This document contains questions from a M.Tech Applied Mathematics exam. It includes questions on various topics in applied mathematics, such as:
1) Finding the binary form of a number, approximating a number, and writing a Fortran program for matrix multiplication.
2) Solving sets of equations using Gauss elimination, finding matrix inverses, and converting eigenvalue problems.
3) Evaluating mixed partial derivatives, Taylor series expansions, and numerically evaluating integrals using Simpson's rule, Gauss-Legendre quadrature, and Adams-Bashforth methods.
4) Solving initial value problems, the transverse deflection of beams, and using finite difference methods to solve PDEs modeling heat transfer.
I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
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I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Differential Equations Homework.
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
Review for the Third Midterm of Math 150 B 11242014Probl.docxjoellemurphey
Review for the Third Midterm of Math 150 B 11/24/2014
Problem 1
Recall that 1
1−x =
∑∞
n=0 x
n for |x| < 1.
Find a power series representation for the following functions and state the radius of
convergence for the power series
a) f(x) = x
2
(1+x)2
.
b) f(x) = 2
1+4x2
.
c) f(x) = x
4
2−x.
d) f(x) = x
1+x2
.
e) f(x) = 1
6+x
.
f) f(x) = x
2
27−x3 .
Problem 2
Find a Taylor series with a = 0 for the given function and state the radius of conver-
gence. You may use either the direct method (definition of a Taylor series) or known
series.
a) f(x) = ln(1 + x)
b) f(x) = sin x
x
c) f(x) = x sin(3x).
Problem 3
Find the radius of convergence and interval of convergence for the series
∑∞
n=1
(x+2)n
n4n
.
Ans. Radius r=2,
√
2 − 2 < x <
√
2 + 2. Problem 4
Find the interval of convergence of the following power series. You must justify your
answers.
∑∞
n=0
n2(x+4)n
23n
.
Ans. −12 < x < 4.
Problem 5
For the function f(x) = 1/
√
x, find the fourth order Taylor polynomial with a=1.
Problem 6
A curve has the parametric equations
x = cos t, y = 1 + sin t, 0 ≤ t ≤ 2π
a) Find dy
dx
when t = π
4
.
b) Find the equation of the line tangent to the curve at t = π/4. Write it in y = mx+b
form.
c) Eliminate the parameter t to find a cartesian (x, y) equation of the curve.
d) Using (c), or otherwise identify the curve.
Problem 7
State whether the given series converges or diverges
a)
∑∞
n=0 (−1)
n+1 n22n
n!
.
b)
∑∞
n=0
n(−3)n
4n−1
.
c)
∑∞
n=1
sin n
2n2+n
.
Problem 8
1
Approximate the value of the integral
∫ 1
0
e−x
2
dx with an error no greater than 5×10−4.
Ans.
∫ 1
0
e−x
2
dx = 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ ... +
(−1)n
(2n+1)n!
+ .... n ≥ 5,
for n=5
∫ 1
0
e−x
2
dx ≈ 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ 1
9.4!
− 1
11.5!
≈ 0.747.
Problem 9
Find the radius of convergence for the series
∑∞
n=1
nn(x−2)2n
n!
.
Ans. R = 1√
e
.
Problem 10
Let f(x) =
∑∞
n=0
(x−1)n
n2+1
.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate the domain of f ′.
Problem 11
Let f(x) =
∑∞
n=0
cos n
n!
xn.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate
∫
f(x)dx.
Problem 12
Using properties of series, known Maclaurin expansions of familiar functions and their
arithmetic, calculate Maclaurin series for the following.
a) ex
2
b) sin 2x
c)
∫
x5 sin xdx
d) cos x−1
x2
e)
d((x+1) tan−1(x))
dx
Problem 13
Calculate the Taylor polynomial T5(x), expanded at a=0, for
f(x) =
∫ x
0
ln |sect + tan t|dt.
Ans. T5(x) =
x2
2
+ x
4
4!
.
Problem 14
Suppose we only consider |x| ≤ 0.8. Find the best upper bound or maximum value
you can for∣∣∣sin x − (x − x33! + x55! )∣∣∣
Same question: If
(
x − x
3
3!
+ x
5
5!
)
is used to approximate sin x for |x| ≤ 0.8. What is
the maximum error? Explain what method you are using.
Problem 15
The Taylor polynomial T5(x) of degree 5 for (4 + x)
3/2 is
(4 + x)3/2 ≈ 8 + 3x + 3
16
x2 − 1
128
x3 + 3
4096
x4 − 3
32768
x5.
a) Use this polynomial to find Taylor polynomials for (4 + ...
