The document contains examination question papers for various subjects including Engineering Mathematics, Discrete Mathematical Structures, and Unix & Shell Programming for third semester BE students of Adichunchanagiri University. Each paper outlines the instructions for answering, detailing the modules and questions to choose from across multiple topics including Laplace transforms, Fourier series, and shell commands. It emphasizes writing legibly and specifically answering the questions as presented.

1. Third Semester BE
Time: 3 Hours
ADICH U NCHANAGI RI UNIVERSITY
n11"
{>cn"t C I
18MAT31
Degree Examination November 2020
(CBCS Scheme)
Max Marks: 100 marks
Sub: Engineering Mathematics - IIl
Q P Code: 60301
" 1'r]'.-o''r''t;;
'"
2. Choose one full question from each module. ,.i!,...i ,,,,. .
3. Your answer should be specific to the questions asked. ..,",,.,
"iu '''
4. write the same question numbers as they appear in this question papgr,, u
5. Write Legibly :. .,n,,.,.,i
Module-1 'lt.:'"'t,''
a Find the Laplace transform of tcosat .,"'.,:;
b ,.,,
..
6marks
A periodic function f(t) of perio d a, a> 0 is defined by f (q _ {E_,
o
,.
t_: y: 7 marks
(-E _1t1a
Show that L[1f (t)] =: tanh(ff). ,i..,.,,,,.
:
c solvethedifferentialequatio"
# *4# + 3.y:r-t with y(0): r:!,(0) using Tmarks
Laplace transforms. *' ',.,.
,,.
,
:,:
'
Or
Find Ff bs(#) ]
(sint,0<r<z
Express f (t) - IriiZt, r 1"t l-'"n
sln3r, t)Zn
in terms of unit step function and hence find their Laplace transform f (t)
Using convolution theorem obtain inverse transformation of
*;!
Module - 2
Find the Fourier series for the function x2 in -r 1 x 1 r .
Find half range sine series of flx): [x 0 < x < 7T/2
ln-x Tt/Z<x<Tr
Express y as a Fourier series up to the first harmonic given.
2a
b
3a
b
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
Or
PTO

2. a Find the Fourier series for the function
n-' in 0 < x I 2n .
?
Hencededucen - 1,-1+ -1......
4357
Expand f (x) : 2x - xz as a cosine half range Fourier series in 0 < x < 2 7 marks
Obtain constant term and the coefficients of the first sine and cosine terms in the Fourier 7 marks
expansion of y from the table
b
c
6 marks
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
Module - 3
Find the Fourier transforms of f (D - { ! f or l:1,
( o for lxl
Find cosine transform of f(x) : e-o*, f or a > 0
=:, Hence evaluate !f,i'ff'Oil
b
c
,'!:''. ' ")
l;. "t+-
l, ..,'
.ict,,*+"..l:
't i.
iii1.. ;i'
usins z transform
.ti
1t _*
-m>0
2',
Solve the difference equation !n+z - 4yn = O,with lo = 0,!t = 2.
0r , ::'
a Find the Fourier sine transform of e-l'|. Hence show that f- #
' .'"
b Obtain the Z-transform of (n + 1) '
, . .
c ObtaintheinverseZ-transform of * .i. ''.,"., ,'
tz-r)(z-2)
Modu,te - 4
a Employ Taylor's method to find y gt x=0.1 given
H
:*' * yz ,y(o):l
b Using fourth order Runge - kuuri..niqthod to find y at x : 0.1
dY -' 'i'
'';;l 'llr': ';:'
given that
fi:3ex *2y , y(0)
=-0,
takingh:0.1
c Given that !: x - y2 and the data y(0):0, y(0.2) :0.02,y(0.4) :0.0795, y(0.6) :0.1J62,
dx .::.
find y(0.8) by using Adam- Bashforth method.
