Engineering
Mathematics III
Question Paper Set
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Model Question Paper (CBCS) with effect from 2016-17
15MAT31
USN
Third Semester B.E.(CBCS) Examination
Engineering Mathematics-III
(Common to all Branches)
Time: 3 Hrs
Max.Marks: 80
Note: Answer any FIVE full questions, choosing at least ONE question from each module
Module-I
  , in    x  0
1. (a) Find the Fourier series expansion of f x , if f x   
 x, in 0  x   .
2
1
1
1

Hence deduce that 2  2  2  ... 
(08 Marks)
.
8
1
3
5
(b) A periodic function f x  of period ‘6’ is specified by the following table over the interval 0,6 :
x 0
1
2
3
4
5
6
9
18
24
28
26
20
9
f x 
Obtain the Fourier series of f x  up to second harmonics.
(08 Marks)
OR
2. (a) Expand the function f x     x  2 as a Fourier series in the interval 0  x  2 .
2
1
1
1
2



...

.
6
12 2 2 3 2
(b) Obtain the Fourier series of f x   x valid in the interval  l ,l .
(05 Marks)
(c) Find the half-range sine series of f x   x  12 the interval 0  x  1.
(05 Marks)
Hence deduce that
(06 Marks)
Module-II
1, for x  1
3. (a) If f x   
, find the infinite Fourier transform of f(x) and
0 for x  1

hence evaluate
sin x
dx
x
0

(b) Find the Fourier sine transform of e
(06 Marks)
x

. Hence show that
(c) Find the Z-transform of (i) cos n & (ii ) sin n
x sin mx
e  m
dx

, m.  0.
0 1  x 2
2
(05 Marks)
(05 Marks)
OR
Page 1 of 3
15MAT31
4. (a) Using Z-transform, solve y n 2  4 y n  0, given that y0  0, y1  2
a  0
2z  14z  1
(b) Find the complex Fourier transform of e  a
(c) Obtain the inverse Z-transform of 18z
(06 Marks)
2
2 2
x
(05 Marks)
(05 Marks)
Module-III
5. (a) Calculate the Karl Pearson’s coefficient of correlation for 10 students who have obtained the
following percentage of marks in Mathematics and Electronics:
Roll No.
Marks in Mathematics
Marks in Electronics
1
78
84
2
36
51
3
98
91
4
25
60
5
75
68
6
82
62
7
90
86
8
62
58
9
65
53
10
39
47
(b) Fit a best fitting parabola y  ax 2  bx  c for the following data:
x
y
1
10
2
12
3
13
4
16
(06 Marks)
(05 Marks)
5
19
(c) Using regula-falsi method compute the real root of the equation xe x  2 ,
correct to three decimal places.
(05 Marks)
OR
6. (a) Fit a curve of the form y  ae bx to the following data:
x
y
77
2.4
100
3.4
185
7.0
239
11.1
(06 Marks)
285
19.6
2
(b) If  is the acute angle between the lines of regression, then show that tan    x y  1  r  .
2
2 

x  y 
r

Explain the significance of tan  when r  0 & r  1.
(c) Find the real root of the equation x sin x  cos x  0 near x   correct four decimal places,
using Newton- Raphson method. Carryout three iterations
(05 Marks)
(05 Marks)
Module-IV
7.
(a) From the data given below, find the number of students who obtained (i) less than 45 marks and,
(ii) between 40 & 45 marks:
(06 Marks)
Marks
No .of Students
30-40
31
40-50
42
50-60
51
60-70
35
70-80
31
(b) Using Newton’s general interpolation formula, fit an interpolating polynomial for the following data:
(05 Marks)
x
f(x)
-4
1245
-1
33
0
5
2
9
5
1335
Page 2 of 3
15MAT31
(c) Using Simpson’s 1 3rd rule, evaluate
1
dx
1 x
2
, taking h  1 6.
(05 Marks)
0
OR
8. (a) From the following table, which gives the distance y (in nautical miles) of the visible horizon for the given
given heights x (in feet) above the earth’ surface, find the value of y at x= 410:
(06 Marks)
x
y
100
10.63
150
13.03
200
15.04
250
16.81
300
18.42
350
19.90
400
21.27
(b) Use Lagrange’s interpolation formula to find f 4 ,given:
x
f(x)
0
-4
2
2
3
14
6
158
 2
(c) Use Weddle’s rule to evaluate
(05 Marks)
cos  d , dividing 0,  2 into six equal parts.

