This document contains instructions and questions for an exam in Analog and Digital Electronics. It is divided into 5 modules. For each module, there are 2 full questions with multiple parts to choose from. Students must answer 5 full questions, choosing 1 from each module. They must write the same question numbers and answers should be specific to the questions asked. Writing must be legible. The questions cover topics like operational amplifiers, logic gates, multiplexers, flip-flops, counters, and more. Diagrams and explanations are often required.

1. 18CS32
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Analog and Digital Electronics
Q P Code: 60302
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a Explain the working of Photodiodes. 7 marks
b Describe the working of Linear Voltage Regulator. 6 marks
c Write a short note on the Liquid Crystal Displays. 7 marks
Or
2 a Derive the equation for Non-inverting amplifier. 8 marks
b Design a inverting amplifier to obtain the following output voltage using OPAMP
Vo = -(0.5V1+0.7V2 + 0.8V3 )
6 marks
c Explain the working of Relaxation Oscillator using OPAMP 6 marks
Module – 2
3 a Simplify the following using K-Map
i) Y=Σ(0,4,5,7,9,12,15) + d(3,10)
i) Y=π(1,4,6,10,13,15) + d(7,9)
10 marks
b Simplify the following using Quine-McCluskyMethod
Y=Σ(0,1,2,4,5,6,8,9,12,13,14) + d(3,10)
10 marks
Or
4 a Write a VHDL/Verilog code for the Full Adder.Draw the wave forms obtained after simulation. 10 marks
b Simplify the following using Quine-McCluskyMethod
Y=Σ(1,4,7,9,12,15) + d(0,7)
10 marks
Module – 3
5 a Explain the working of 8:1 MUX. Write the circuit for the same using basic gates 7 marks
b Implement the following using 3:8 decoder and using OR gates
F1=Σ(0,1,5)
F2=Σ(4,5,7)
F3=Σ(2,4,6)
6 marks
c Explain 7 segment decoder. 7 marks
PTO

2. Or
6 a Realize a 8:1 MUX using 2:1 MUX. 7 marks
b Explain the working of Priority Encoder 7 marks
c Write a short note on Programmable Logical Devices 6 marks
Module – 4
7 a Explain the SR Flip-Flops using NAND gate. 6 marks
b What is an edge triggerd Flip-Flop. 6 marks
c Write a HDL code for JK Flip Flop. Draw the simulated waveform also. 8 marks
Or
8 a Write the working of 4 bit Serial In Serial Out Shift Register using D flip Flop. 7 marks
b Explain the working of Universal Shift Register. 8 marks
c Draw a 4 bit Ring Counter using D Flip Flop. 5 marks
Module – 5
9 a Explain the working of Asynchronous UP-DOWN counter. 10 marks
b Design a counter using JK Flip Flop to count 7 to 0 10 marks
Or
10 a Explain the working of Binary Ladder for converting 4 bit data to its analog equivalent using
OPAMP.
10 marks
b Describe the working of Analog-to-Digital converters. 10 marks
*****

3. 1 | P a g e
18CS33
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Data Structures using C
Q P Code: 60303
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a Define Pointer. With examples, explain pointer declaration, pointer initialization and use of
the pointer in allocating a block of memory dynamically.
6 marks
b Explain structure and Union with suitable example. 6 marks
c Write a program to implementing binary search to find an element in an array. 8 marks
Or
2 a Write a C program for pattern matching using pointers. 8 marks
b Write a program using structures with following fields NAME, ROLL NO, Marks in M1,
M2, M3 and find total and average. Read any N records and print all the records and also
print the record who is having second highest total with all the fields.
10 marks
c Write a short note on string operations. 4 marks
Module – 2
3 a Define Queue. Write a function for both INSERT ( ) and DELETE ( ) functions. 8 marks
b Write the postfix form of the following expression.
i. ( a + b ) * d + e / ( f + a * d ) + c
ii. ( ( a / ( b – c + d ) ) * ( e – a ) * c )
iii. a / b – c + d * e – a * c
6 marks
c Differentiate between Iteration and Recursion. 6 marks
Or
4 a Write an algorithm to convert infix to postfix expression and apply the same to convert
following expressions from infix to postfix.
i. a / b – c – d * c – a * c
ii. ( a – b ) – c / d $ n e
12 marks
b Define Stack. Give the C implementation of PUSH and POP operation using array. 8 marks
Module – 3
5 a
1) Write a program in C to implement insert front, delete front and display functions using
circular double linked list?
10 marks
PTO