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
CW RADAR, FMCW RADAR, FMCW ALTIMETER, AND THEIR PARAMETERSveerababupersonal22
It consists of cw radar and fmcw radar ,range measurement,if amplifier and fmcw altimeterThe CW radar operates using continuous wave transmission, while the FMCW radar employs frequency-modulated continuous wave technology. Range measurement is a crucial aspect of radar systems, providing information about the distance to a target. The IF amplifier plays a key role in signal processing, amplifying intermediate frequency signals for further analysis. The FMCW altimeter utilizes frequency-modulated continuous wave technology to accurately measure altitude above a reference point.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
6th International Conference on Machine Learning & Applications (CMLA 2024)
Applied mathematics 40
1.
2.
3.
4.
5.
6.
7. 08MTP/MAU/MFD11
USN
First Semester M.Tech. Degree Examination, Dee.09/Jan.1O
Applied Mathematics
Time: 3 hrs.
Max. Marks:100
Note: Answer any FIVE full questions.
1 a. Find the binary form of the number
193. (06 Marks)
b. The number 3.1415927 is approximated as 3.1416. Find the following:
i) error ii) relative error and iii) number of significant digits of the approximation.(04 Marks)
8 0
If A=
[ write Fortran program for the multiplication of ther
1
2
-
3
5 6
4
, B 2c. =
L -1
7
4
matrices A & B.
Generates the functions f;(7,.) , i = 0, 1, 2, 3 and 4 for the tridiagonal matrix
3 -2 0 0
-2 5 -3 0
A 0 -3 7 -4
0 0 -4 9
r4 3
2 a. Find the L,, L2, L, and L, norms of the matrix A = 17 6, (06 Marks)
b. Find the solution of the following set of equations using the Gauss elimination method:
2x, - x2 + x3 = 4 , 4x, + 3x2 - x3 = 6 and 3x, + 2x2 + 2x3 = 15 (06 Marks)
4 -1 1
E
c. Find the inverse of the matrix A = (-1 6 - 41 using the relation [A]-' = [U]-' ([U]-' )T .
1 -4 5
3 a. Convert the eigen value problem l
5000 -500
00 5 00 ]{x2} ^[ 0 5J{x2}[1 0
to a standard eigen value problem.
J
b. Find the Eigen values and the Eigen vectors of the matrix
12 6 -6
A= 6 16 2 by using Jacobi method.
-6 2 16
c.
(10 Marks)
(08 Marks)
(08 Marks)
(04 Marks)
(08 Marks)
a. Evaluate the mixed partial derivative &2 2 of the function f(x, y) = 2x4x3 using central
differences at x = I and y = 1 with a step size Ax = Ay =
0.1 (12 Marks)
b. Find the first three derivatives of f(x) = x z e-4x and express Taylor's series expansion
(
08 Marks)
atx=1.
8. 08MTP/MAU/MFD11
b
crnti tic the value of the integral I = f (x)dx, where
a
0.84885406+31.51924706x - 137.66731262x2 + 240.55831238x3
-171.45245361x4 + 41.95066071x5
J - 0.0 and b = 1.5 using Simpson's 1/3 rule with different step sizes. ( 08 Marks)
date the integral I = J ye2vdy using the Gauss-Legendre quadrature . (04 Marks)
0
I n
aiuate the integral I= J J f Sxy3z2dxdydz using the two-point Gauss-Legendre
ddrature rule. (08 Marks)
nu inc solution of the initial value problem y' = y + 2x -1 ; y(O) =1 in the interval
I using Adams - Bashforth open formulas of order 2 through 6. (10 Marks)
nd the solution of the initial value problem.
2x + v y(0) = 1, at x = 0.4 using the fourth order Adams predictor-corrector
hod with h = 0.1. (10 Marks)
ditterential equation governing the transverse deflection of a beam w(x) subjected to a
,^:'tnbutcd load, p(x) as shown in the Fig.7(a), is given as
z z
22 (EI dx p(x)
i e l = Young's modulus and I= area moment of inertia of the beam.