(05 Marks)
0
Module-V
9. (a) Derive Euler’s equation in the standard form viz.,
 y
1
(b) Find the curve on which the functional
2
f
d  f 
    0.
y dx  y  
(06 Marks)

 x 2 y  dx with y0  0, y1  1 ,
0
can be extermized.



(c) If F  3xyi  y 2 j , evaluate
from 0,0 to 1,2 .
(05 Marks)
 
2
 F  dr where C is the arc of parabola y  2x
C
(05 Marks)
OR
10. (a) Verify Green’s theorem in the plane for
 xy  y dx  x dy where C is the closed curve bounded
2
2
c
by y  x & y  x .
(b) Using Stoke’s theorem, evaluate
2
 






  F  nˆdS where F  3 yi  xzj  yz 2 k and S is the surface of
(06 Marks)
s
the paraboloid 2 z  x 2  y 2 bounded by z  2 .
(c) Prove that geodesics on a plane are straight lines.
(05 Marks)
(05 Marks)
*****
Page 3 of 3
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELGAUM
SYLLABUS FOR 2015 -2019
ENGINEERING MATHEMATICS-III
(Common to all Branches)
Course Title: Engineering Mathematics - III
Credits: 04
Contact Hours/Week : 04
Exam. Marks : 80
Exam. Hours : 03
Course Code : 15MAT31
L-T-P : 4-0-0
Total Hours: 50
IA Marks : 20
Course Objectives:
The objectives of this course is to introduce students to the mostly used analytical and
numerical methods in the different engineering fields by making them to learn Fourier series, Fourier
transforms and Z-transforms, statistical methods, numerical methods to solve algebraic and
transcendental equations, vector integration and calculus of variations.
MODULE
MODULE-I
Fourier Series: Periodic functions, Dirichlet’s condition, Fourier Series of
periodic functions with period 2π and with arbitrary period 2c. Fourier series of
even and odd functions. Half range Fourier Series, practical harmonic
analysis-Illustrative examples from engineering field.
MODULE-II
Fourier Transforms: Infinite Fourier transforms, Fourier sine and cosine
transforms. Inverse Fourier transform.
Z-transform: Difference equations, basic definition, z-transform-definition,
Standard z-transforms, Damping rule, Shifting rule, Initial value and final value
theorems (without proof) and problems, Inverse z-transform. Applications of ztransforms to solve difference equations.
MODULE- III
Statistical Methods: Review of measures of central tendency and dispersion.
Correlation-Karl Pearson’s coefficient of correlation-problems. Regression
analysis- lines of regression (without proof) –problems
Curve Fitting: Curve fitting by the method of least squares- fitting of the curves
of the form, y = ax + b, y = ax2 + bx + c and y = aebx.
Numerical Methods: Numerical solution of algebraic and transcendental
equations by Regula- Falsi Method and Newton-Raphson method.
MODULE IV
Finite differences: Forward and backward differences, Newton’s forward
and backward interpolation formulae. Divided differences- Newton’s
divided difference formula. Lagrange’s interpolation formula and inverse
interpolation formula (all formulae without proof)-Problems.
Numerical integration: : Simpson’s (1/3)th and (3/8)th rules, Weddle’s rule
(without proof ) –Problems.
MODULE-V
Vector integration:
Line integrals-definition and problems, surface and volume integralsdefinition, Green’s theorem in a plane, Stokes and Gauss-divergence
theorem(without proof) and problems.
Calculus of Variations: Variation of function and Functional, variational
problems. Euler’s equation, Geodesics, hanging chain, problems.
RBT
Levels
L1, L2
&
L4
No.
of Hrs
10
L2, L3
&
L4
10
L3
10
L3
10
L3 & L4
10
L2 & L4
Page 1 of 1
CBCS STUDENTS ARE ADVISED TO REFER THEIR
SYLLABUS
COMPARE & STUDY
THE FOLLOWING QUESTIONS PAPERS ARE FROM 2010
SCHEME
10MAT31
USN
Important Note : 1. On completing your answers, compulsorily draw diagonal cross lines on the remaining blank pages.
2. Any revealing of identification, appeal to evaluator and /or equations written eg, 42+8 = 50, will be treated as malpractice.
Third Semester B.E. Degree Examination, June 2012
Engineering Mathematics – III
Time: 3 hrs.
Max. Marks:100
Note: Answer any FIVE full questions choosing atleast two from each part.
PART – A
1 a. Obtain the Fourier series for the function
 2x
1   ,    x  0
1 1 1
2
and deduce 2  2  2  ........ 
.
(07 Marks)
f (x )  
2x
1 3 5
8
1 
, 0x