4. 2 | P a g e
b Explain the following:
i. Doubly linked list.
ii. Linked representation of sparse matrix.
10 marks
Or
6 a Write a C program to implement STACK operations using single linked list. 10 marks
b What is a linked list? Explain the types of linked list with diagram. 10 marks
Module – 4
7 a
2) What is a Tree? Explain these terms by taking an example.
3) i. Root node
4) ii. Leaf node
5) iii. Degree
6) iv. Siblings
7) v. Depth of a tree
10 marks
b Write a function to insert an item into a binary search tree based on direction. 6 marks
c Construct a binary search tree having the following sequences.
i. Preorder seq ABCDEFGHI
ii. Inorder seq BCAEDGHFI
4 marks
Or
8 a What is binary search tree? Draw the binary search tree for the following list 14, 5, 6, 2,
18, 20, 15, 19, 3, 16.
10 marks
b Explain the following with suitable example:
i. Strictly binary tree.
ii. Complete binary tree.
iii. Expression tree.
iv. Almost complete binary search tree.
v. Skewed tree.
10 marks
Module – 5
9 a What is Hashing function and what are its types explain with example. 10 marks
b Define Files and explain the following:
i. Opening and Closing of files.
ii. Input and Output operations of files.
10 marks
Or
10 a Explain the following with example.
i. Directed graph
ii. Multi graph
iii. Complete graph
iv. Cyclic graph and Acyclic graph
8 marks
b Differentiate between Static and Dynamic hashing. 6 marks
c What is BFS? Briefly explain the traversal of BFS with example. 6 marks
*****

5. 1 | P a g e
18CS34
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Discrete Mathematical Structures
Q P Code: 60304
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a Find the possible truth values for p, q and r if
i) p→(q ˅ r) - FALSE
ii) p ˄ (q → r) - TRUE
5 marks
b Establish the validity of the following argument
∀x, [p(x) ˅ q(x)]
Ǝx, ¬p(x)
∀x, [¬q(x) ˅ r(x)]
∀x, [s(x) → ¬r(x)]
∴ Ǝx ¬s(x)
6 marks
c Negate and simplify:
i) ∀x [ p(x) ˄ ¬q(x)] ii) Ǝx [ (p(x) ˅ q(x)) → r(x)]
4 marks
d Prove the following logical equivalence using the laws of logic.
[¬p ˄ (¬q ˄ r)] ˅ (q ˄ r) ˅ (p ˄ r)  r
5 marks
Or
2 a Prove that for any three propositions p,q,r [p → (q ˄ r)]  [(p → q) ˄ (p → r)] 5 marks
b Prove the following logical equivalence using the laws of logic.
(p →q) ˄ [¬ q ˄ (r ˅ ¬q)]  ¬ (q˅p)
5 marks
c Determine the truth value of the following statements if the universe comprises all non- zero
integers:
i) Ǝx, Ǝy [ xy = 1]
ii) Ǝx, ∀y [ xy = 1]
iii) ∀x, Ǝy [ xy = 1]
iv) Ǝx, Ǝy [(2x + y = 5) ˄ (x - 3y = -8)]
v) Ǝx, Ǝy [(3x – y = 17) ˄ (2x + 4y = 3)]
5 marks
d Find whether the following argument is valid or not:
No Engineering student of 1st
or 2nd
semester studies logic
Anil is an Engineering student who studies logic
∴ Anil is not in second semester
5 marks
PTO

