12 mmd11 applied mathematics - june, july 2013

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.
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)
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ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 12MMD/MDE/MCM/MEA/MAR/MST11

2. 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)
* * * * *
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