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July 3rd, 2014, 11:35 AM
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Join Date: Mar 2012
Re: M.Sc Maths modal question papers of Alagappa University Distance Education

As you want to get the M.Sc Maths modal question papers of Alagappa University Distance Education so here is the information of the same for you:

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1. (a) If H and K are subgroups of G prove that HK is a subgroup of G if and only if HK = KH.
(b) If G is an abelian group of order o(G), and if p is a prime number, such that does not divide o(G) prove that G has a subgroup of order .
2. (a) Prove that the number elements conjugate to a in G os the index of the normalizer of a in G.
(b) Prove that the number of p-Sylow subgroups in G, for a given prime, is of the form 1 + .
3. (a) Let G be a finite group and suppose that G is a subgroup of the finite group M. If M has a p-Sylow subgroup Q prove that G has a p-Sylow subgroup
P such that for some .
(b) Discuss the number and nature of the 3-Sylow subgroups ans 5-Sylow subgroups of a group of order 32 . 52.
4. (a) If R is a commutative ring with unit element and
M is an ideal of R, prove that M is a maximal of R if and only if R/M is a field.
(b) If G is the internal direct product of prove that for , , and if then ab = ba.
5. (a) If R is a Euclidean ring and with is not a unit in R prove that d(a)<d(ab).
(b) Prove that the ideal A = (a0) is a maximal ideal of the Euclidean ring R if and only if a0 is a prime element of R.
6. (a) If V is n-dimensional over F and if (V) has all its characteristic roots in F prove that T satisfies a polynomial of degree n over F.
(b) If (V) is such that (vT, v) = 0 for all , prove that T = 0.
7. (a) If N is normal and AN = NA, then AN* = N*A.
(b) Prove that the element is algebraic over F if and only if F (a) is a finite extension of F.
8. (a) Prove that a polynomial of degree n over a field can have at most n roots in any extension field.
(b) Prove that the fixed field of G is a subfield of K.

Time : Three hours Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a) State and prove Fixed-point theorem.
(b) Prove that compact subsets of metric spaces are closed . (10 + 10 = 20)
2. State and prove the Bolzano-Weierstrass theorem.
3. (a) State and prove chain rate.
(b) State and prove Generalised Mean-Value theorem.
(10 + 10 = 20)
4. Assume on an open set S in , and let . If the Jacobian determinate for some point a in S then prove that there are two open sets and and a uniquely determined function g such that :
(c) is 1 – 1 on X.
(d) g is defined on Y, and for every .
(e) .
5. (a) Prove that if is a refinement of P, then and .
(b) Show that .
(c) Show that R on [a,b] iff for every there exists a partition p such that (6 + 6 + 8 = 20)
6. Assume that each term of is a real-valued function having a finite derivative at each point of an open interval (a,b). Assume that for atleast one point in (a,b) the sequence converges. Assume further that there exists a function g such that uniformly on (a,b). Then such that :
(a) There exists a function f such that uniformly on (a,b).
(b) For every in (a,b) the derivative exists and equals .
7. (a) Prove that the outer measure of an interval is its length.
(b) Prove that is are measurable, so is . (10 + 10 = 20)
8. State and prove monotone convergence theorem

Time : Three hours Maximum : 100 marks
Answer any FIVE questions.
Each question carries 20 marks.
1. (a) State and prove the existence theorem.
(b) Let be any solution of
on an interval I containing a point . Then prove that for all in whose , .
(10 + 10 = 20)
2. (a) Prove that the two solutions of are linearly independent on an interval I if, and only if, for all x in I.
(b) Solve the degendre equation. (10 + 10 = 20)
3. Find the general integrals of the linear partial differential equations
(b) .
(10 + 10 = 20)
4. (a) Prove that .
(b) Find a particular integral of the equation .
(c) Find a particular integral of the equation . (8 + 6 + 6 = 20)
5. (a) Obtain a linear polynomial approximation to the function on the interval using the least square approximation with .
(b) For the solution of the system of equations set of the method. Find the relaxation factor and the rate of convergence. (10 + 10 = 20)
6. (a) Consider the system of equations

Use the Gauss Seidal iterative method and perform three iterations.
(b) Obtain the Chebyshev polynomial approximation of second degree to the function on .
(10 + 10 = 20)
7. (a) The following table of values is given
x : -1 1 2 3 4 5 7
f(x) : 1 1 16 81 256 625 2401
Using the formula and the Richardson extrapolation, find .
(b) Find the approximate value of . using
(i) Trapezoidal rule
(ii) Simpson’s rule. (10 + 10 = 20)
8. (a) Explain Euler’s method and solve the initial value problem
with and 0.05 on the interval .
(b) Use the Numerov method to solve the initial value problem , , with . (10 + 10 = 20)

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Alagappa University
Tamil Nadu ‎
04565 226 250

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