Anna University Chennai 2010-3rd Sem B.Tech Information Technology / , anna university ster - Question Paper
a Fourier series in ihe interval
1. Find the constant term m he e: (7T, 7r).
B.E./B.Tech.Degree Examinations, November/rrecemlj,2$10 Regulations 2008
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Third Semester Common to all branches MA 2211 Transforms and Partial Differential Equations Time: Three Hours J Maximum: 100 Marks
Answer ALL Cjjiest!
Part A -
2. Find the root mean square vMuel$LJ(x\ = $? ixwfu, I).
f
3. Write the Fourier transformaix.u f
\
4. Find the Fourier sine transform a > 0.
5. Form the partial differential equation by eliminating the arbitrary function from
tegral of (D2 - 2DD' + Da) z =
6. Find the ps
7. Write doWn thftl
ible solutions of one dimensional heat equation.
8. Give three fssiblemtighs of two dimensional steady state heat flow equation.
9. EMinethe tulyi step sequence. Write its Z-transform.
10. madreTOiation by eliminating the arbitrary constant A from yn = A-Zn.
Part B - (5 x 16 = 80 Marks) (a) (i) Find the Fourier series expansion of f(x) =
111 7T2
Also, deduce that + -i- r? + oo =
1 O u o
(ii) Find the Fourier series expansion oi f(x)
' X < 7T *
x for J 2-k x for I
|
1 x2 i#the infeBvitlj |
 |
(6)
OR
Obtain the half range cosine series for f(x) x in (0; 7r). (8)
Find the Fourier series as far as the second harmonic to-represent the function f(x) with period 6, given in the following table.
11. (b)
(0
()
x 0 1 2 3 4 5 f(x) 9 18 24 28 26 20;
\.
(8)
(8)
(a)
(i) Derive the ParsevaPs identity for Furier'Transforms.
(ii) Find the Fourier integral representation af defined as
12.
e"- for x > 0
(B)
 |
/,;R |
(b)
12.
(8)
(i) Find the Fourier sine transform of (x, 0 < x < I
/(*) = ] 2-z, 1 < 2; < 2 -x>2
dx
I
(ii) Evaluate
using Fourier cosine transforms of e~ax and
4Jfc2+)(+&z)
e Jt
(8)
R~,. Tjw ..mW
(gJJbrri| the PDE by eliminating the arbitrary function <j> from <f>(x2 + y2 + i?, cz) = 0. (8)-
13.
(ii) Solve thepartial differential equation x2(y~ z)p + y2(z-x)q ~ z2(x y). \ / (8)
\
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I.
\
\
OR
13. (b) (i) Solve the equation [D3 + D2Dr 4Z>>'2 4Z>'3] = cos(2x + y).
(ii) Solve [2D2 - DD - Tf2 + 6D + 3D']z = are1'. 4?***
14. (a) A tightly stretched string of length 21 is fastened at both ends.
the string is displaced by a distance lb' transversely and he string is rJ|pased from rest in this position. Find an expression for the transverse displac%nentj? o/ the string at any time during the subsequent motion*' Q&T
OR \ % >
14. (b) A square plate is bounded by the lines x = 0, y = 0, x 20nd Jf 20. Its
faces are insulated. The temperature along the upper horizontaWdge is given by
u(;r,20) a;(20 ),0 < x < 20
while the other two edges are kept at Qf. Find fifStsteady state temperature distribution zn he plate. I \ (16)
15. (a) (i) Find the -transform of cosnO and! snvllMice dlduce ihe -transforms
of cos(n + 1)0 and aRsinn#i - ; ' (10)
;S- Z(Z l)
/ii) Find the inverse -transform-irs.by residue method.
%*# \% W %
(6)
%
15 (b) (i) Form the difference|qua|B frorrTjhe relation yn = a + b 3n.