6. 2 | P a g e
Module – 2
3 a By Mathematical induction. Prove that for every positive integer n, the number
An = 5n
+ 2.3n-1
+ 1 is a multiple of 8.
5 marks
b A certain question paper contains 3 parts A,B,C with 4 questions in part A, 5 questions in
part B and 6 questions in part C. It is required to answer 7 questions selecting atleast 2
questions from each part. In how many ways can a student select his 7 questions for
answering.
5 marks
c Find the co-efficient of a2
b3
c2
d5
in the expansion of (a+2b-3c+2d+5)16
5 marks
d Find the number of integer solutions of x1+x2+x3+x4+x5 = 30 where x1≥ 2, x2 ≥3,
x3 ≥ 4, x4 ≥ 2, x5 ≥ 0.
5 marks
Or
4 a Prove by mathematical induction that,
1 . 2+2 . 3+3 . 4+....+n(n+1) = 1/3 n(n+1)(n+2)
5 marks
b How many arrangements are there for all letters in the word SOCIOLOGICAL. In how many
of these arrangements i) A and G are adjacent? ii) all the vowels are adjacent?
5 marks
c Find the co-efficient of
i) x9
y3
in the expansion of (2x-3y)12
ii) x12
in the expansion of x3
(1-2x)10
5 marks
d A sequence {an} is defined recursively as a1=7 and an=2an-1 + 1 for n ≥2. Find an in explicit
form.
5 marks
Module – 3
5 a Let f: R→R be defined by f(x)= 3x-5 for x>0
-3x+1 for x≤0
Determine f(-1),f(5/3),f-1
(1),f-1
(-3),f-1
(3)
5 marks
b State pigeon hole principle. Prove that if 30 dictionaries contain a total of 61,327 pages, then
at least one of the dictionary must have at least 2045 pages.
4 marks
c For A={1,2,3,4,5} and B={w,x,y,z}, let a function f : A→B be given by
f={(1,w),(2,x),(3,x),(4,y)}. Find the images of the subsets A1={1}, A2={2,3},
A3={1,2,3} under f.
3 marks
d Let A={1,2,3,4} and B={1,2,3,4,5,6}
i) Find how many functions are there from A to B. How many of these are one-to-one
and onto?
ii) ii) find how many functions are there from B to A. How many of these are one-to-one
and onto?
8 marks
Or
6 a Let f and g be functions from R to R defined by f(x)=ax+b and g(x)=1-x+x2
. If
(gof)(x)=9x2
-9x+3, determine a,b.
5 marks
b Let A={1,2,3,4,6,12} On A define the relation R by aRb if and only if “a divides b” Draw
the digraph.
4 marks
c Verify the following function is one-to-one or onto. f(a)=a2
, a is any real number. 3 marks
d Let A= {1, 2, 3} and B = {2, 4, 5}. Determine the following: i) Number of binary relations
on A. ii) Number of relations from A to B iii) Number of relations from A to B that
contain (1,2) and (1,5). iv) Number of relations from A, B that contain exactly 5 ordered
pairs. v) Number of binary relations on A that contain at least 7 ordered pairs.
8 marks