'.,.:)::.
a Using nrqdified Euler's method find y(0.2) given that
X: x - y2 with yt0) :1 taking
h=P.l."i','..,,''
b'.fSl.f3't (y-x)dx-(y+x)dyfory(0.2) given thaty=Latx=0 initially, taking
h+0.1. by applying Runge-Kutta Method of order 4.
c Apply Milne's Predictor and Corrector formula to compute y(0.4) given
H:rr. - ,
with
x 0 0.1 0.2 0.3
v 2 2.010 2.040 2.090
;bv,
,l
.dx
x 0 2 3 4 5
f(x) 4 8 l5 7 6 2

3. Module - 5
Find y(0.1) by using Runge-Kutta method of 4th order,
given yu - x'y'-2xy = 1,y(0) =lqnd -y'(0) = 0.
Find the extremal of the functional ktr' - y'' - Zysinx) dx with y(0) : o ,y(:): 1
A heavy cable hangs freely under gravity between two fixed points. Show that the shape of the
cable is a catenary.
Or
I 0 a Apply Milne's method to solve y' -- | - 2W' given the following table of initial values
,l ti
,4., i
'1,i :1,_.'i
!;
]. , ,-:
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
;:'
Compute y (0.8) numerically ,,.
,:t:;
Derive Euler's equation in the Standard form { - *. t*):0
i"'"''"'l:
0Y dx 'dYt
Prove that the shortest distance between two points in a plane is a:ritfaight line joining them.
X 0 0.2 0.4 0.6
Y 0 0.02 0.0795 0.r762
v' 0 0.1996 0.3937 0.5689
***r8,k
'i:.:' ,
t..

4. ADICHUNCHANAGIRI UNTVERSITY 18CS34
BE Third Semester Examination November Z0Z0
(CBCS Scheme)
Tirne: 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 ..,:1,:,,,.,,..;,"
:,:,
3'YouranSwershouldbespecifictothequestionsasked.
4. write the same question numbers as they appear in this question pdper,.....,.." "'::,.
5. Write Legibly .1:.,,.;1r,,...-1
Module - 1 ,;t'r,,,,,,;.u'':::'
,*i17,.. " ;l
I a Determine whether the following compound statements are tautology &;o, 7 marks
(i) [(p- q) A (q - r)] - (p - r) .. ;.'' ',,
(ii) tfu v q) n (p - r) A (q - r) -+ r :: "',,,,..''
b Prove the logical equivalence by using laws of logic.
:
6 marks
1p-+ q)A[-aA(r v -q)] e -(qVp)
c Establish the validity of the following arguments. 7 marks
p
P'-+ r
', ..,', P1(qV-rr)
'"'-r:. r...' -rQ V -rS
'. : .'. 5;
Or
2 a Verifu the principle of dualiq, for the following logical equivalence 6 marks
[-(P n q) - -p v (-p v q)] ,i (---,pv il.
b Establish the validity of the following argument 7 marks
p+q
q -+ (rAs)
-rrV(-rfVu)
pAt
,.. u
t (i) Define open sentence and quantifiers with an example each. 7 marks
(ii) Negate and simplify the following:
(a) vxlp(x)A-q(x)l
(b) lxlp(x)Vq(x)l - r(x)
PTO
l lDtnn

5. 4a
b
c
Module - 2
a Using mathematical induction,
Prove that 4n< (n'-l) for all positive integer n26.
b For the Fibonacci sequence F0, Fr, andF2...
prove that F":l lftf)' - (r4"t
------" Vs[ z ) z )]'
c Find the number of arrangement of the letters in TALLAHASSEE which have no
adjacent A's
Or
Find an explicit formula for an:an-1+dn, &r:4 for nZ2.
,ir 't' l:.
I ,,,,,,
Prove)[., :^ - a- yn e Z+. ..,.,:,, ,]
-t-r i(i+1) n+7
Find the coefficient of a2b3c2ds in the expansion of (a+2b-3c+2d+5)16 ;:,''' t" ..:'
Module - 4
'1,- :];",';'' -I.Yllrt'l tllt -
rt
t' ?'t'j"
7 A For A:{a,b,c,d,e}, the Hasse diagram for the poset (A,R) is shown below:
Module - 3
a Determine whether or not each of the following relations is a functibn."lf 'a
relation is
(i) {(x,y) I x,y e R, y:3x*1}, a relation from R to R
(ii){(x,y) I x,y e Q, *'+ y2:7, a relation from e to.Q.
b If f: R -+ R is defined by, (x): x2+5 find f 1({6}),f ,([G,7j),f
'([-4,5]),
f '([6,10]), f 11ps,.o;;.
