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

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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