This document contains 8 questions on topics in applied mathematics for a Masters degree examination. The questions cover a range of numerical methods including Muller's method, Bairstow's method, Gauss-Seidel method, cubic spline interpolation, and the Numerov method. Other questions address linear algebra topics such as basis vectors, subspaces, linear transformations, and inverses. Students are asked to prove theorems about graphs, vector spaces, and matrix eigenvectors. Calculations involve solving systems of equations, finding roots of polynomials, and maximizing profit under constraints.

1. 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)
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ImportantNote:1.Oncompletingyouranswers,compulsorilydrawdiagonalcrosslinesontheremainingblankpages.
2.Anyrevealingofidentification,appealtoevaluatorand/orequationswritteneg,42+8=50,willbetreatedasmalpractice.
USN 12EPE/EPS/ECD/EMS11

2. 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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