Sub.

Code: matics-III
Date: MAY 63350/73203
Dayand
Time:10:30
AM To ,03-05-2024
01:00 PM Total Marks: 70
1.
Instructions: Figures to the right indicate
full
Inst.: 1) marks
Special Attemptanythree
Attempt any three questions from question
questions from number 1 to 4
2)Use of question number 5
3)AssumeNon-Programmable caleulator is to 8
suitable data if allowed
necessary

o) Solve the following LD cquations

[12]
a. Solve (D²-6D + 5)y = sin2x
(6]

b.
Solve
(D + 10D2 + 9)y = sin2xcosx
[6]

Q2) Solve the following [11]

a. Find fromthe following values of the demand and the corresponding price of a (6]
commodity, the degree of correlalion between the demand and price by
computing Karl Pearson's caeficientof correlation.
Demand in quintals 65
66 67 67 68 69 70 72

Price in Paise per kg 67 68|65 68 72 72 69 71

[5]
b.
Fit a straight line
V = a+ bx
to the following data with as independent

variable.
b)
8.0 9.0 10.0
6.0 7.0
0.5 0.0
1.5 1.0
2
[11)

Q3) Solve the following

[5]
dt va
a. of lo
Laplace transform
Find the [6]
5s-15s - 11

b.
of
(s + 1)\s - 2)°
transform
Find the Inverse Laplace [12]

the following P.T.0.

04) Solve any two from
 QP-1
a.
Solve -3r+5y = sin (logx) [6]

b
Fit a curve of the form }= Co to the following data using the method of least l6]
square.
2 3 4 5 6
Y 34.385 79.0855 181.90 418.36 | 962.23

Use Laplace transform to solve

3 +2y = e3t where )'=1 at t= 0 [6]

Q5) Solve the following.

[11]

a.
Findthe directional derivative of =xy“ + yz at (2,-1,1) in the 15]

direction 2i+j+3k
b.
b)Ifa is a

0,iü) curl ( x
constant

)2ä
vector and xi + ý + zk, Prove that L}Va.) i)V.( x ) [6]

Q6) Solve the following.

[12]
a.
X 0<xS1 [71
f(x) =
Obtain fourier series for (1 -xl<S2
b.
(1, <x<1 0 [5]
f)=} X,
Obtain half range cosine series for 1<xS2
Q7) Solve any one from the following.
[11]
a.
 QP-172
Solve the Laplace's
iterations
cquation +=0 dy² for the follosing datz
up to tuo [11]
4 16

14

12

0.5 2.0

b. A string is stretched and

fastened to two poirnts distance I apart. Motion is [11]
started by displacing the
string in the form = Kxi- X)where k
constant is relcased from
rest .
find yl X, t )the vertical
displacement
is

if

Q8) Solve any two from the following.

[12]
a. Show that the vector
= (r2+yz)i +(y²
F
+ xz)j+ (z²+ xy)k is
[6]
Hence find scalar potential
irrotational.

b.
Obtainfourier series forf(x) =x,0 <*< 2n [6]
C.

Solve the partial

differential cquation
=2+z ôy by the method
[6]
separation of variables of
 S.Y. B.Tech.(Mechanical Engg.) (Semester- Srn)(CBCS
Examination, January-2024
ENGINEERING MATHEMATICS-M
834 Sub. Code:73203

Day and Date:Tuesday, 02 - 01 -2024 ToratVares

Time 10.30a.m. to 1.00p.m.

Instructions 1) Attempt any threequestions from each seation

:
2) Figuresto the right indicate full marks.
3) Use ofnon-programmable calculator is allowed

4) Assume suitable data if necassary.

SECTION
Q1) Solve following LD equations.

a) (D² – 6D+5) y =
sin 2
b) (D'-1)y =(re+ 2)

Q2) Solve the following.

(cm) and yield of witear(un
a) Following are the results of Rainfall
of I an
of different states. Find regression equation
6 6.0
X 3.6 4.5 5.8 2.8 4.7

55 13
Y 38 47 67 30

b) Fita straightline y =a+ bx to the following data

70
|6.0
2 1.5
8.0

1.0
9.010.0

0.5 0.0
108
PIO
 SP-11

Q3) Solvethe following. (5]

a) Evaluate Jo |e sin2t dt. [6]

transform of
b) Find inverseLaplace

ts+4s +44
44)(s-s-2)

from the following.

