Madhya Pradesh Bhoj (Open) University, Bhopal

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1 Subject : Advanced Abstract Algebra Q.1 State and prove Jordan-Holder theorem. Q.2 Two nilpotent linear transformations are similar if and only if they have the same invariants. Q.3 Show that every finite extension of a field is algebraic. Q.4 State and prove Hilbert basis theorem. Q.5 State and prove Fundamental structure theorem for finite generated modules. Over a principal ideal domain. I Q.1 A group is solvable if and only if for some integer Q.2 State and prove Schuler s Lemma. Q.3 State and prove Fundamental theorem of Galois theory. Q.4 State and prove Wederburn-Artin theorem. Q.5 Reduce the following matrix to a rational canonical form: [ ]

2 Subject : Real Analysis Q.1 State and prove fundamental theorem of calculus. Q.2 State and prove Weierstrass approximation theorem. Q.3 State and prove inverse function theorem. Q.4 Let is defined by { ( ) Then show that Q.5 State and prove Minkowski inequalities. I Q.1 State and prove Riemann s theorem. Q.2 Obtain the radius of convergence R of the power series, where. Q.3 State and prove implict function theorem. Q.4 Let be any set, and a finite sequence of disjoint measurable sets, Then show that Q.5 State and prove Holder inequalities. ( [ ])

3 Subject : Topology Q.1 Show that countable union of countable sets is countable. Q.2 State and prove Lindelof s theorem. Q.3 State and prove Urysohn s lemma. Q.4 Show that a subset of is connected iff it is an interval. Q.5 Write short notes on nets, filter, ultra-filters and homotopy of paths. I Q.1 Define closed set, neighbourhood, interior and exterior in topological spsce. Q.2 Show that a second countable space is always first countable. Q.3 Show that every - space is Tychonoff space. Q.4 State and prove Urysohn metrization theorem. Q.5 Show that every filter is contained in an ultra filter.

4 Subject : Complex Analysis Q.1 State and prove Morera theorem. Q.2 State and prove Cauchy residue theorem. Q.3 Show that the set of all bilinear transformations form a non-abelian group under the product of transformations. Q.4 State and prove Harnack s inequalities. Q.5 State and prove Borel s theorem. I Q.1 State and prove Maximum modulus principle. Q.2 Evaluate the residues of at the poles where Q.3 State and prove Montel s theorem. Q.4 State and prove Duplication formula. Q.5 State and prove Great Picard theorem.

5 Subject : Advanced Discrete Mathematics Q.1 Describe Universal quantifier and Existence quantifier with example. Q.2 Show that every chain is a lattice. Q.3 Show that a tree of vertices has edges Q.4 Define finite state machine with example. Q.5 State and prove Kleene s theorem. I Q.1 Show that is tautology. Q.2 In Boolean algebra, Show that if and then Q.3 Derive Euler s formula for connected planner graph. Q.4 Define Moore machine and Mealy machine. Q.5 Show that the language { } is not a finite state language.

6 Subject : Differential Equation Q.1 Solve simultaneous differential equation Q.2 Apply Picard method to the IVP and find the successive approximations. Q.3 State and prove Poincare-Bendixion theorem. Q.4 Solve Sturm Liouville problem Q.5 Find out complete solution of Q.1 Solve simultaneous differential equation I Q.2 State and prove existence theorem. Q.3 Define foci, nodes and saddle points. Q.4 Solve Sturm Liouville problem Q.5 Solve

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