This document contains 8 questions regarding various topics in applied mathematics for a first semester M.Tech degree examination. The questions cover topics such as significant figures, accuracy, precision, and round off errors; solving equations using methods like Regula-Falsi, Newton's formula, Bairstow method, and Graeffe's root squaring; numerical integration using Romberg's method; solving systems of linear equations using Gauss-Jordan and triangularization methods; eigen values and vectors using inverse power and Rutishauser methods; linear transformations and their properties; least squares solutions; and constructing orthogonal sets using Gram-Schmidt process. Students are required to answer any 5 full questions out of the 8 questions provided.

1. 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)
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ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 10MMD/MDE/MCM/MEA/MAR11

2. 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)
*****
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