, ",",.
t L.t f: R -r'R be defined by, f(x):t t' - L' '';r0
Find f ,
(- 1 0), ., ;; ;;-;'il;.,t,,i{ il="gl, under f for
each of the following intervals:.[-5,-i1,,1i5,61,
. t,.'t''tl '', 0r
a Define the Cartesian produ.ct o{,Wb sets. For any three non empty sets A, B, c
Prove that (i) A x (B;c|: (AxB)-(Axc) (ii) A x (Bnc): (AxB) n (Axc)
b Let f: R -+ R, g: R -+ R be defined by (x):x2 and g(x):x+5.
Determine fo g aritll;go,f.
( x*7, x< 0
Let f: R --+ R is defined by, f1x) :|-Zx + 5,0 < x < 3
: ( ,-i,,x)3
Find f1(-10), fl(0), ft1+) and also determine pre images under f for each of the
.',.. following intervals: [-5,-l], [-5,0], l-2,41).
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
6 marks
7 marks
7 marks
6 marks
6 marks
8 marks
.,"_:,
-,r:_, ;a"'
:'j' '';. -. r'
1,. .i
Let A:{ 1,2,3,4} and let R the relation defined by R:{(x,y)lx,y€A, xSy}.
Determine whether R is reflexive, symmetric, antisymmetric or transitive.
ln how many ways can the 26 letters of the English alphabet be performed so that
letters CAR,DOG,FLIN,BYTE occurs?
6 marks
5 marks
none of the 8 marks

6. 9a
Or
Let A:{ 1,2,3,4,6,8,12} and R be the partial ordering on A defined by aRb
if adivides bthen
i) Draw the Hasse diagram of the POSET(A'R)
ii) Determine the relation matrix for R
iii) Topologically sort the Poset (A'R)
Define an equivalence relation and equivalence class with an example'
ln how many ways can one arrange the letters in the word CoRRESPONDENTS so that
i)There is no consecutive identical letters?
iitt","r" is exactly 2 pairs of consecutive identical letters?
iii)there are at least i pairs of consecutive identical letters?
Module - 5
Define and give an example for each of the following
i) Graph
ii) Dirlcted Graph and Undirected Graph
iii) Walk, OPen and Closed Walk
iv) Trail and Circuit
Show that the following graphs are isomorphic' .::
l0aDisctrssKonisbergbridgeplgblemandthesolutionoftheproblem.
bProvetlratineverygraph;:11re'numberofverticesofodddegreeiseven
!l ,,
c Define prefix code.,Obtainiah optimal prefix code for the message
ROAD IS GOOD indlcPte the code
.:. 'k '*:&
*
8 marks
4 marks
8 marks
l'.
''t''"
6 marks
6 marks
6 marks
8 marks
6 marks
c Obtain a,., ojtin]al prefix code for the message MISSION SUCCESSFUL indicate the code' 8 marks
Or
3lFage

7. ADICHUNCHANAGIRI UNIVERSIry
18CS35
Third semester BE Degree Examination November 2020
(CBCS Scheme)
Time: 3 Hours Max Marks: 100 marks
Sub: Unix and Shell Programming
QPCode:60305 ,.,
Instructions: 1. Answer five full questions. . .,''li '
2. Choose one full question from each module' ''i'',,.,.i
3. Your answer should be specific to the questions asked. ,i'"''',,.
''
4" write the same question numbers as they appear in this question paprpfif..- i
5. Write LegiblY '- 'i:t"''+
Module- l .,+:?'
