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1 I. (a) Determine the number of positive integers n where 1 and n is not divisible by 2 or 3 or. (b) How many integers between 1 and 2000 are divisible by 2, 3, or 7? (c) Let S = {1, 2, 3, 4,, 6, 7, 8, 9}. Determine whether or not each of the following is a partition of S. i) {{1, 3, }, {2, 6}, {4, 8, 9}} ii) {{1, 3, }, {2, 4, 6, 8}, {, 7, 9}} iii) {{1, 3, }, {2, 4, 6, 8}, {7, 9}} iv) {{S}} (d) It is known that at the University, 60% of the professors play tennis, 0% of them play bridge. 70% jog, 20% play tennis and bridge, 30% play tennis and jog, 40% play bridge and jog. If someone claimed that 20% of professors jog and play bridge and tennis, would you believe the claim? Why? (a) Draw truth tables for the following logical statements: i) ( ) ( ) ii) ( ( )) (b) Prove the following logical equivalence using laws of logic only: i) [( ) ( )] ( ) ii) ( ) [ ( )] ( ) (c) Determine whether ( ) and ( ) ( ) are logically equivalent using truth tables. (d) Prove that (3n-1) = ( ) using Mathematical Induction. III (a) Let A = {1, 2, 3} and R and S are relations on A (b) [ ] and [ ] Find Let A = {1, 2, 3, 4} Determine whether the relation R whose matrix [ ] is Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric or Transitive. (c) Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 1), (2, 3), (3, 4)} Find transitive closure of R using Warshall s Algorithm. (d) Draw Hasse Diagram for the given relation matrix defined on set A = {1, 2, 3, 4, } Is it a chain? [ ]

2 IV (a) Obtain CNF and DNF for the following: i) ( ) ii) ( ) (b) Let A = {a, b, c, d, e} and B = {x, y, z}. Determine whether relation R from A to B is a function, give its range. i) R = {(a, y), (b, z), (c, x), (d, z), (a, z), (b, x)} ii) R = {(a, x), (b, y), (c, x)} iii) R = {(a, x), (b, z), (c, y), (d, y), (e, z)} iv) R = {(a, x), (b, x), (c, x), (d, x), (e, x)} v) R = {(a, z), (b, y), (c, z), (c, y), (c, x),(b, x)} (c) Let A = B = C. Let defined by ( ) ( ) Find f o g(x), g o f(x), f o f(x), g o g(x), f o f o f(x) (d) Let f, g, h be the functions mapping set A = {1, 2, 3, 4} into itself. f = {(1, 2), (2, 1), (3, 1), (4, 4)} g = {(1, 2), (2, 4), (3, 1), (4, 3)} h = {(1, 1), (2, 3), (3, 1), (4, 3)} Find f o g, g o h, g o g, h o h V. (a) Using Nearest neighbor method find Hamiltonian circuit for the graph shown in figure stating with vertex a. What is the weight of this path? (b) Define: i) Graph ii) Degree iii) Loop iv) Isolated Vertex v) Adjacent Vertices (c) How many nodes are necessary to construct a graph with exactly 6 edges in which each node is of degree 2? (d) Show that the maximum number of edges in a simple graph with n vertices is ( ) VI (a) Find the solution of that satisfies (b) Find the solution of (c) Find the solution of that satisfies (d) Find total solution of

3 I. (a) Write step involved in DDA algorithm. Consider the line coordinates A(,) and B(13,9).Determine the line segment using DDA algorithm (b) Describe the components of a typical computer graphics display system. Explain the construction & working of CRT displays. (c) Define Image. Also write a difference between bitmap and vector image. (d) Derive steps involved in Mid point circle drawing algorithm. I (a) Develop a single transformation matrix which does the following i) Scale by 2, 3 in x and y direction ii) Rotates by an angle 30 0 in clockwise direction iii) Translates 4 units in the xdirection (b) Derive the matrix of rotation by an angle Ө in clockwise direction. (c) Derive the transformation matrix to magnify the triangle A(0,0),B(1,2),C(3,2) to twice its size. Draw the same. (d) Write note on 2D Shear transformation. (a) Shear a unit cube situated at origin in xz plane by factor 2, 3 in x and y direction respectively. (b) What is homogeneous coordinate system? Write scaling, translation 3D matrix in homogeneous form. (c) Rotate an object about z axis with coordinates A(1,1,), B(4,1,1),C(,1,4) and D(6,0,0) by and angle 90 degree. (d) Reflect the object about the x=0 plane, where object is having vector coordinates A[ ], B[ ], C[ ],D[ ]. (a) Explain why the Sutherland-Hodgman algorithm works only for convex polygon regions. also suggest modifications to clip concave polygon. (b) Explain Inside-outside test method test with example (c) Write a short note on 2D Viewing pipeline. (d) Clip line p1p2 having coordinates p1 (-10, 0),p2 (30,80 ) against window (Xmin= -30,Ymin= 10), (Xmax= 20,Ymax= 60) using Cohen-Sutherland line clipping algorithm. V. (a) Generate a B-spline curve having polygon vertices p1(1,1), p2(4,3), p3(8,),(8,10) { X=[ ] and N 1,4 (u)=(1-u) 3,N 2,4 (u)=(1-u) 2 *u, N 3,4 =(1-u)*u 2,N 4,4 =u 2 } (b) Explain Painter s algorithm for hidden surface removal. Also give its advantages and disadvantages. (c) Write short note on Bilinear surfaces. (d) Explain following terms i) Spane line coherence ii) Object coherence iii) frame ccoherence iv) depth coherence. (a) What is shadow? Explain different types of shadows. (b) How to put realism in a scene?explain 2 types of colour model. (c) Explain how phong shading technique is works. Also state advantages and disadvantages of phong shading technique (d) Explain term visibility invisible surfaces. Explain two approaches used to determine hidden surfaces