Fig.7(a)
n,uiate the boundary value problem for a uniform beam i ) fixed at the both ends and ii)
;.phv supported at both ends . ( 08 Marks)
,.main the shooting methods. ( 12 Marks)
tai rod of length 1 in is initially at 70°C. The steady-state temperature of the left and
ends of the rod are given as 50°C and 20°C respectively. Using a2 = 0.1 m2/min,
U 'm and At = 0.3 min, determine the temperature distribution in the rod for 0 5 t <- 0,
rfl ( rank-Nicholson method. (10 Marks)
luminium plate of size 0.3mxO.3m is initially at the temperature 30°C. If the adjacent
)C the plate are suddenly brought to 120°C and maintained at the temperature, derive
quations necessary for the determination of the time variation of temperature in the
using the alternating direction implicit method.
K = 236W/m-k. C = 900 J/kg-K, p = 2700 kg/m3 , Ax = Ay = 0.1 M. (10 Marks)
9.
10.
11.
12.
13.
14.
15. East West Institute of Technology
PG Library & Information Centre
Suggestions at: library.ewit@gmail.com
16.
17.
18.
19.
20.
21.
22.
23. First Semester M.Tech. Degree Examination, June/July 2013
Applied Mathematics
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions.
1 a.
b.
A bungee jumper with a mass of 68.1 kg leaps from a stationary hot balloon. Use
m
cV
g
dt
dv
to compute the velocity for the first 12 secs of free fall by employing a finite
difference technique. The drag coefficient is 12.5 kg/s and g = 9.81. Given V(0) = 0. Employ
a step size of 25 for calculation. (10 Marks)
Explain in brief: i) Significant figures ii) Inherent error iii) Truncation error
iv) True percentage relative error (10 Marks)
2 a.
b.
Explain Regular Falsi method to find the root of the equation, show that the roots lie in the
interval (2.7, 2.8). Use it to find three approximations for 02.1xlogx)x(f 10 with four
decimals in each computation. (10 Marks)
Use both the standard and modified Newton-Raphson method to evaluate the multiple root
of 3x7x5x)x(f 23
with an initial guess of ,0x0 perform three iterations in each
case. (10 Marks)
3 a.
b.
Use Muller’s method with guesses x0, x1 and x2 = 4.5, 5.5, 5 to determine a root of the
equation 12x13x)x(f 3
with a permissible error of 1%. (10 Marks)
Find all the roots of the polynomial 06x11x6x 23
by Graeffc’s root square method.
Perform three iterations. (10 Marks)
4 a.
b.
A rod is rotating in a plane. The following table gives the angle radians through which the
rod has turned for various values of the time t secs. (10 Marks)
t 0 0.2 0.4 0.6 0.8 1.0 1.2
0 0.12 0.49 1.12 2.02 3.20 4.67
Calculate the angular velocity and angular acceleration of the rod when t = 0.6 second.
Use Romberg’s method to compute
1
0
x1
dx
I correct to three decimal places. (10 Marks)
5 a.
b.
Determine the inverse of the matrix
353
134
111
using the partition method. Hence find the
solution of the system of equations 1xxx 321 ,
6xx3x4 321 ,
4x3x5x3 321 (10 Marks)
Solve the system of equations by Gauss elimination method,
2xxxx 4321
5xx2xx2 4321
7x4x3x2x3 4321
5x2x3x2x 4321 (10 Marks)
1 of 2
ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice. USN 10MMD/MDE/MCM/MEA/MAR11
24. 10MMD/MDE/MCM/MEA/MAR11
6 a.
b.
Use Jacobi method to find all the eigen values and the corresponding eigen vectors of the
matrix. Perform four iterations
121
212
121
. (10 Marks)
Find all the eigen values of the matrix using Rutishauser method,
231
212
111
A iterate till
all the elements of the lower triangular part are less than 0.05 in magnitude. (10 Marks)
7 a.
b.
Define the matrix with examples for, i) Linear transformation ii) Shear transformation iii)
Super position transformation. (10 Marks)
Let
742
213
111
A ,
7
2
3
C
Define a linear transformation :T 3
3
by T(x) = AX. Determine if C is in the range of transformation of T. (10 Marks)
8 a.
b.
Find a QR factorization of
111
111
011
001
A using Gram-Schmidt process. (10 Marks)
Find a least square solution of AX = b for
1001
1001
0101
0101
0011
0011
A ,
1
5
2
0
1
3
b . (10 Marks)
* * * * *
2 of 2
25. First Semester M.Tech. Degree Examination, Dec. 2013 / Jan 2014.
Applied Mathematics
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions.