1,ri '
10 marks
i a Describe the features of the LINIX j'"",.
b Explain the following commands with suitable examples: ::. "',,,,. l0 marks
i. cal and who . ii""'','.
ii. echo and printf. .,q;i;"'i
.'
Or ii._,,r,
"O.
2 a With the help of examples, explain the knqu;irtg the user terminal, displaying its 10 marks
characteristicsandsettingcharacteristics.''r''::,''
b r. Dif]"erentiate between internal and extefnal commands. 10 marks
ii. Discuss tho significance of tl're /etc/passwd and ietclshadow files.
,)
Module - 2
a L.is1 and explain the basic categorieq'of files.
b L.xplain the directory commands used in the LNIX'
rt 'i:,:ij'
a Write the INIX conir,naiid for the following:
i. List ali"fhe files in print working directory which are having exactly 5 character in
theii'iitdfraine and any number of characters in their extension.
-t.....:'
ii. fo ccipy all files stored in /home/bgsit/cs with .c ,.cpp and .java extensions to progs
directory in the current directory.
iii. r,To delete all files containing .c,.html,.js and .pl in their filename extensions.
iv. To remove all file u,ith three-character extension except .out from the curent
directory.
b Explain the ls command and its options.
Module - 3
5 a Explain how you can switch from one mode to another mode in vi editor.
h Illustrate how Input and output redirection u,orks in LNIX standard files.
10 marks
10 marks
l0 marks
10 marks
10 marks
10 marks
PTO
Llf',rlt:

8. Or
a List and explain shell wild cards of Ln{IX with examples.
b Explain the features of the pipe and tee command with suitable examples.
Module - 4
a Write a shell script to accept an option from the terminal and to display the following menus. 10 marks
MENU
1. List of files
2. Processes ofuser
3. TodaY's Date
4. Quit to LINIX
b Explain the shell features of while and for control statements with syntax'
Or
a Explain the following commands with suitable examples.
i. cut
ii. Paste
iii. sott
iv. umask
.,1 "
..::il._
.: i"
'.4 i!
) l'. -:,'
i 'r;
,,.., .i.. .'
i 1'1-"
10 marks
10 marks
10 marks
I0 marks
l0 marks
10 marks
l0 marks
10 marks
l0 marks
'it"r.
tl ;l'
'1
ii r,. ,
,1]!.1 i
1tr
'i!
!r_.J"!
!a'
,". i::. 1
;.. "'41:^.t,. , ,t
1! ni ''
'!, "a
,:,:r,..,1,
't,
ri:
l0a
Describe the significance of hard links and symbotic lintsrlt.uisik.
Module - !"',,,...
''*"
Explain the mechanism of process creation andidHqr.€i,en the details about ps command
wiih its options. ij:;,:i!
it".tl
, :;.
Explain string handling function in perl Wjfh,e*ailrnles'
t;:,'.
.ij
1.''"'tttt' 't"
Or
Explain the following perl script functiijns with suitable examples:
i. push ( ) and PoP t )
ii. sPlit Q'ahd'join 1;
Briefly explain the foJlo,wlrrg:'
i . C ro r1.,and'l.c."omtab
ii. Associative arraYs.
'{'.4" ,F"
ii '*, ,"
.- r'ri
_.rri,i. !"q, r!,
i.. .t1nr.
'"".1,*.*'
,,i.
a.
t !4";*
(
4,1
'.- "{r:,..
lj
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Third Semester B.E. Degree Exami44iio%, Jan./Feb .2021
a"rt' I -
Discrete Mathematic,aJ,.Structu res
Time: 3 hrs. -*.,, l'uTU
"
* )''l.$: lVlax' Marks: 8o
iL q.:"
':i il 'lr'
Note: Answer any FIVEfull questions,
"#qTft
ONEfull questionf{om each module.
'
!..
la.Definethefollowingt,,*,*itr,"rti.[nffiConjuncti9a,u}.i..ii;ruo,oto*y
iiD Quantifiers iv) Proposition,,
' v) Conditional or lrirblication
vi) Dualofstatement. 4%P- 'ru (06Marks)
b. Prove the validity of tne fotpwing arguments : ,*.1
*
i) p --+ r d* : ii) (-p v - q) -4. (i,X O
-p+e %# r-+t ii.'""
