NBHM Research Scholarships Screening Exam paper

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1.2 Let S7 denote the symmetric group of all permutations of the symbols
f1; 2; 3; 4; 5; 6; 7g. Pick out the true statements:
a. S7 has an element of order 10;
b. S7 has an element of order 15;
c. the order of any element of S7 is at most 12.

1.3 Let C(R) denote the ring of all continuous real-valued functions on
R, with the operations of pointwise addition and pointwise multiplication.
Which of the following form an ideal in this ring?
a. The set of all C1 functions with compact support.
b. The set of all continuous functions with compact support.
c. The set of all continuous functions which vanish at in

3.2 Let X and Y be metric spaces and let f : X ! Y be a mapping. Pick
out the true statements:
a. if f is uniformly continuous, then the image of every Cauchy sequence in
X is a Cauchy sequence in Y ;
b. if X is complete and if f is continuous, then the image of every Cauchy
sequence in X is a Cauchy sequence in Y ;
c. if Y is complete and if f is continuous, then the image of every Cauchy
sequence in X is a Cauchy sequence in Y ;

3.3 Which of the following statements are true?
a. If A is a dense subset of a topological space X, then XnA is nowhere
dense in X.
b. If A is a nowhere dense subset of a topological space X, then XnA is
dense in X.
c. The set R, identied with the x-axis in R2, is nowhere dense in R2.

3.4 Which of the following metric spaces are separable?
a. The space C[0; 1], with the usual `sup-norm' metric.
b. The space `1 of all absolutely convergent real sequences, with the metric

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1.1 Compute (√3 + i)14 + (√3 − i)14 (Hint: Use De Moivre’s theorem).

1.2 Let p(x) be the polynomial x3 −11x2 +ax−36, where a is a real number.
Assume that it has a positive root which is the product of the other two roots. Find the value of a.

1.3 Identify which of the following groups (if any) is cyclic:
(a) Z8 ⊕ Z8
(b) Z8 ⊕ Z9
(c) Z8 ⊕ Z10.

1.4 In each of the following examples determine the number of homomor phisms between the given groups:
(a) from Z to Z10;
(b) from Z10 to Z10;
(c) from Z8 to Z10.
1.5 Let S7 be the group of permutations on 7 symbols. Does S7 contain an
element of order 10? If the answer is “yes”, then give an example.

1.6 Let G be a finite group and H be a subgroup of G. Let O(G) and O(H) denote the orders of G and H respectively. Identify which of the following statements are necessarily true.
(a) If O(G)/O(H) is a prime number then H is normal in G.
(b) If O(G)=2O(H) then H is normal in G.
(c) If there exist normal subgroups A and B of G such that H = {ab | a ∈
A, b ∈ B} then H is normal in G.

1.7 Which of the following statements are true?
(a) There exists a finite field in which the additive group is not cyclic.
(b) If F is a finite field, there exists a polynomial p over F such that p(x) = 0 for all x ∈ F, where 0 denotes the zero in F.
(c) Every finite field is isomorphic to a subfield of the field of complex numbers.

1.8 Let V be a vector space of dimension 4 over the field Z3 with 3 elements.
What is the number of one-dimensional vector subspaces of V ?

1.9 Let V be a vector space of dimension d < ∞, over R. Let U be a vector subspace of V . Let S be a subset of V . Identify which of the following statements is true:
(a) If S is a basis of V then U ∩ S is a basis of U.
(b) If U ∩ S is a basis of U and {s + U ∈ V/U | s ∈ S} is a basis of V/U
then S is a basis of V .
(c) If S is a basis of U as well as V then the dimension of U is d.

1.10 Let M(n, R) be the vector space of n × n matrices with real entries.
Let U be the subset of M(n, R) consisting {(aij) | a11 + a22 + ... + ann = 0}.
Is it true that U is a vector subspace of V over R? If so what is its dimension?

1.11 Let A be a 3 × 3 matrix with complex entries, whose eigenvalues are
1,i and −2i. If A−1 = aA2 + bA + cI, where I is the identity matrix, with a, b, c ∈ C, what are the values of a, b and c?

NBHM Research Scholarships Screening Exam paper 1

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