1 a. Explain in brief : i) Significant figures ii) Accuracy iii) Precision iv) Round off
errors v) Truncation errors. (10 Marks)
b. Find the sum of 0.1874 × 104
and 27.8 × 10-1
and obtain its chopping off and rounding off
approximations in four – digit mantissa. Find the error and relative error in each case.
(10 Marks)
2 a. Applying Regula – Falsi method, find a real root of the equation xlog10 x – 1.2 correct to 3
decimal places. (10 Marks)
b. Derive Newton’s formula to find N . Hence find 28 correct to 4 decimal places.
(10 Marks)
3 a. Perform two iterations of the Bairstow method to extract a quadratic factor x2
+ px + q from
the polynomial x3
+ x2
– x + 2 = 0. Use the initial approximation p0 = -0.9 , q0 = 0.9.
(10 Marks)
b. Find all the roots of the polynomial equation x3
– 6x2
+ 11x – 6 = 0, using Graeffe’s root
squaring method (Squaring 3 times). (10 Marks)
4
a. Use Romberg’s method to compute
1
0
2
x1
dx
, correct to four decimal places, by taking
h = 0.5, 0.25 and 0.125. (10 Marks)
b. Find the derivative of f(x) = -0.1x4
– 0.15x3
– 0.5x2
– 0.25x + 1.2 at x = 0.5 using the high
accuracy formula (Forward difference of accuracy and back ward difference of accuracy of
O(h2
). Take h = 0.25). (10 Marks)
5 a. Apply Gauss – Jordan method to solve the equation x + y + z = 9 ; 2x – 3y + 4z = 13 ;
3x + 4y + 5z = 40. (10 Marks)
b. Solve the system of linear equations
x1 + x2 + x3 = 1
4x1 + 3x2 – x3 = 6
3x1 + 5x2 + 3x3 = 4 by using triangularization method. (10 Marks)
6 a. Find the smallest eigen value in magnitude of the matrix
A =
210
121
012
, using the four iterations of the inverse power method. (10 Marks)
b. Find all the eigen values of the matrix A =
21
34
using the Rutisha user method.(10 Marks)
1 of 2
ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 10MMD/MDE/MCM/MEA/MAR11
26. 10MMD/MDE/MCM/MEA/MAR11
7 a. Define a linear transformation T : R2
→ R2
by T(X) = AX =
1
2
x
x
, where A =
01
10
.
Find the images under T of u =
3
2
, v =
2
3
and u + v =
5
5
. Interpret the result
graphically. (10 Marks)
b. For T(x1, x2) = (3x1 + x2, 5x1 + 7x2 , x1 + 3x2) , show that T is a one - to – one linear
transformation. Does T map R2
onto R3
? (10 Marks)
8 a. Find the least square solution of the system Ax = b for
A =
11
20
04
, b =
11
0
2
. (10 Marks)
b. Construct an orthogonal set for A =
111
111
011
001
. Using Gram – Schmidt process. (10 Marks)
*****
2 of 2
27.
28.
29. First Semester M.Tech. Degree Examination, June/July 2013
Applied Mathematic
Time: 3 hrs. Max. Marks:100
Note: 1. Answer any FIVE full questions.
2. Missing data, if any, may be suitable assumed.
1 a.
b.
Perform two iterations with Muller method for log10x – x + 3 = 0, x0 = 0.25, x1 = 0.5,
x2 = 1. (10 Marks)
Perform one iteration of the Bairstow method to extract a quadratic factor x2
+ px + q from
the polynomial x4
+ x3
+ 2x2
+ x + 1 = 0. Use the initial approximation p0 = 0.5 and q = 0.5.
(10 Marks)
2 a.
b.
Solve the equation
t
u
t
u
2
2
subject to the conditions u(0.t) = u(1, t) = 0
1x)x1(2
x0x2
)0,x(u
2
1
2
1
Carryout computation for 3 levels, taking h = 0.1 and k = 0.001. (10 Marks)
Solve 2
2
2
2
t
u
x
u
subject to the boundary conditions u(0,t) = 0 = u(1, t) t 0 and the initial
conditions, 0
t
)0,x(u
, u(x, 0) sin x, 0 x 1 by taking h = ¼ and k = ⅕. Carryout
second level solutions in the time scale. (10 Marks)
3 a.
b.
Solve the following system of equations :
20x + y – 2z = 17
3x + 20 y – z = – 18
2x – 3y + 20z = 25
Using Gauss – Seidel method directly and in error format. Person two iterations. (10 Marks)
Explain Jacobi method to find the eigen value and corresponding eigen vectors of a real
symmetric matrix and find all the eigen values and corresponding eigen vectors of the matrix
321
232
123
A
Iterate till off – diagonal elements in magnitude are less than 0.4. (10 Marks)
4 a.
b.