,4 *.+ &* **1, _ (06 Marks)
c. Find the p,gf1B, of each ofthe followiffiefuantified statements :
D vxrffi)t(x > y)
- ((x - v) , 9)J
-.
ii) "Wr,,]rfOtx,
y) n q(x, y))
;1S) v)J. ,_,,*$ (04 Marks)
-":"" ,,' "
a. prove that the foltowing 9q.{lp'6und rrorSrl,onr,uB,iffiotor,.r
' Fii!"'
i) [p^(p-*q)] --q-ffi* ii) {p-*1q--,ji1iL- {(p-,0 id..-,r)}. (05Marks)
b. Prove the following byuding laws of logic S',, *
'i) [(p-q)n(- g,,f(rv-9)].+-.,(q*p) *.'
ii) [pvqv(--&. -q^r)]<+(py-g9i). *u*'" (g5Marks)
c. Slrow that "1f.ffi$""6n odd integer ttlo#h,} I I is an even inte'!6r" by 0 Direct proof
ii) An indir.e._gt"$roof iii) Proofih$contradiction-g*;- . (06 Marks)
-.f*ff ,l',i ,.s" Module-2 "481"
.'. --rf J.,E .',/
a ** il$ *t (o6Marks)
c. Iinl lhegeifi1{tp,
of each.ofth: f"1tg_*r$.irantified statements :
.:;+l
...irir ."
"* :,Ea ,/k1;
a. Prof$$e following by
ffi@&{htical Inductionl :
,!: * ,i + 33 + . . ...* n''= ilg.J)l' .
''
t'
*:.'*b *lodl ) I
-*{a$ - "&-" L - .l ;
wJ,'i + 2'+ 3'+ .....* n'= l : I . *f*i"o (05 Marks)
*$ff ,.%flL 2 J /' -
e,i}$.i) How many a,$agrg&nents are possilJe for all the letters in the word SOCIOLOGICAL?
' ir.) l" how mantb.fithese arrangeme4ts A & G are adjacent?
iii) ln how riilnf"of these a..a".F#i6frts all the voweli are adjacent? t06 Marks)
c. Asequendeftfp?] is definedreifff,pfvelybyar = 4,&n= &n-r * n for n> 2. Finda, in explicit
fnrm 'r ! tnt tt
15CS36
(05 Marks)
4t;;'" oR
a. If Fi's are the FibonacrclYiumbers and Li's are the Lucas numbers, prove that
L*+ - Ln = 5 f"*: for all integers n > 0. (06 Marks)
b. A certain collgsbiifriestion paper contains 3 parts A, B and C with 4 questions in part A, 5
questions in pffiB & 6 questions in part C. lt is required to answer 7 questions by selecting
atleast 2 questibns from each part. In how many different ways can a student solve thi
questio#;ginPer?
':'
.s.
,*.'"{@
o6r -r4:
I of3
(06 Marks)

10. c. Find the coefficient of : " ft
'
i*-*;f;t i"iil;.purrio,of(x+ y + z)7. r'' d'$-'';')'"
iD l;3 * t' ,o in ihe expansion of (v r- w * x+ y #,#r#''
...- *u ''J
Module-3:!,, *
Let A: {a, b, c, d} and B: {2,4,5,7).Detet$-lfri he following:
;:
i) I A x B1. ..*.+,
+r'J ,,;!
ir) Number of relations from A to B. ffia, ;,,iliil*'
iii) Number of relations from A to B tl&#ntain (a, 4) and (c, 7).@it
iv) Number of relations from A to Bjhafbontain exactly six ordqledpairs.
15CS36
(04 Marks)
lsan
(05 Marks)
5a.
b.
v) Number of binary relations orAtffat contain atleast fog{g,f,3 ordered pairs. (06 Marks)
Let t, g, h be tunctions from, ffiff'6fre by f(x) = x2 , g14ffi + 5 and h(x) : ",1* ., .
