illustrate with examples.

(b) What is a constant? How to declare a constant explicitly in a C ++ program?

(c) What is enumeration? Illustrate with an example.

(d) Write about new and delete operators. Discuss their importance in C ++ programming. (e) How to declare a pointer in C ++ ? Explain the use of & operator. (f) Discuss the differences between call by value and call by reference. Illustrate with

example.

(g) What are the different types of access specifiers in a class?

(h) Write a program in C++ to calculate factorial of a number with the help of a function.


(a) What is a constructor? Write the rules for declaring a valid constructor.

(b) What is meant by inheritance? Describe the types of inheritances supported by C++?

(c) What is polymorphism? Explain any one way in which polymorphism is implemented in C++.

(d) Explain the advantages and disadvantages of initializing data members through a default constructor.

(e) What is meant by operator overloading? Write the rules of overloading a binary operator?

(f) What is a pure virtual function? Illustrate with an example.

(g) Write a program to add two complex numbers by overloading + operator.

(h) Write a program to increment a data member by overloading ++operators.


(a) Write a short note on any one of the following functions: i. good() ii. bad()

(b) Explain the various modes to open a file with examples.

(c) Explain the keywords try, catch and throw with the help of an example.

(d) Create a function template to calculate the area of a rectangle (area=l*b). Write a program to test the function template.

(e) Write a program to illustrate array access beyond bound exception.

(f) Illustrate a vector with an example.

(g) Write a program to accept a line “I know how to create a file” and write the line to a text file.

(h) What is a class template? Illustrate with examples.


(a) Write a program in C++ to illustrate pointer arithmetic.

(b) What is a static member? Illustrate with the help of an example.

(c) What is a copy constructor? How it is different from default constructor?

(d) Explain any one of the basic concepts of object oriented programming with examples.

(e) What is a function template? Illustrate with examples.

(f) Write a program to illustrate Divide by zero exception.

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CC3R3

Time: 2½ hrs. Marks:75 Note: 1. All questions are compulsory with internal options.

2. Figures to the right indicate full marks.


(a) What are the various kinds of operators used in C ++? Illustrate with examples.

(b) How to declare a pointer in C ++? Explain the use of & operator.

(c) Discuss how Double dimensional array is stored and accessed from the memory

(d) Write about new and delete operators Discuss their importance in C ++ programming. (e) A retail shop is offering discount on total amount of purchase. If the total is between 100

to 500,a discount of 5%,above 500 to 1000 a discount of 10% ,above 1000 to 5000 a discount of 20% and above 5000 a discount of 30% is provided. Calculate the total cost paid by the customer.

(f) Write a program to swap or interchange the contents of two variables using a function in C++.

(g) What are the different types of access specifiers in a class?

(h) What is function overloading? Write the rules of function overloading in a program.


(a) What is a friend function? How it is used to access the members of a class.

(b) Discuss single inheritance in C++ with an example.

(c) Write a program which creates two classes emp_personal and emp_salary which contains the following information:- Write a program to accept data in both the classes and display the data using a common friend function.

(d) Explain the merits and demerits of runtime binding over compile time binding.

(e) Illustrate the rules of overloading a binary operator with the help of friend function.

(f) What is a virtual base class? Illustrate with an example.

(g) Write a program to add two complex numbers by overloading + operator.

(h) Create a resistance class with a data member ohm. Add two resistances by overloading + operator.


(a) Write a short note on the following functions: (i) fail() (ii)eof()

(b) Explain the various modes to open a file with examples.

(c) What is an exception handler? Describe the keywords used to handle the exceptions in

Emp_personal

Empid:int

Empname:char[10]

Address :char[20]

Phono :long

void getd();

friend Void disp(emp_personal

e,emp_salary s);

Emp_salary

Basic:int

HRA:float

IT:float

void getd();

friend Void

disp(emp_personal

e,emp_salary s);

C++.

(d) Create a function template which calculates the square of a number (integer, float and double).

(e) What are the benefits of using a template in a C++ program? Illustrate a function template with examples.

(f) Write a program in C++ to read the contents of an existing file and print it on the screen.

(g) Write a program to illustrate Divide by zero exception.

(h) Write a program to illustrate array access beyond bound exception.


(a) Write a program to illustrate function overloading to print the area of a square, rectangle and triangle.

(b) What is a static member? Illustrate with the help of an example.

(c) Illustrate multiple inheritance concepts with an example.

(d) What is meant by run time binding? Write an OOP in C++ to illustrate run time binding.