Obtain the cubic spline approximation for the function defined by the data with M(0) = 0.
x 0 1 2 3
f(x) 1 2 33 244
M(3) = 0. Hence find an estimate of f(2.5). (10 Marks)
Use the Numerov method to solve the initial value problem u = (1 + t2
)u, u(0) = 1, u(0) = 0
t [0, 1] with h = 0.2. (10 Marks)
1 of 2
ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 12EPE/EPS/ECD/EMS11
30. 12EPE/EPS/ECD/EMS11
5 a.
b.
A factory uses 3 types of machines to produce tow types of electronic gadgets. The first
gadget requires in hours 12, 4 and 2 respectively on the 3 types of machines. The second
gadget requires in hours 6, 10 and 3 on the machines respectively. The total available time in
hours on the machines are 6000, 4000, 1800. If two types of gadgets respectively fetches a
profit of rupees 400 and 1000 find the number of gadgets of each type to be produced for
getting the maximum profit. Use graphical method. (10 Marks)
Solve the following minimization problem by simplex method :
Objective function : P = -3x + 8y – 5z
Constants : – x – 2z 5
2x – 3y + z z
2x – 5y + 6z 5
x 0, y 0, z 0. (10 Marks)
6 a.
b.
Prove that a simple graph with n vertices and k components can have at most (n – k)
(n – k + 1)/2 edges. (10 Marks)
Define Euler circuits. Prove that a connected graph G has an Euler circuit if and only if all
vertices of G are of even degree. (10 Marks)
7 a.
b.
c.
Let V be a vector space which is spanned by a finite set of vectors B = {1, 2, - - - - m}.
Then prove that any independent set of vectors in V is finite and contains no move than ‘m’
elements. (07 Marks)
Determine whether B = {(1, 2, 1), (3, 4, –7), (3, 1, 5)} is a basis of V3(R). (06 Marks)
If W1 and W2 are finite dimensional subspaces of a vector space V. Then prove that
W1 + W2 is finite dimensional and dim W1 + dim W2 = dim (W1 W2) + dim (W1 + W2).
(07 Marks)
8 a.
b.
Find the range, null space, rank and nullity of linear transformation T : V3(R) V2(R)
defined by T(x,y, z) = (y – x, y – z). Also verify the Rank – Nullity theorem. (10 Marks)
Show that the transformation T : V3(R)V3(R) defined by
T(x1,x2,x3)=(x1 + x2 +x3, x3 –x2, x3) is non – singular and find its inverse. (10 Marks)
* * * * *
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31.
32.
33.
34.
35. First Semester M.Tech. Degree Examination, June/July 2013
Applied Mathematics
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions.
1 a.
b.
Explain in brief:
i) Inherent errors
ii) Rounding errors
iii) Truncation errors
iv) Absolute, relative and percentage errors. (10 Marks)
If 3
2
z
xy4
f and the errors in x, y, z are 0.001, find the maximum relative error in f at
x = y = z = 1. (10 Marks)
2 a.
b.
Explain secant method for finding a root of the equation f(x) = 0. By using the Regula-Falsi
method, find the root of the equation 2.1xlogx 10 that lies between 2 and 3. (10 Marks)
Find the multiple root of the equation 064x16x36x11x 234
, that lies near 3.9.
(10 Marks)
3 a.
b.
Find two iterations of the Muller method to find the root of the equation
01x5x)x(f 3
in (0, 1). (10 Marks)
Perform two iterations of the Bairstow method to extract a quadratic factor qpxx2
from
the polynomial 060x40x20x5x 234
, use the initial approximations p0 = 4,
q0 = 8. (10 Marks)
4 a.
b.
From the following data obtain the first and second derivatives of xlogy e i) at x = 500,
ii) at x = 550. (10 Marks)
Evaluate the integral
5.0
0
dx
xsin
x
using Romberg’s method, correct to three decimal places.
(10 Marks)
5 a.
b.
Use Cholesky’s method to solve the system
1zyx
5z3yx3
10z5y2x (10 Marks)
Using the partition method, solve the system of equations:
4z3y4x2
1zy
2zy2x2 (10 Marks)
1 of 2
ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 12MMD/MDE/MCM/MEA/MAR/MST11
36. 12MMD/MDE/MCM/MEA/MAR/MST11
6 a.
b.