7. 3 | P a g e
Module – 4
7 a Using expansion formula, find the rook polynomial for the board shown below. 6 marks
b Show that the set of positive divisors of 36 is a POSET and draw its Hasse diagram. Hence
find its i) least element ii) greatest element.
7 marks
c In how many ways can one arrange the letters in the word CORRESPONDENTS so that
i) There is no pair of consecutive identical letters?
ii) There are exactly two pairs of consecutive identical letters?
7 marks
Or
8 a In how many ways can the 26 letters of the alphabet be permuted so that none of the patterns
spin, game, path or net occurs.
6 marks
b Let A={1,2,3,4,5} Define a relation R on AxA by (x1,y1)R(x2, y2) if and only if
x1+y1= x2+y2. i) Verify that R is an equivalence relation on AxA.
ii) Determine the equivalence classes [(1,3)],[(2,4)],[(1,1)].
7 marks
c Four persons P1,P2,P3,P4 who arrive late for a dinner party find that only one chair at each
of five tables T1,T2,T3,T4 and T5 is vacant. P1 will not sit at T1 or T2, P2 will not sit at T2,
P3 will not sit at T3 or T4, and P4 will not sit at T4 or T5. Find the number of ways they can
occupy the vacant chairs.
7 marks
Module – 5
9 a Construct an optimal prefix code for the symbols a,b,c,d,e,f,g,h,i,j that occur with respective
frequencies 78,16,30,35,125,31,20,50,80,3
6 marks
b Show that the below graphs are isomorphic. 6 marks
c Suppose that a tree T has two vertices of degree 2, four vertices of degree 3 and three vertices
of degree 4. Find the number of pendant vertices in T.
4 marks
d Define the following with example. i) Spanning subgraph ii) Induced subgraph 4 marks
Or
10 a Obtain an optimal prefix code for the message “LETTER RECEIVED”. Indicate the code. 6 marks
b Determine the order |V| of the graph G = (V, E) in the following cases.
i) G is a cubic graph with 9 edges.
ii) G is regular with 15 edges.
iii) G has 10 edges with 2 vertices of degree 4 and all other vertices of degree 3.
6 marks
c Is there a simple graph with 1,1,3,3,3,4,6,7 as the degrees of its vertices. 4 marks
d Define a tree. Prove that a tree with two or more vertices contains at least two leaves. 4 marks
*****

8. 4 | P a g e

9. 1 | P a g e
18CS35
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Unix and Shell Programming
Q P Code: 60305
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a Describe the role played by the kernel and shell in the UNIX architecture. 10 marks
b Explain the following with suitable examples:
i. Internal and External commands.
ii. Command arguments and options.
10 marks
Or
2 a With the help of examples, explain the following commands.
i. Printf ii. Passwd iii. Who iv. Echo
10 marks
b Explain the significance of the man with keyword option and whatsi. 10 marks
Module – 2
3 a What is FILE? List and explain the categories of files. 10 marks
b List and explain the directory commands used in the UNIX. 10 marks
Or
4 a Explain the following with suitable examples:
i. absolute and relative paths ii. Absolute and relative permissions.
10 marks
b Which of the command is used to listing directory contents? Explain its options. 10 marks
Module – 3
5 a With a neat diagram, explain the different modes of vi editor. 10 marks
b What are the 3 standard files supported by UNIX and also give the suitable Illustration how
Input and output redirection works in UNIX.
10 marks
Or
6 a Describe the concepts of shell interpretive cycle in UNIX. 10 marks
b Explain the features of the grep command and its options. 10 marks
Module – 4
7 a What is shell script? Explain the shell feature of test command and its shortcut. 10 marks
b With the help of example, explain the here(<<) document, trap and filters command. 10 marks
PTO

10. 2 | P a g e
Or
8 a Explain the following commands.
i) Exit ii. umask iii. Head iv. tail
10 marks
b Describe the significance of the file inodes and inode structure in UNIX. 10 marks
Module – 5
9 a Explain the mechanism of process creation and also given the details about process states. 10 marks
b List and explain any three string handling function in perl. 10 marks
Or
10 a Explain the following perl script functions with suitable examples:
i. chop ( ) and chomp ( )
ii. split ( ) and join ( )
10 marks
b Briefly explain the file handles and handling file in perl. 10 marks
*****

11. Page 1 of 2
18DIPMAT-301
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Additional Mathematics I
Q P Code: 60306
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. Write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a Express 1-i√3 in the polar form and hence find its modulus and amplitude. 6 marks
b
Find the real part of .
sin
1
sin
1


i
i

 6 marks
c Define dot product between two vectors A and B Find the angle between the vectors
.
6
3
2
,
5 k
j
i
B
k
j
i
A 