(e) Compare ifstream, ofstream and fstream classes? Why it is used? Illustrate with examples.

(f) Illustrate a class template with examples.

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CM3R5

Time: 2½ hrs. Marks:75 Note: 1. All questions are compulsory with internal options.

2. Figures to the right indicate full marks.

Q. 1 (A) Answer the following: (Attempt any one) (08)

(a) State Fubini’s theorem in first form & verify for − ��R where R = { , / ≤ ≤ , ≤ ≤ }

(b) Find the mass,first moments and center of mass of triangular lamina which is bounded by co-ordinate axes and line = − and the density function is � x, y = + x + y.

Q. 1 (B) Answer the following: (Attempt any three) (12) (a) Convert (1, 1, 1) given in cartesian co-ordinate system into spherical co-ordinate

system.

(b) Set up the double integral that gives the area between y = & = .

(c) Evaluate with the polar form +√ − 2 � � .

(d) Find the average value of � , = − 6 on the rectangle D={(x,y)/0 ≤ x ≤ ,

-1 ≤ ≤ }.


(a) Prove : If � is integrable on [ , ] then |�| is also integrable on [ , ] further | � � | ≤ |� |� .

(b) Let � = , � = [ , ] , = { , , , } & = { , , , , }. Find � , � , � , � , � �, , � �, .

Q. 2 (B) Answer the following: (Attempt any three) (12)

(a) Let �: [ , ] → � defined by � = + . Prove that � is integrable & evaluate � � .

(b) Let �: [ , ] → � be defined by � = �� < < = � = Prove that � is integrable and � � = .

(c) Prove : Let �: [ , ] → � be a bounded function & ⊆ be two partitions of [ , ] then (i) � , � ≤ � , � ≤ � , � ≤ � , � (ii) � , � − � , � ≤ � , � − � , � .

(d) Define: (i) upper sum (ii) lower sum (iii) Riemann sum of a bounded function.


(a) State & prove mean value theorem & hence find average value of � = √ , ∈ [ , ].

(b) Prove : If �: [ , ] → � is a Riemann integrable function on [ , ] and there is a function � such that �′ = � ∀ ∈ [ , ] then � �� = � − � .


(a) Show that � , = � + , + � , + .

(b) Define : (i) gamma function (ii) beta function.

(c) State & prove Leibnitz rule.

(d) Examine convergence of �√ − .


(a) Evaluate � � .

(b) Define : (i) mass (ii) moments (iii) center of mass.

(c) Let �: [ , ] → � defined by � = �� = � � ∈ �

= otherwise.

Check whether � is integrable on [ , ].

(d) Prove : Let �: [ , ] → � be a bounded function & let P & Q be any two partitions of [ , ].Then � , � ≤ � , � .

(e) If (i) � = √sin �2 �� (ii) � = �2 �� > , then find f’(x).

(f) Prove : � , = ⌈ ⌈⌈ + .

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CM3R7




(a) Define determinant of matrix using Laplace expansion. Hence expand the determinant of

matrix A by 2nd row and also by 3rd column A=[ ].

(b) Define Vandermonde determinant and hence solve

(a) | | (b) | |.

Q. 1 (B) Answer the following: (Attempt any three) (12) (a)

Find the value of determinant | | using permutation.

(b) Check whether the following vectors are linearly dependent or independent =< , − , > , =< − , , − > , =< , − , > =< , , > , =< − , , > , =< , , − >.

(c) Check whether the following vectors form a parallelepiped

(i) − + , + − , − +

(ii) =< , , − > , =< , , > , =< , , >.

(d) Solve & check the uniqueness of the solution + = , + − = − , − = .


(a) Using Gram-schmidt process construct an orthonormal basis of ℝ 3 from { , , , , , , , , }.

(b) If V is an inner product space then prove that the distance of v from u satisfies following properties :(i) , (ii) , = = (iii) , = ,

(iv) , , + , , w is any vector.


(a) If & are any two vectors in an inner product space, then show that < , >= ‖ + ‖ − ‖ − ‖

(b) Determine whether the following set of vectors is orthogonal { , , , , − , , , }

(c) Suppose & are any two orthogonal vectors in an inner product space then prove that ‖ + ‖ = ‖ ‖ + ‖ ‖

(d) Define : (i) orthogonal vector (ii) orthonormal vector (iii) projection vector.


(a) Prove : �: → is an isomorphism iff �� = {��} & �� = .