Determine(ho(go0)(x)ai'ft(ft."e)o0(x).VerifythaJ"EnGo0=(hog)of(05Marks)
Let A = 11,2,3,4) and.,,,,., ..$.
N
R: {(1, 1), (1, 2) ,(?a);|,;p,2) ,(3,1), (3, 3) , (t}d)',, (4,
equivalence relation? Fj,fl-{-tne corresponding partiflon on A.
t o*.*o
6 a. Provethat if fL46.B, g: B -- C arep)1erffible funetions, the go f :A--+ C is invertible
and (g o 0 l = f' o g-'. "qqF (05 Marks)
),.!.J:.:.:.
b. LetA-:,9[1]?;3,4] andR= {(1, l)-(H2),(2,3),(3,4)}, S =J(3, l),4,4),(2,4),(1,4)
b" rqldhrjfi,y on A. Determine the.Eti:,lations R o S , S-o4$, * and Sz. Write down their
matri'&$i ,r,;; ; .*".;"-.,
' (05 Marks)
c. Consider the Hasse diagram of a POSET (A, R) gpen:i.n Fig. Q6(c).
i) Determine the relation matrix R ii) Constiubt ihe digraph for R*'
iii) Write maximal, minimal, greatest:rd].+fl.ments.
?
' (06Marks)
o u th^rif,i! .*1
- ;,.. /:: .... ;.. '
,^fu. ,2" "4
FiefQ6tc) #ffi )r i
*.:flF ._, ?p
u'"
Modol"_4 4,h:
7 a. Ho#ffiy integers betwffil?flnd 300 (inclusive) are r) divisible by atleast one of 5, 6, 8?
ii) divisible by none of 5, 6, 8? *.u. (0s Marks)
b. r$termine in how p3ny ways can the lg[1ffp in the work ARRANGEMENT be arranged so
f;:j":$atthere are exactly two pairs of consebitive identical letters. (06 Marks)
i1='Find the Rook polgomial for 3 x 3-"Qoriid by using the expansion formula. (05 Marks)
d' oR
i""'%
8 a. A person inffits Rs 100,000 al 12% interest compounded annually : i) Find the amount at
the end of l't , 2nd , 3'd yeq1.. iil Write the general explicit formula iii) How long will it
take to double the investment. (06 Marks)
Solve the recurrence rd lbn ?n+z-8ana1 * l6an = 8(5") + 6(4), where n > 0 and h: 12,
ot : 5. , L' (05 Marks)
I
An apple , a banatiL a mango and an orange are to be distributed to four boys 81, Bz, Br and
B+. The Boys*Ei,;dhd Bz do not wish to have apple, 83 does not want banana or mango and
Ba refuses o$hge. In how many ways the distribution can be made so that no boy is
displeasedJ s (05 Marks)
'.e.,,.,.1:'ul' 2 of 3
,
i,,,iii
;:iliu
b.
c.

11. 9a.
b.
c.
"
''"''''il
1scs36
Module-S .* W
Define the following : i) Complete Graph ii)r,g:irrlite Graph iiD Isolated Vertex
iv) Regular Graph v) Subgraph. fi".;q'.; u"'' (05 Marks)
Let G = (V, E) be simple graph of order lvl= ii'"anA size lEl
: m. If G is a bipartite graph,
prove that4m<n'. ."r , (05Marks)
bonstruct an optimal prefix code for the-@Ols & , o, P, t,Y, zthaLoginur with frequencies
iir) The degree of eveqryerte? in a cycle is two.
" d
Define Isomorphism.,!$;i the two graphs are Isffitlt
10 a. Prove the following :
i) A path with n vertices
ii) If a cycle has n verti
*ij;fe
vl
V3
Vr,
t/a
(06 Marks)
(04 Marks)
(06 Marks)
",(1.:2i.[
'
J,r#:8, ,.;..Yt*:"
'.,
Fie. Ql0(tj..ffi "Figspl0(b
(ii)) .
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3 of3