8 marks
Or
2 a If   ).
(
,
2
3
,
3
2 B
A
and
B
A
that
show
k
j
i
B
k
j
i
A 






 are orthogonal. 6 marks
b Show that the position vectors of the vertices of a triangle
k
j
i
and
k
j
i
k
j
i 4
4
3
5
3
,
2 




 , form a right angle triangle.
6 marks
c If
.
,
.
4
8
,
4
3
2 b
a
find
also
b
to
lar
perpendicu
is
a
that
prove
then
k
j
i
B
k
j
i
A












8 marks
Module – 2
3 a
With usual notation, prove that 






dr
d
r


tan
6 marks
b Find the pedal equation of the curve : rm
=am
cosm𝛉. 6 marks
c
Prove that 











 
y
x
y
x
u
where
u
yu
xu y
x
3
3
1
tan
,
2
sin , using Euler’s theorem,
8 marks
Or
4 a Find nth
derivative of sinx sin2x sin3x 6 marks
b
if




































2
2
2
2
2
2
1
y
u
x
u
u
y
u
x
u
that
show
e
u xy
8 marks
c
If 













)
,
,
(
)
,
,
(
.
,
,
w
v
u
z
y
x
J
find
uvw
z
z
y
v
z
y
x
u
6 marks
PTO

12. Page 2 of 2
Module – 3
5 a
Evaluate i) dx
x
x
ii
dx
x 5
2
0
2
0
3
6
cos
sin
.
cos
 
 
6 marks
b Evaluate
ay
x
parabola
the
and
a
x
ordinate
the
axis
x
by
bounded
gion
the
is
R
where
dx
dy
xy
A
4
2
,
Re
,
2




6 marks
c Obtain the reduction formula for ∫ 𝑠𝑖𝑛𝑛
𝑥𝑑𝑥 8 marks
Or
6 a
Evaluate ∫ 𝑠𝑖𝑛10
𝜋
2
0
𝑥 𝑑𝑥
6 marks
b Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦
𝑅
over the region bounded by x=0, y=0 and x+y=1 6 marks
c Evaluate ∫ ∫ 𝑥𝑦. 𝑑𝑦. 𝑑𝑥
√𝑥
𝑥
1
0
8 marks
Module – 4
7 a A particle moves along a curve vector r = cos2t i+sin2t j+t k, where t is time .Find the
velocity and acceleration at time t=π/8 along √2i+√2j+k
6 marks
b If 𝐹
⃗ = 2𝑥2
𝑖 + 3𝑦𝑧𝑗 + 𝑧𝑥2
𝑘 then find i) ∇𝑋(∇𝐹
⃗) ii) ∇(∇𝑋𝐹
⃗⃗⃗⃗⃗⃗) 8 marks
c Show that the vector
.
,
2
2
2
al
irrotation
is
G
F
that
show
xyk
zxj
yzi
G
k
z
j
y
i
x
F











6 marks
Or
8 a Find the velocity and acceleration of a particle moves along the curve
.
2
)
4
3
(
)
4
(
3 3
2





 t
time
at
k
t
j
t
t
i
t
r

6 marks
b i. if )
1
,
1
,
1
(
,
2
2
2
2
2
2
at
find
x
z
z
y
y
x 
 


 6 marks
c Show that 𝐹
⃗(𝑦 + 𝑧)𝑖 + (𝑧 + 𝑥)𝑗 + (𝑥 + 𝑦)𝑘 is irrotational. 8 marks
Module – 5
9 a Solve 𝑦(2𝑥 − 𝑦 + 1)𝑑𝑥 + 𝑥(3𝑥 − 4𝑦 + 3)𝑑𝑦 = 0 6 marks
b Solve 𝑦(2𝑥𝑦 + 1)𝑑𝑥 − 𝑥 𝑑𝑦 = 0 6 marks
c
solve x
y
x
y
dx
dy
sec
tan 2