(b)

Let �: ℝ → ℝ is linear transformation defined as L [ ] = [ +++ ] (i) find a basis for Ker L(ii) find a basis for Img L (iii) verify rank-nullity theorem.


(a) Define linear transformation and check whether the following is linear transformation �: ℝ → ℝ defined as L [ ] = [ ].

(b) Prove : A square matrix A is invertible iff it can be written as the product of elementary matrices.

(c) Check whether the following linear transformation is invertible or not � ∶ ℝ → M × defined as L , , , =[ ]

(d) Find the row rank & column rank for the following matrix [ − ].


(a) Find the value of k for which the following vectors are linearly dependent < , , , >, < , , , >, < , k, , >, < k, , , >.

(b) Solve by Cramer’s rule: + = , + = , + = .

(c) Define unit vector & find unit vector along , , .

(d) Find projection vector & component of u along v (i) = , − , , = , , (ii) = , , − , , = , − , , .

(e) Solve the following system of linear homogeneous equations + + + = , + − − = , − + + = .

(f) Define : (i) row space (ii) row rank (iii) column space (iv) column rank.

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CM3R6


Note: 1. All questions are compulsory with internal options. 2. Figures to the right indicate full marks. 3. Use of scientific calculator is allowed.


(a) Show that the set of integers Z is countable.

(b) Let S(n,k) denote the number of partitions of a nonempty set A with n elements into k parts then prove that:

1. S(n,1)=S(n,n)=1 2. S(n,k)=S(n-1,k-1)+kS(n-1,k) ; 2≤k≤n-1

Q. 1 (B) Answer the following: (Attempt any three) (12) (a) How many different three letters initials can people have? Also find how many of them

have no repetition in their initials? And how many of them begin with alphabet S.

(b) Find the value of S(10,4) using recursive formula.

(c) Show that if 10 points are selected at random in a triangle with side 3 units then there are atleast two points which are 1 unit apart.

(d) A train required 15 hours to complete the journey of 972 kms. It is known that the speed of the train was 50 kms/hr in first three hours and 40 kms/hr in last 3 hours. Show that the train must have travelled at least 234 kms. within a certain period of 3 consecutive hours.


(a) State Vandermonde’s identity and prove it using combinatorial proof.

(b) Find the number of positive integers from 1 to 160 which are divisible by 2 not by 3 or 5.


(a) Find the coefficient of x2y8z in the expansion of (2x-3y+z)11. Also find the number of terms and sum of all coefficients in the expansion.

(b) How many ways in which 5 men and 5 women stand in a row so that no two men and no two women are adjacent to each other?

(c) Using prime factorization theorem, find the value of ∅ (1325).

(d) Find the number of derangements of the integers {1,2,…….,7,8} 1. Beginning with the integer 1,2,3,4 in some order. 2. Beginning with the integer 5,6,7,8 in some order.


(a) Solve the recurrence relation an=-2an-1+2an-2+4an-3, a1=0, a2=2, a3=8.

(b) An employee joints an organization in 1999 with a starting salary of Rs.10,000. Every year this employee receives an increment of Rs.500 plus 5% of the salary of the previous year.

1. Set up the recurrence relation for the salary of this employee n years after 1999. 2. Find an explicit formula for the salary of this employee n years after 1999.


(a) Find the five terms for the given recurrence relation bn=5bn-1+3 , b1=1

(b) Solve the following using backtracking method cn=cn-1-2 , c1=0

(c) Solve the following recurrence relation bn=4bn-1-4bn-2 , b1=1, b2=7

(d) Define the following 1. Transposition. 2. Linear recurrence relation.


(a) How many different subsets of N9 having exactly 4 numbers and each number from N9 belongs to exactly 6 subsets. Write all these subsets.

(b) Find the value of S(6,4) using explicit formula.

(c) Let n be a positive integer. Prove that

1

22

2

12

1

2

n

n

n

n

n

n

(d) How many integral solutions are there of a+b+c+d=17 if 1≤a≤3, 2≤b≤4, 3≤c≤5, 4≤d≤6?

(e) Let A={1,2,3,4,5,6}. Compute the following permutations and determine whether they are odd or even.

P1=

562143

654321 , P2=

645132

654321

1. P1 o P2 2. P1

-1 3. (P2 o P1)

-1 4. (P1 o P2) o P2 5. P2

-1

(f) Let A={1,2,3}, then write all permutations of A.

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time: 2½ hrs. marks: 75 all questions are compulsory...

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