Q4) Attempt any Two
[6]

r-x+4y = dy cos(logx).
a) Solve d? dx
Fit the exponential
curve y =ab'to the following data
b) 5
2 3
[6]
2.2974 2.4623| 2.2974
2.0 2.1435
assuming
Laplace transform
LD equation using
c) Solve following

=2 & y' (0)=3

dy 3.-+2y=2e
dt
[6]

y(0)

SECTION - II

questions.
Q5) Attempt thefollowing
hence
irrotationaland
then show that curlVF)
is

a) If F=+y'+2? [6]
¢such that Vo=F
find the scalar potential function

Show that the surfaces

x +y +z=12and +y-26areorthogonal
[5]
b)
4).
tthe point (1,1,
 SP-11
Q0) Attemptthe following questions.

a) Obtain the Fourier series for the function f() =,

2r-x,
b) Find Pouner series tor fr) =sin ar, -t <I<I. (5]

Q7) Atteupt anyone trom the following.

Solvethe Laplace equation u, + =0

for the following square mesh
with boundary values as shovwn is figure by Gauss-Siedal interative method
by pertorming two iterations. [11]

S.7 12.1 12.5

17
1329 = ia
21
Us

21.9

13.5
11.1 17 19.3

b) A tightlystretched string with fixed end points x=0 andx=lis initially

at rest in its equilibrium position. If it isby giving to each of
set vibrating
its points avelocity r (l-x). Find thedisplacement ofthestring at any
distancex from one end at any time t.
[11]

Q8) Attempt any two from the following.

a)
f7wi+y+ and r then prove that f(r) -f"(n)+-fn).

b) [6]
Obtain the half range cosine
series offx) =xin the interval (0, 2). [6]
c) Using method of separation of variables, Cu
solve 2 +u with
u(x, 0) = 6e-sr,
[6]
 Total Marks: 70
Day an4 1Wae,1EEMBER 1722124
nica
Instruhisns
Sup
iG
Co

4)4hend aryhres MS fron 31 oQ4 Anc Aend anythree auestions

[12]

[11]

[6]

as independent [5]
Fit a stracit lire72 b2 tothe following data with

1965 15 1351

45 2155 222

[11]
03) Shvethefollowngs
[6]
of 'o
Findthe Laplace trarsfom

[5]
b. Lapace transiorm of s-s-8s
Find the Inverse

[12]
Q4) Sohve ary Two
[6]

Sohve
-12y = ziogr

P.T.0.
 QP-258
[6]
b.
Fit a curve of the form y= ar°to the following data
X1 2 4 5 6
y120 90 60 20 11 5
6]
C. y (0)=1
+44 dt + 8y=1where y(0) = 0and
Use Laplacetransform to solve de²

Q5) Solve the following [11)

questions

a.

nd the directional
normal to the surface x
derivativa

logz +4=
f 0 xy2 +yz3
y² at (-1,2 ,1)
at (2,-1,1)in the direction l

b. [5]

Ifaisaconstant vector andr=xi+v+ zk, Provethat

).(a.F)= a? ,i)V. x )= 0

Q6) Solve the following

questions [12]

a.
(l-x,0<xsi [
Obtain fourier series for f(x)

0,1-21
b.

x,0<x^ [5)
Obtain half range sine series for
f(x) =

Q7) Solve any One

[11]
a. ove the lapiace's equation
for the folowing data us to No
[11]
87 12:1 I2:8 9.0

17.00
.0
L9 2):9
3
186
2 1?.o 19.7
b. A string is
stretched and fastened to two points distance
apart. Motion is [11]
started byy displacing the stringin the form Y = kx (l-x)
where k is constant is

released from rest.find y( x,

t)the vertical displacement
if a::
 Seat|
No. 0r

Solve any Two

QP-2589
S)
[12]

a. Show that the vector ( cosx+ z)t + (2ysin x- 4)3rg+2)k

is [6]
irotational. Hence find scalar potential

b. Obtainfourier seriesfor
*,-KI [6]
C.
Solve the partial dìferential equation [6]
by the method of separation
of variables

End Of Question Paper

ortantNote for Chief Exam Officer /SRPD
Coordinator /Sr Supervisor/ Student
QuestionPaper may be distributed for
following Subjects as common code.