4 I. (a) Explain the following Functions with syntax and example i) MONTHS_BETWEEN ii)next_day iii) SQRT iv) SIGN v)trunc (b) What is a predicate? Explain its different types (c) What are the guidelines for GROUP BY clause (d) What is a view? Explain its types with example I (a) What is a privilege? Explain its types with example. (b) Explain multiple column subqueries with suitable example (c) What is set operators? Explain its type with example (d) What is correlated subquery? Explain scalar subquery with example (a) Explain the advantages of PL/SQL (b) What is an identifier? Write a note on Bind variable. (c) Write a short note on comments in PL/SQL (d) Explain PL/SQL Block structure. (a) Explain IF-THEN-ELSE statement with example. (b) write a PL/SQL program to reverse the nos from 10 to 1 using FOR loop (c) Explain any collection methods with example. (d) What are cursors? Write a short note on classification of cursors. V. (a) What is a procedure? Explain with syntax. (b) List the advantages of stored functions and procedure. Also write the syntax for dropping a function. (c) What are packages? What are its advantages? (d) create a procedure raise_salary to update the emp table, raise the salary by 10% using IN parameter. (a) Explain the execution flow of SQL. (b) Define triggers. List the guidelines for designing triggers. (c) Distinguish between statement level and row level triggers. (d) Explain the INSTEAD of trigger with syntax.

5 I. (a) What is object oriented programming? How is it different from procedure oriented programming? (b) Explain the basic concepts of OOPs. (c) Explain the benefits of OOP. (d) How does object oriented approach differ from object based approach. I (a) Can a class contain multiple constructors? If yes, explain with an example. (b) Short note on destructor. (c) Explain static member functions with an example. (d) What is a class? Explain the general form of class declaration. (a) Explain overloading of unary operator. (b) Explain this pointer in detail. (c) Explain overloading of increment operation. (d) What are friend functions? Write a program using friend function to find largest of 2 numbers. (a) Explain multilevel inheritance with an example. (b) How are constructors used in derived class? (c) Explain read() and write() function. (d) Explain how to open and close a file with an example. V. (a) Explain showpos, showpoint flag with an example. (b) What is a stream? Explain C++ stream classes. (c) How can we design our own manipulators? (d) Both cin and getline() function can be used for reading a string. Comment. (a) What are function templates? Write a program using function template 2 swap 2 integer and 2 float numbers. (b) What is a vector? What are the functions of vector class? (c) Explain maps with an example. (d) Explain use of function objects in algorithms.

6 I. (a) Explain operating system with diagram (b) Write a short note on i) Assembler ii) Linker (c) Discuss real time system (d) Write a short note on cluster system. I (a) Discuss kernel concept (b) Discuss layered approach structure (c) Discuss system call with example (d) Write a short note on system boot (a) Discuss peterson s algorithm (b) Discuss process states (c) Explain multi-threading models (d) Discuss threading issues (a) Discuss concept of swapping (b) Assume there are 4 fixed partitions 1k, 2k, 4k, 8k in the main memory for user. Separate memory is allocated for operating system is 200k then show the memory map of MFT with single queue as well as memory map for MFT with multiple queue. 1. Job 1 need 3.7k memory 2. Job 2 need 1.2 k memory 3. Job 3 need 2k memory 4. Job 4 need 7k memory. Job 2 over 6. Job 3 over 7. Job needs 6 k memory (c) Explain demand paging algorithms (d) Find out page removal algorithms for 3 frames and 4 frames. If reference strings is as follows 1,2,3,4,1,2,,1,2,3,4, Use MRU algorithms. V. (a) Discuss function that is performed on file. (b) Discuss necessary condition required for deadlock creation. (c) Write a short note on i) Bit vector ii) Linked list (d) Discuss deadlock prevention (a) Discuss security problem (b) Discuss DOS attack. (c) Explain virus and it s types. (d) Explain password and it s rules and characteristics.

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