8 marks
Or
10 a 0
)
(
)
( 3
2



 dy
x
y
dx
y
x
Solve 7 marks
b Solve (4𝑥𝑦 + 3𝑦2
− 𝑥) 𝑑𝑥 + 𝑥(𝑥 + 2𝑦)𝑑𝑦 = 0 6 marks
c Solve (2𝑥 + 𝑦 + 1)𝑑𝑥 + (𝑥 + 2𝑦 + 1)𝑑𝑦 = 0 7 marks
*****

13. 1 | P a g e
18EC33
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Computer Organization and Architecture
Q P Code: 62303
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a With help of a block diagram Explain functional units of a Digital Computer 10 marks
b What is a bus? Explain Single Bus structure 10 marks
Or
2 a Perform addition and subtraction on the following pairs of numbers represented in
2’s complement format. In each case, verify whether overflow has occurred or not. The
numbers are represented using 7-bits including the sign bit.
i) +25 and +38 iv) +33 and+51
ii) –24 and +63
iii)–12 and –40
10 marks
b Write a note on IEEE standard for floating point numbers 10 marks
Module – 2
3 a Define addressing modes? Explain any five addressing modes with an example for each. 10 marks
b Explain interfacing of Keyboard and Display using program controlled I/O 10 marks
Or
4 a Explain how data is exchanged between a calling program and a subroutine 10 marks
b With the help of examples explain i) logical instructions ii) Shift and Rotate instructions 10 marks
Module – 3
5 a Discuss the sequence of steps involved in handling an interrupt from a single device 10 marks
b Explain briefly about Nesting Interrupts, Vectored interrupts and Simultaneous request
handling
10 marks
Or
6 a Write a note on bus arbitration 10 marks
b Write a note on working of 2 channel DMA controller 10 marks
Module – 4
7 a Explain the Read/Write operation of an SRAM cell designed using CMOS with the help of
a neat diagram
10 marks
PTO

14. 2 | P a g e
b Write a note on ROM and its various types 10 marks
Or
8 a With a neat block diagram explain Virtual Memory Organization 10 marks
b With a neat diagram explain Magnetic Disk Principles 10 marks
Module – 5
9 a Discuss the need for gating signals with an example 10 marks
b With the help of a neat sketch, explain three-bus organization of the processor 10 marks
Or
10 a Explain in detail and with necessary step involved in execution of instruction Add(R3), R1 10 marks
b Explain the hardwired control unit organization 10 marks
*****

15. Page 1 of 2
18MAT31
Third Semester BE Degree Examination January 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Engineering Mathematics - III
Q P Code: 60301
Instructions: 1. Answer five full questions.
2. Choose one full question from each module.
3. Your answer should be specific to the questions asked.
4. write the same question numbers as they appear in this question paper.
5. Write Legibly
Module – 1
1 a
Find the Laplace transform of
cosat−cosbt
t
.
7 marks
b A periodic function f(t) of period a , a > 0 is defined by
f(t) = {
E 0 < t < a/2
−E
a
2
< t < a
Show that L[f(t)] =
E
s
tanh(
as
4
) .
6 marks
c Solve the differential equation
d2y
dt2 + 4
dy
dt
+ 4 y = e−t
with y ( 0 ) = 0 = y′
( 0 ) by using Laplace
transforms.
7 marks
Or
2 a Find L−1 [ cot−1
s ]. 7 marks
b
Express f(t) = {
sint, 0 < 𝑡 < π
sin2t, π < 𝑡 < 𝜋
sin3t , t > 2π
.
in terms of unit step function and hence find their Laplace transform f (t).
6 marks
c Using Convolution theorem obtain inverse transformation of
s2
(s2+ a2)(s2+ b2)
. 7 marks
Module – 2
3 a Find the Fourier series for the function
π−x
2
in 0 < x < 2π .
Hence deduce that
π
4
= 1 −
1
3
+
1
5
-
1
7
+ ……
7 marks
b
Find half range sine series for f(x) = {
1
4
− x 0 < x < 1/2
x −
3
4
1/2 < x < 1
.
7 marks
c Express y as a Fourier series upto the first harmonic given.
x 0 π/3 2π/3 π 4π/3 5 π/3 2π
y 1.98 1.30 1.05 1.30 - 0.88 - 0.25 1.98
6 marks
Or
4 a Obtain Fourier series for the function f(x)=│x│ in –π ≤ x ≤ π
hence deduce that
π
8
2
=∑
1
(2n−1)2
∞
n=1 .
7 marks
b Expand f ( x ) = 2x -1 as a cosine half range Fourier series in 0 < x < 1. 6 marks
PTO