154) B.Tech.CBCS (73203) Engineering

Mathematics-|| Part 2 SEM 3
01)Bachelor of Engineering (63350) Engineering Mathematics-IlIl Part 2 SEM 3
D1) Bachelor Engineering Part 2 SEM 3
of (77734) Engineering Mathematics-IlI

673
 No.
Total No. of Pages:3
SHIVAJI UNIVERSITY KOLHAPUR
S. Y.B.Tech. (Mechanical Engg) (Semester-II) (CBCS)

8 March -2023 (Supplement Examination)

ENGINEERING MATHEMATICS-III
Sub. Code:73203

Day and Date :Thursday, 05-10-2023 Total Marks :70

Time : 10.30 a.m. to 1.00 p.m.

Instructions : 1) Attempt any three questions from each section.

2) Figures to the right indicate full marks.

3)Use of Non-Programmable calculator is allowed.

4) Assume suitable data if necessary.

SECTION-A

Q.1) Solve following LD equations.

a) (D + 10D2 +9) y =sin 2x cos x [6]

b) (D'+1)y = x
[6]

Q.2) Solve the following.

a) Followings are the results of No. of workers (X) and production

(Y
and write conclusion, [6]
Calculate coefficient of correlation

55 58 58 57 56 60 54 59 57
X 46

158|167 167|160165168 |165 166 168 166

Y

b) Fit a straight line y =a+ bx to the following data.

X 2.6 2.7 |2.8 | 2.9 3.0

y 0.4 0.3 0.20.I|0.0

Q.3) Solve the following.

a) Evaluate

e sin'tdt
 of
inverse Laplace transform
b) Find
s+1
s³+4s²+8s
the following.
Attempt any two from
Q.4)
a) Solve
dy
2'y + 2x-2y = x. sin(log r)

Fit the exponential

curve y =ab tothe followingdata
b)

X 1 2 33 4
27 81 243
y 3

LD equation using Laplace transform

c) Solve following

assuming y(0) =0& y'(0) =0

-=-8t
di dt

SECTION-II

the following questions

Q.5) Attempt 6,
the surfaces
a) Ifthe angle between
y+l=z (0,,2) is
r+arz +byz =2 and z2+y
+ at

the constants a and b.

then find

b) If =+y +2-3zyz, find grad .and show tht gradeis

irrotational.

Q.6) Attempt the following questions

f(z)=- as Fourierseries in the interval

7
a
a) Expand the function

-T<I<n.
 5) Find the half rangecosine series for f(z)= (r-1)? in theinterval

Q.) Atempt any one from the following

[11]
a)Solve the Laplace equation u, +u
=0 forthe following square mesh with
boundaN llvalues as shown in the following

2
figureby Gauss-Seidel method.
Carout three iterationsby assuming u, 0. =
1000 1000
1000
000

2000 500

2000

1000 500

b) The vibration of an elastic string is governed by the partial differential

equation The length of the string is and the ends are fixed.The
initial velocity is zero andthe initial deflectionis y (x,0)=2sinx +4 sin 3x.
Find the deflection y (x, t) of the string for t0
Q.8) Attempt any two from the following

a)Ifi -0i+0ai+ ak is constant vector and F- zi+j+ then prove that

i)div(Fx) =0and ii) curl (ãx)=2@ [6]

0sr<1
b) Obtain the Fourier series for the function fl) =
l12-srsa 16]

c) Using method of separation of variables solve the partial

[6]

differentialequation 4 with u(0.u) =8e-y

 SB - 256
No. Pages :3
2365
Total of
Seat

Na.
S.Y. B.Tech. (Mechanical Engineering) (Semester - ID(CBCS)
Examination, January - 2023
ENGINEERING MATHEMATICS - II
Sub.Code :73203

Day and Date :Friday, 20 - 01 - 2023 Total Marks:70

Time : 10.30 a.m.to 1.00 p.m.

