Mathematics (MA) 2007

1

2007 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2007

Let \(u(x, y) = f(xe^y) + g(y^2 \cos(y))\), where \(f\) and \(g\) are infinitely differentiable functions. Then the partial differential equation of minimum order satisfied by \(u\) is

A

\(u_{xy} + xu_{yx} = u_y\)

B

\(u_{yy} + xu_{xx} = xu_x\)

C

\(u_{yy} - xu_{xx} = u_x\)

D

\(u_{yy} - xu_{xx} = xu_x\)

2

2007 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2007

Which of the following inequality is NOT true for \(x \in (\frac{1}{4}, \frac{3}{4})\)?

A

\(e^{-x} > \sum_{j=0}^2 \frac{(-x)^j}{j!}\)

B

\(e^{-x} < \sum_{j=0}^3 \frac{(-x)^j}{j!}\)

C

\(e^{-x} > \sum_{j=0}^3 \frac{(-x)^j}{j!}\)

D

\(e^{-x} > \sum_{j=0}^4 \frac{(-x)^j}{j!}\)

3

2007 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2007

Let \(P_n(x)\) be the Legendre polynomial of degree \(n\) and let \(P_{n+1}'(0) = -\frac{m}{m+1} P_{n-1}'(0), m = 1, 2, ... .\) If \(P_n'(0) = -\frac{5}{16}\), then \(\int_{-1}^1 P_n^2(x) dx =\)

A

\(\frac{2}{13}\)

B

\(\frac{2}{9}\)

C

\(\frac{5}{16}\)

D

\(\frac{2}{5}\)

4

2007 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2007

The linear map \( T \) is

A

one to one and onto

B

one to one but NOT onto

C

onto but NOT one to one

D

neither one to one nor onto

5

2007 · Unclassified

Mathematics (MA) 2007

Consider \(\mathbb{R}^2\) with the usual topology. Let \(S = \{(x, y) \in \mathbb{R}^2 : x \text{ is an integer}\}\). Then \(S\) is

A

open but NOT closed

B

both open and closed

C

neither open nor closed

D

closed but NOT open

6

2007 · Unclassified

Mathematics (MA) 2007

Suppose \(X = \{\alpha, \beta, \delta\}\). Let \(\mathfrak{T}_1 = \{\phi, X, \{\alpha\}, \{\alpha, \beta\}\}\) and \(\mathfrak{T}_2 = \{\phi, X, \{\alpha\}, \{\beta, \delta\}\}\). Then

A

both \(\mathfrak{T}_1 \cap \mathfrak{T}_2\) and \(\mathfrak{T}_1 \cup \mathfrak{T}_2\) are topologies

B

neither \(\mathfrak{T}_1 \cap \mathfrak{T}_2\) nor \(\mathfrak{T}_1 \cup \mathfrak{T}_2\) is a topology

C

\(\mathfrak{T}_1 \cup \mathfrak{T}_2\) is a topology but \(\mathfrak{T}_1 \cap \mathfrak{T}_2\) is NOT a topology

D

\(\mathfrak{T}_1 \cap \mathfrak{T}_2\) is a topology but \(\mathfrak{T}_1 \cup \mathfrak{T}_2\) is NOT a topology