By employing the Given’s method, reduce the matrix
322
221
212
A
to tridiagonal form and hence find its largest eigen value. (10 Marks)
Find all the eigenvalues of the matrix
231
212
111
A
using the Rutishauser method. (Carryout 5 steps) (10 Marks)
7 a.
b.
Find a spanning set for the null space of the matrix
48542
13221
71163
A (10 Marks)
Let
6873
3752
1242
A ,
0
1
2
3
u and
3
1
3
v
i) Determine if u is in Nul A, could u be in Col A?
ii) Determine if v is in Col A, could v be in Nul A? (10 Marks)
8 a.
b.
Find a least-squares solution of the system AX = b where
42
13
51
A ,
3
2
4
b . (10 Marks)
If
1
5
2
u1 ,
1
1
2
u2 ,
3
2
1
y and w = span {u1, u2}, then find orthogonal projection of
y onto w. (10 Marks)
* * * * *
2 of 2
37. First Semester M.Tech. Degree Examination, Dec.2013/Jan.2014
Applied Mathematics
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions.
1 a.
b.
Define with suitable examples
i) Inherent error.
ii) Significant figures.
iii) Truncation error.
iv) True percentage relative error. (10 Marks)
A parachutist of mass 68.1kgs jumps out of a stationary hot air balloon, use v
m
c
g
dt
dv
to
compute velocity v prior to opening the chute. The drag coefficient is equal to 12.5kg/s.
Given that g = 9.8. Use analytical method to compute velocity prior to opening the chute,
calculate the terminal velocity also. (Take a step size of 2 secs for computation). (10 Marks)
2 a.
b.
Explain bisection method. Use regular Falsi method to find a third approximation of the root
for the equation tanx + tanhx = 0, which lies between 2 & 3. (10 Marks)
Explain modified Newton-Raphson method. Use modified Newton-Raphson method to find
a root of the equation x4
– 11x + 8 = 0 correct to four decimal places. Given x0 = 2.
(10 Marks)
3 a.
b.
Use Muller’s method with guesses of x0, x1 and x2 = 4.5, 5.5, 5 to determine a root of the
equation f(x) = x3
– 13x – 12. Perform two iterations. (10 Marks)
Find all the roots of the polynomial x3
– 6x2
+ 11x – 6 using the Graffe’s Root square
method by squaring thrice. (10 Marks)
4 a.
b.
Obtain a suitable Newton’s interpolation formula to find the first derivative. Use it to
evaluate at x = 1.2 from the following table: (10 Marks)
x: 1.0 1.5 2.0 2.5 3.0
y: 27 106.75 324.0 783.75 1621.00
Apply Romberg’s integration method to evaluate
2.1
0
x1
dx
taking stepsize h = 0.6, 0.3, 0.15.
(10 Marks)
5 a.
b.
Solve the following set of equations by Crout’s method:
2x + y + 4z = 12
8x – 3y + 2z = 20
4x + 11y – z = 33. (10 Marks)
Determine the inverse of the matrix
353
134
111
by using the Partition method. (10 Marks)
1 of 2
ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 12MMD/MDE/MCM/MEA/MAR/MST11
38. 12MMD/MDE/MCM/MEA/MAR/MST11
6 a.
b.
Find all the eigen values and eigen vectors of the matrix by
A
124
232
421
by Jacobi’s method. Perform two iterations. (10 Marks)
Find the smallest eigen value in magnitude of the matrix
210
121
012
A
using four iterations of the inverse power method. (10 Marks)
7 a.
b.
c.
Define a linear transformation T: 2
2
by
1
2
2
1
x
x
x
x
01
10
)x(T .
Find the images under T of
3
2
v
1
4
u and
4
6
vu . (07 Marks)
Let T : n
m
be a linear transformation and let A be the standard matrix. Then prove
that: T maps n
onto m
iff the columns of A span m
. (06 Marks)
For T(x1, x2) = (3x1 + x2, 5x1 + 7x2, x1 + 3x2). Show that T is a one to one linear
transformation. Does T maps 2
onto 3
. (07 Marks)
8 a.
b.
Let W = span {x1, x2} where
2
2
1
xand
0
6
3
x 21 x. Construct an orthonormal basis
{v1, v2} for w. (10 Marks)
Find a least-squares solution of the inconsistent system
11
0
2
b
11
20
04
AforbAx . (10 Marks)
* * * * *
2 of 2