Instructions: 1) Attempt any three questions from each section.

2) Figures to the right indicates full marks.

3) Use of non-programmable calculator is allowed.

4) Assume suitabledata if necessary.

SECTION -I

Q1)Solve the following:

d'y -2y=e'+ cos x,

Solve
a)
d d? dx

b) Solve (D'-2D'+D)y=x tx.

Q2) Solvethe following:

coefficient of
of regression and the
a) Find equationsto the
the lines

data:
correlation forthefollowing

4 5 7 8 10
1 2

y 10 | 12 16|28 25 36 41 49 40 50

b Fit a curve of the form y=a to the following data:

4 6 10
X

30.128 81.897 222.62

4.077 11.084

PIO.
 SB -256 bodacy

Q3)Solve the following:

sin 3t
5]
e

a) Find

b) Using convolutiontheorem
find inverse rm
Laplace transform ofof (+a)6

Q4) Attempt any two from the following:

[6]
a Solve x?. 9-3y=(xlog x)' .
dx

second degree paraboliccurve to the following data:

b) Fit a

22.5 33.5 4

c)
y
1

1.1| 1.3

Using Laplace
1.5

1.6 2 2.73.4 4.1

E E465+Csr
transforms,find the solution"of initial value
asx+bsx+S3
problem

(+2)(->()-2> (x) 0.y() Y() 0 and y"(0) 6-[6]

SECTION- II

Q5) Attempt the following questions:

a Prove that (y'-+3yz-2x)i +(3xz+2ry))j +(3.ary- 2xzt+ 2z)k is

solenoidaland irrotational. [6]

b) Find the directional derivative of 6 =x'y +y'z+z'x at the point P(1,2, 1) in

the direction of the nomal to the surface+y-zx=l at Q(1, 1, 1).51

06)Attempt the following questions:

Obtain the Fourier series

a)
sto represent f(:)=(7-x.0<x<2r
4 (71

b) Express / (x)-!<x<l as Fourier

-2
 SB -256
the following:
07) Attempt any one from
Solve the Laplace equation ,,+ y =0 for the following square mesh
with boundary values as shown in the figure by Gauss-Seidel method.
(Carry out two iterations). [11]

14

12
us

10
U2 U3

b) Solve the
conditions:
differential equation
0.5

-
2.0 4.5 8

Subject to the following

[11]

u is finite for all t.

u=0 for = 0, n for

x all t.

i u=NX-x for t=0and between x = 0 and x = n.

Q8) Attempt any two from the following:

a) If7xi + y +zk and r +y+zl, prove that V

-1
Obtain the Fourier series of f(x)= x in the interval -ISxSI and
b)

1 1 1 1
(6]
hence show that
723' +....:
6

c) Solve the equation dx? t0 + forthe following data by Gauss-Seidel

[6]
method. Carry out two iterations.
1000 1000
1000 1000

2000 500
U1

2000

1000 500
 SB -256
Seat Total No. of Pages :3
No. 2973
S.Y. B.Tech. (Mechanical Engineering)(Semester- III) (CBCS)
Examination, January - 2023
ENGINEERING MATHEMATICS - III
Sub. Code:73203

Day and Date :Friday, 20 - 01-2023 Total Marks :70

Time: 10.30 a.m. to 1.00p.m.

Instructions : 1) Attempt any three questions from each section.
2) Figures to the right indicates full marks.

3) calculator is allowed.
Useof non-programmable
4) Assume suitable data if necessary.

SECTION - I

Q1) Solve the following:

[6]
Solve d'y_d+4-2ye +cos.
a) dx' d? dx
[6]
b) Solve (D'-2D°+ D)y =x* +x.

