1. A bungee jumper with a mass of 68.1 kg leaps from a hot air balloon. Using a finite difference technique and step size of 0.25 seconds, the velocity is computed for the first 12 seconds of free fall, taking into account drag. 2. The document explains various types of errors in calculations such as significant figures, inherent error, truncation error, and true percentage relative error. It also describes the Regular Falsi and Newton-Raphson methods for finding roots of equations. 3. The problems involve using numerical methods such as Muller's method, Graeffe's root-square method, Romberg's method, Gauss elimination, Jacobi method, QR

1. 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

2. 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