16. Page 2 of 2
c Obtain constant term and the coefficients of the first sine and cosine terms in the Fourier expansion
of y from the table.
x 0 1 2 3 4 5
f(x) 9 18 24 28 26 20
7 marks
Module – 3
5 a Find the Fourier sine transforms of f(x) =
1
x(1+x2)
. 7 marks
b Find the Fourier cosine transform of f(x) =
1
1+x2. 6 marks
c Solve the difference equation yn+2 + 6yn+1 + 9yn = 2n
, with yo = y1 = 0 by using z transform. 7 marks
Or
6 a Find the Fourier sine transform of e−|x|
Hence show that ∫
xsinmx
1+x2
∞
0
dx =
π
2
e−m
, m > 0 . 7 marks
b Obtain the Z-transform of 2n + sin ( nπ / 4 ) + 1. 6 marks
c Obtain the inverse Z-transform of
2z2+3z
(z+2)(z−4)
. 7 marks
Module – 4
7 a Employ Taylor’s method to find y at x=0.1 and 0.2 correct to four places of decimal in step size of 0.1
given the linear differential equation
dy
dx
- 2y = 3ex
whose solution pasess through the origin.
7 marks
b Using fourth order Runge – Kutta method to find y at x = 0.1 given that
dy
dx
= 3ex
+ 2y , y( 0 ) = 0 ,
taking h = 0.1.
7 marks
c Given that
dy
dx
= x – y2
and the data y(0) = 0, y(0.2) = 0.02, y(0.4) = 0.0795 , Y(0.6) = 0.1762 ,find
y(0.8) by using Adam- Bashforth method.
6 marks
Or
8 a Using modified Euler’s method find y(20.2) and y(20.4) given that
dy
dx
= log10(
x
y
) with y(20) = 5
taking h=0.2.
7 marks
b Solve : ( y2
− x2
)dx = (y2
+ x2
)dy for x = 0 (0.2) 0.4 given that y = 1 at x = 0 initially, by
applying Runge-Kutta Method of order 4.
7 marks
c Apply Milne’s Predictor and Corrector formulae to compute y(1.4) correct to four decimal places.
given
dy
dx
= x2
+
y
2
with
x 1 1.1 1.2 1.3
y 2 2.2156 2.4649 2.7514
6 marks
Module – 5
9 a
By Runge-Kutta method, solve
d2y
dx2 = x (
dy
dx
)
2
− y2
for x=0.2 correct to four decimal places, using the
initial conditions y=1 and y′ = 0 when x=0.
7 marks
b Solve the variational problem ∫ (12𝑥𝑦 + 𝑦′2
)𝑑𝑥
1
0
under the conditions 𝑦(0) = 3 and 𝑦(1) = 6. 7 marks
c A heavy cable hangs freely under gravity between two fixed points. Show that the shape of the cable
is a catenary.
6 marks
Or
10 a Apply Milne’s method to solve
d2y
dx2 = 1+
dy
dx
given the following table of initial values.
Compute y (0.4) numerically.
X 0 0.1 0.2 0.3
Y 1 1.1103 1.2427 1.399
y′ 1 1.2103 1.4427 1.699
7 marks
b Derive Euler’s equation in the Standard form
∂f
∂y
-
d
dx
(
∂f
∂y′
) = 0 7 marks
c Prove that the shortest distance between two points in a plane is a straight line joining them. 6 marks
*****

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