02) Solve the following:

and the coefficient of
to the lines of regression
a Find the equations [6]
data:
correlationfor the following
7
3 4 6 8 9 10
1 2
25 36 49 40 50
10 12 16 28 41

[5)
the form y=ar to the following data:
b) Fit a curve of

6 10
X 2
81.897 222.62
11.084 30.128
4.077

PT.0.
 SB -256

thefollowing: 151
Q3)Solve
sin'3

a) Find

inverse
Laplacetransformof
(+6]
theoremfind
Using convolution
b)

the following:
any two from [6]
Q4)Attempt
(xlog x)*.
a) Solve r-x-3y
dx
data:
[6]
following
curve to the
parabolic
a second degree
b) Fit
33 13.5 T4
x1|
X 15| 2
1.6
|2.5|

22.7 3.4 4.1

value problem
y1.1| 1.3
of initial
find the solution
transforms,
0and y(0) 6.[6]
0.y() y()
Using Laplace
c)
2y()
()+2y"(x)-Y(x)-
SECTION - II

following questions:
Q5) Attempt the +(3y -2xz +2z)k
is

+(3xz +2.1xy)j
a) Prove that (y'-z+3yz-2x) [6]
irrotational.
solenoidaland 1) in
the point P(1,2,
derivative of o=x*y+ y'z+e'x at

Find the directional

x² +
b) Q(1, 1, 1).(51
the surface
-'x=l
y² at

the direction ofthe normal to

following questions:
Q6) Attempt the
seriesto represent
f(*)=(-x) 0<x<2r.(7])
a)

b)
Obtain the Fourier

Express f ()-,-<x<l
as Fourier series
. (5]

-2
 SB-256
07) Attempt any one from the following:
Solve the Laplace equation
a) tuyy =0
for the following square
mesh
with boundary values as
shown in the figure by Gauss-Seidel
method.
(Carry outtwo iterations).
[11]
16

14

12
U6

10
u1 U

0.5 2.0 4.5

b) Solve the differential equation

dt a2 Subject to the following

conditions: [11]
) uis finite forallt.

u=0for x=0, for all t.

iü) u=IX-x fort=0and between x =0 and x = n.

Q8) Attempt any two from the following:

a) If f=xi+yj +ak and r=/ +y'+2', prove that V r=10 3

b) Obtain the Fourier series of f(x)=x in the interval -<x<I and

hence shou that

1

+'46+
1 1 1
[6]

c)
C) Solve the equation t=0 for the following data by Gauss-Seidel
[6]
method. Carry out two iterations.
1000 1000
1000
1000

500
2000

2000

1000 S00
 sin'

tdt
-
Evaluate
a)

following.
the
Solve
Q.3)

0.0
0.1
0.2
y0.40.3

2.62.7
3.0
X

2.9
following
2.8
straight
data.
the
bx
line

to
=at
Fit

y
b)

165l66
18166
l65
167

168
l60
167|
Y|158

57
54
57
58
58
55
46
X

concusion.
corelation
coefficient
Calculate
and

write
of

production
workers
results
Followings
and
No.
of

ot
the
are

()
a)

following.
the
Solve
Q.2)

(
I)y

x
(D
t
=

2
sin

N
cos
+9)
10D
(D+

y
b) a)

equations.
following
LD
Solve
Q.1)

SECTIÒN-A
necessary.
suitable
Assume
if
data
4)

allowed. caleulator
Non-Programmable
of
Use

is

nmarks. indicate
Figures
tull
the

right
to

section.
questions Attempt
Instructions
each
from
any

three
3) 2) 1)

10.30 Time:
to

p.m.
1.00
a.t.

Mlarks Total
05-10-2023
:Thursdsy,
70
and
Day

Date
:

73203
Code:
Sub.

MATHEMATICS-II
ENGINEERING
Examination)
(Supplement
281

-2023
March

(CBCS)
(Semester-l) (Mechanieal B.Tech.
Engg)
Y.
S.