b. Find the half range cosine series for the function f(x) = (x – 1)2 in 0 < x < 1
(06 Marks)
Obtain
the
constant
term
and
the
coefficient
of
the
first
sine
and
cosine
terms
in
the
Fourier
c.
expansion of y as given below.
(07 Marks)
x 0 1 2 3 4 5
y 9 18 24 28 26 20
2
a. Express the function
1, | x | 1
as a Fourier integral and hence evaluate
f (x )  
0, | x | 1
b. Find the sine and cosine transform of f(x) = e– ax , a > 0
e as
c. Find the inverse Fourier sine transform of
.
s

sin x
dx .
x
0

(07 Marks)
(06 Marks)
(07 Marks)
3
a. A tightly stretched string with fixed end points x = 0 and x = l is initially at rest in its
equilibrium position. If it is vibrating giving to each of its points a velocity x(l - x), find the
displacement of the string at any distance x from one end and at any time t.
(07 Marks)
b. Find the temperature in a thin metal bar of length 1 where both the ends ate insulated and the
initial temperature in bar is sin x.
(07 Marks)
2
2
c. Find the solution of Laplace equation,  u   u  0 , by the method of separation of
x 2 y 2
variables.
(06 Marks)
4
a. Fit a parabola y = a + bx + cx2 to the following data:
(07 Marks)
x -3
-2
-1
0
1
2
3
y 4.63 2.11 0.67 0.09 0.63 2.15 4.58
b. A fertilizer company produces two products Naphtha and Urea. The company gets a profit
of Rs.50 per unit product of naphtha and Rs.60 per unit product of urea. The time
requirements for each product and total time available in each plant are as follows:
Plant
A
B
Hours required Available hours
Naphtha Urea
2
3
1500
3
2
1500
The demand for product is limited to 400 units. Formulate the LPP and solve it graphically.
(06 Marks)
c. Solve the following using Simplex method:
Maximize Z = x1 + 4x2
Subject to constraints – x1 + 2x2  6 ; 5x1 + 4x2  40 ;
1 of 2
x j  0.
(07 Marks)
10MAT31
5
6
PART – B
a. Use Regula-falsi method to find a root of the equation 2x – log10x = 7 which lies between
3.5 and 4.
(06 Marks)
b. Solve by relaxation method.
10x – 2y – 2z = 6 ;
– x + 10y – 2z = 7 ;
– x – y + 10z = 8
(07 Marks)
c. Use the power method to find the dominant eigenvalue and the corresponding eigenvector of
 2 1 0 
the matrix A   1 2  1 with the initial eigenvector as [1 1 1]T.
(07 Marks)


