Entranceexammaths2015, Questões do prova de Cálculo. Universidade Federal de Goiás (UFG)
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Entranceexammaths2015, Questões do prova de Cálculo. Universidade Federal de Goiás (UFG)

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CEU MATHEMATICS ENTRANCE EXAM, 2015

DIRECTIONS: There are 10 problems, 5 problems from Analysis, and 5 from Algebra (10 points each). You have 3 hours for the 10 problems. No books or notes. To receive full credit your solutions must be clear, complete and correct.

ANALYSIS EXAM

(1) Assume that f is differentiable and F (x, y) = yf(x2−y2). (Here F : R2 → R and f : R→ R.) Prove that y2 · ∂F

∂x +xy · ∂F

∂y = xF .

(2) Compute ∑∞

n=1 1

n2+n . Determine whether

∑∞ n=1 2

− 1 n is conver-

gent or not.

(3) Suppose that f : R→ R is a continuous periodic function with period T > 0, that is, f(x+T ) = f(x) for all x ∈ R. Show that for any a ∈ R

1

t

∫ a+t a

f(s) ds −→ 1 T

∫ T 0

f(s) ds

as t −→∞.

(4) Define the Fourier coefficients of a continuous function f on [−π, π] as

an = 1

π

∫ π −π f(t) cosntdt, (n = 0, 1, 2, . . .) and

bn = 1

π

∫ π −π f(t) sinntdt, (n = 1, 2, . . .).

Suppose that the Fourier coefficients of the contimuous func- tions f and F are given as an, bn and An, Bn respectively. Show that

1

π

∫ π −π f(t)F (t)dt =

1

2 a0A0 +

∞∑ n=1

(anAn + bnBn).

(5) Suppose that f is continuous and ∫ B f = 0 for any subinterval

B ⊂ [a, b]. Show that in this case f = 0 on [a, b].

1

2

CEU MATHEMATICS ENTRANCE EXAM, 2015

ALGEBRA EXAM

(6) Let ϕ be the linear transformation of R3 whose matrix with respect to the standard basis of R3 is

A =

0 1 20 1 2 0 1 2

 . Determine a basis in which the matrix of ϕ is diagonal or show that no such basis exists.

(7) For a ring R a derivation is a map D : R → R such that D(a + b) = D(a) + D(b) and D(ab) = D(a)b + aD(b). Let now R = Z[x] be the ring of integer polynomials. For which polynomials f(x) ∈ R does there exist a derivation D of R such that D(x2 + 1) = f(x)?

(8) Let G = {r ∈ Q | 0 ≤ r < 1} be the set of non-negative rationals smaller than 1. Define

a ◦ b = { a+ b, if a+ b < 1; a+ b− 1, if a+ b ≥ 1.

Verify that G is a group with this operation. Show that for any finite set {g1, . . . gn} ⊆ G there is a proper

subgroup H ⊂ G, H 6= G such that {g1, . . . gn} ⊆ H. (9) Let M5(Z) denote the ring of 5×5 matrices with integer entries.

(a): Does M5(Z) have a subring isomorphic to Z[x], the ring of integer polynomials?

(b): Let I = (x2(x − 1)3) / Z[x] be the ideal generated by x2(x− 1)3 in Z[x]. Does M5(Z) have a subring isomorphic to the factor ring Z[x]/I?

(10) Let v1, v2 ∈ Rn be two vectors in a Euclidean space. Suppose that for every integer k ∈ Z we have |v1| ≤ |v2+kv1|. Show that for every k1, k2 ∈ Z, (not both 0) we have |v1| ≤ |k1v1 + k2v2|.

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