KOLHAPUR UNIVERSITY SHIVAJI

Pages Total
of
No,

3
No.

296

289
Sest

SE
-
: :
 of
b) Find inverse Laplace transform

s+1
s3 + 4s2+8s
the
Q4) Attempt any two from following.

a) Soe

28 b)
rt2r-2y
Fit the exponential curve y
= x. sin(log r)

=ab' to the followingdata

X 1 2 3 4 5
y 3 9 27 81 243

c) Solve following LD equationusing Laplace transform

assuming y(0) =0& y'(0) =0

SECTION-II
Q.5) Attempt the followingquestions

a) If the angle between the surfaces

n+azz + byz =2and '+y+y+l=z at (0,1,2) is

conthen find the constants

a andb.
b) If
=g+y+2-3ryz,
irrotational.
find grado . and shw
81
tht grade is

Q.6) Attempt the following

questions

a) Expand the function f(z) =r-as a Fourier series in the interval

 b) Find the half rangecosine series for flz) (r- 1 in the interval

0<TS1.
Q.7) Attempt any one from the following [111

a)Solve Laplace cquation u,, t u

the -0
forthe following squzre mesh with
boundary ll values as shown in the following figure by Gauss-Seidel mehod

Carry out three iterations by assurning u, 0.

1000 1000
1000 00

200 500

2000

1000 500

b) The vibration of an elastic string is governed by the partial differential

equation The lenigth ofthe string is and the ends are fixed. The

initial velocity is zero and the initial deflectionis y (x,0)-2sinx +4 sin 3x.
Find the deflectiony (x, t) of the string for t>0
Q.8) Attempt any two frem the following

a Ifimeji+ 0aj +ak is constant vector and +hen prove that

i) div(Fx ) =0 and i) curl (ã x )=2 [6]

b) Obtainthe Fourier seriesfor

the function
))(2-2srs2 [6]

c) Using method of separation of variables solve the partial [6]

differentialcquation 4 a with u(0, y) = 8e

 06)
b) b) a)
B) 0
Derive

+4 the
Clasify

rAy].
=u
u(0.

y)
0:
Solve

+u
=
u(4.
dxoy
y)
+4
Prepare

8 =0

following
the
in

Crank-Nicholson
+2y:
partial
mesh

formula
dxdr
OR and
for
Oy 0<x<4

differential
y(x,0)
+2=0

perform
and
Parabolic
FOUR
y
x/2:

equations:
(1+*(5+2r)(4+x)-0
(x,
4)
equations.
0sys4

=x
10703
iterations.
given

[4] 14] [8] with that

SP-44)
 tha

D) b) 05)
b) b)
a)
(n The
d 2) gven
Also
point.
Use
Find
Fínd
Find The by

Given
find
the its tablc vertical

the Power
g+y yV43.1

d
Corresponding
Runge-Kutia
value below

Trapezoidal
other
+y+ distance
477

!
ofy

approxinate
Evaluteby
Eigen method rulc

acucleration
to Given,y
(0. method. at revcals to
Eigen y'=0 valuc 12
vector
(0) 1) 52.1
the using

=1 by 1, find x0009 covercd

for
find and ofy
I by
value.

the
OR Taylor's
y(0)
- az

).1
564

60.?
1.3

14
scc.
velocity

of
OR

Simpon's
thc

140000

rocket

in

|140000-21000
dominant
distance
3/%°
body
series
one
t
from
following
rule.

step during overed %

(Larger)
matrix
method
by the

Eigen t
from
tot

corret

to
one
value
using tirne

'1 2 3)
Fourtn tt
and

the
15] order 16 speified.
30 i
nd
SE-07
decirmal
 -4
cquations.
Parabolic formula Crank-Nicholson
for
b)

14|
Derive

OR
ÔxÔy

+4y)=sin
-+4

+(x*

y)
(x
8'u

+ equation: following classify

classification
Partial Second
Equations Differcntial
Give
b)

of

and
Order
)

10921
Numerical
calculations
iterations
Perform

rough

temperatures
method.

and

T'wo

different
intermediate
only.
Find

the

wall
of

at

mesh

using
points
+

-=0

conduction
heat
of

which

conduction
heat

Equation
trans

takes

place
fer

maintained
is due

different
to

temperatures.
tlhe
per

dillerence
temperatureg
Laplace
Due
as to

in

surfaces
wall
are The
a)

at made

following.
is

up

bricks
of

as

shown
in
Q6)

Solve

The

wall
the

figure.

of
SE-07