 0  1 2 
a. The following data is on melting point of an alloy of lead and zinc where t is the temperature
in Celsius and P is the percentage of lead in the alloy, tabulated for P = 40(10)90
(i.e., P from 40 to 90 at intervals of 10). Find the melting point of the alloy containing 86%
of lead.
P 40 50 60 70 80 90
t 180 204 226 250 276 304
(07 Marks)
b. Using Lagrange’s formula, find the interpolation polynomial that approximates to the
functions described by the following table:
x
0 1 2
5
f(x) 2 3 12 147
and hence find f(3).
(07 Marks)
5
rd
1
dx
c. Evaluate 
, by using Simpson’s
rule, taking 10 equal parts. Hence find log 5.
3
4x  5
0
(06 Marks)
7
a. Solve the partial differential equation
2 U  2U
 2  10 x 2  y 2  10
2
x
y
over the square with side x = 0, y = 0, x = 3, y = 3 with u0 on the boundary and mesh length
h = 1.
(07 Marks)
2
U  U
, subject to the conditions

b. Solve the heat equation
t x 2
 2 x for 0  x  1 / 2
U(0, t) = u(1, t) = 0 and u ( x ,0)  
2(1  x ) for 1 / 2  x  1
Taking h = 1/4 and according to Bender Schmidt equation.
(06 Marks)
c. Evaluate the pivotal values of the equation u tt = 16 uxx taking h = 1 upto t = 1.25. The
boundary conditions are u(0, t) = u(5, t) = 0, u t(x, 0) = 0 and u(x, 0) = x2(5 – x). (07 Marks)

8

2
a. If U (z)  2z  5z 4 14 , evaluate u 2 and u3.
( z  1)
1
.
(n  1)!
c. Solve the yn+2 + 6yn+1 + 9yn = 2 n with y0 = y1 = 0 using Z-transforms.
b. Find the Z-transform of i) sin(3n + 5)
ii)
*****
2 of 2
(06 Marks)
(07 Marks)
(07 Marks)
Course Outcomes: On completion of this course, students are able to:
1. Know the use of periodic signals and Fourier series to analyze circuits and system
communications.
2. Explain the general linear system theory for continuous-time signals and digital signal
processing using the Fourier Transform and z-transform.
3. Employ appropriate numerical methods to solve algebraic and transcendental equations.
4. Αpply Green's Theorem, Divergence Theorem and Stokes' theorem in various applications in
the field of electro-magnetic and gravitational fields and fluid flow problems.
5. Determine the extremals of functionals and solve the simple problems of the calculus of
variations.
Question paper pattern:
• The question paper will have ten full questions carrying equal marks.
•
Each full question consisting of 16 marks.
•
There will be two full questions (with a maximum of four sub questions)
from each module.
•
Each full question will have sub question covering all the topics under a module.
•
The students will have to answer five full questions, selecting one full question from
each module.
Graduate Attributes (as per NBA)
1. Engineering Knowledge
2. Problem Analysis
3. Life-Long Learning
4. Accomplishment of Complex Problems
Text Books:
1. B.S. Grewal: Higher Engineering Mathematics, Khanna Publishers, 43rd Ed., 2015.
2. E. Kreyszig: Advanced Engineering Mathematics, John Wiley & Sons, 10th Ed., 2015.
Reference books:
1. N.P.Bali and Manish Goyal: A Text Book of Engineering Mathematics, Laxmi Publishers,
7th Ed., 2010.
2. B.V.Ramana: "Higher Engineering M athematics" Tata McGraw-Hill, 2006.
3. H. K. Dass and Er. RajnishVerma: "Higher Engineerig Mathematics",
S. Chand
publishing, 1st edition, 2011.
We links and Video Lectures:
1. http://nptel.ac.in/courses.php?disciplineID=111
2. http://wwww.khanacademy.org/
3. http://www.class-central.com/subject/math
Page 2 of 2
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