\ * t* (8) (ii) Solve yn+2 + 4?yn+i -f 3t&= 25fwith 7/0 = 0 and y\ = 1, using Z-transform. v (8)
B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2010 Third Semester
Computer Science and Engineering CS 2202 DIGITAL PRINCIPLES AND SYSTEMS DESIGN (Common to Information Technology)
(Regulation 2008)
Time : Three hours "% Maximum : 100 Marks
Answer ALL questions PART A (10 %20 Marks)
1. Find the octal equivalent of hexadecimal number AB.CD.
2. State and prove the consensus theorem.
3. Compare the serial andp;allel adder.
4. Define look ahead carry addition.- .....
5. Define priority encode?.
6. Write a dataflow description of a 2-tQtl line Mux using a conditional operator.'
7. Differentiate Moore and Meajy circuit models.
8. What are the applications of shift registers?
9. What is meant by critical race?
10. What are the tvoes of hazards?
/ * PART B (5 x 16 = 80 Marks)
11. (a) f Simplify the following 5 variable Boolean expression using McCluskey
method.
"2? = Srrt (0; 1, 9, 15, 24, 29, 30) + d (8, 11, 31).
Or
(b) Determine the minterm sum of product form of the switching function.
jF = X (0, 1, 4, 5, 6, 11, 14, 15, 16, 17, 20-22, 30, 32, 33, 36, 37, 48, 49, 52, ' 53, 59, 63).
12. (a) Realize a BCD to Excess-3 code conversion circuit starting fromgts truth, table. \ %.
Or
%
/
(b) Design a full adder and subtractor using NAND and NOR respectively.
/
13. (a) (i) Define Multiplexer
(ii) Implement the following Boolean function using 8:
F(A, B, C, D) = ABD + ACD + BCD + ACD
Or
(b) Implement the switching functions zx - ab de + ab cde + be + de z2 = ace.
z3 = be + de + cde + bd ... ....
z4 = ace + ceusing a 5X8X4 programmable logic array.
14. (a) Design a clocked sequential machine using T flip flops for the following
state diagram. Use state reduction if possible. Also use straight binary state assignment.
(b) Using RS-FFs design a parallel counter which counts in the sequence
000,111, 101, 110, 001, 010, 000.......
15. (a) Design a T flip flop from logic gates.
\ r
(b) Find a static and dynamic hazard free realization for the following function using
(i) NAND gates. /
(ii) NOR gates
/T"' | /a,6, c,d) = 2m (1,5,7,14,15).
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PART B (5 x 16 = 80 Marks) f
11. (a) (i) Derive an ADT to perforin insertion and deletion in a singly linked
list. (8)
(ii) Explain cursor implementation of linked lists. Write % the essential operations. - % (8);
0r \ /' (b) (i) Write an ADT to implement stack of size' N using an array. The elements in the stack are to be integers.? The operations to be supported are PUSH, POP and DISPLAY, 'fake into account the exceptions of stack overflow and stack underflow. # (8)
(ii) A circular queue has a size of 5 and has 3 elements 10, 20 and 40 where F = 2 and R = 4. After inserting 50 and 60, what is the value of F and R. Trvmg to insert 30 at this stage what happens? Delete 2 elements from the queue and: insert 70, 80 & 90. Show the sequence of steps with necessary diagrams with the value of F & R.
12. (a) (i) Write an ADT to construction AVL tree. ; (8)
(ii) Suppose the following sequences list nodes of a binary tree T in preorder and inorder, respectively :
Preorder : A, B, D, C, E, G, F, H, J
Inorder : D, B, A, E, G, C, H, Fr J
Draw the diagram of the tree. (8)
% 0r
(b) (i) Write an ADT for performing insert and delete operations in a Binary Search Tree. (8)
(ii) Describe in detail the binary heaps. Construct a min heap tree for ... -*the following :
| 5,2, 6,%.1, 3, 8, 9, 4 (8)
13. (a) ; (i) ' Formulate an ADT to implement separate chaining hashing scheme.' (8)
f? (ii) Show the result of inserting the keys 2, 3, 5, 7, 11, 13, 15, 6, 4 into y? ail initially empty extendible hashing data structure with M = 3. (8)
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