BABSc, B.Com & BCA Questions _III - Nalanda Open University

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12. (a) Let M be a closed linear subspace of a Hilbert space H.Let x be a vector not in M and let d be the distance from x to M. The show that there exists a unique vector y 0 in M such that x − y0 = d . (b) Construct a Banach space which is not a Hilbert space.
Nalanda Open University Annual Exam-2010, Bachelor of Science (Mathematics) Hons, Part-III Paper-VI Time: 3Hrs Full Marks: 75 Answer Six questions, selecting at least one question from each group. Group-A 1. (a) Define centre of a group. Show that the centre of a group G is a normal subgroup of G. (b) Let f be a function on the group G defined by f(x) = x -1 for each x in G. Show that f is an automorphism if and only if G is abelian. 2. (a) Let f R T be an isomorphism of the ring R onto the ring T. If O and O ' are the zero elements of R and T respectively, show that f(O) = O' and f(-a) = -f(a) for all a in R. (b) Show by an example that if I and J are ideals in R, then I U J is not an ideal in R. 3. Let R be a commutative ring with unity 1. Let M be an ideal in R. Then prove that R / M is a field if and only if M is a maximal ideal in R. 4. Show that any ring can be embedded in a ring with unity. Group-B 5. For any cardinals α, β, γ show that β γ β + γ (i) α . α = α (ii) αβ γ α γ β γ b g = 6. (a) If E is any set then show that Card P(E) = 2 CardE where P(E) denotes the power set of E. (b) If α and β are cardinal numbers such that α β β α . Prove that α β . 7. (a) Prove that the order type of the set of all real numbers in the open interval (a, b) with usual order is λ . (b) State and prove Zorn's lemma. Group-C 8. (a) Find the mean and variance of a Binomial distribution. (b) 6 die are thrown 729 times. How many times you expect at least 3 dice to show a five or six. 9. (a) Find an explicit formula for the Fibonacci numbers. (b) Illustrate how generating function can be used to get the number of r- combinations from a set with n elements when repition of elements is allowed. Group-D 10. (a) Show that the function f ( z) = xy is not analytic at the origin although the Cauchy - Riemann equations are satisfied at the point. (b) Construct the analytic function f(z) = u + iv where u = x 3 - 3xy 2 + 3x+1 11. State and prove Laurent's theorem. 12. (a) Explain different types of singularities with examples. CotπZ 2 (b) Find kind of singularities of ( z − a) at Z=a and Z = ∞ . ****
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BABSc, B.Com & BCA Questions _III - Nalanda Open University

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