Text 2

Excerpts from:

Peek-a-boo

Astronomers get some new toys to play
with

FEW scientists believe that the space shuttle has helped their profession. Mostly, it has been used to convey astronauts to a space station that has produced little worthwhile research and to launch satellites that might have been put into orbit more cheaply by old-fashioned, throwaway rockets. But it has done one thing to assist astronomers. It has allowed what is probably their most famous instrument to be repaired and upgraded. That instrument is the Hubble space telescope, which took the picture of the Carina nebula shown above, and has snapped more than half a million other images over the years. Now, as the shuttle programme draws to its close (the final launch will take place next year), Hubble is to be given its last makeover by the crew of Atlantis. On top of that, if the week has gone well, two other satellites intended to probe the universe’s earliest days will have been launched.

The mission to Hubble, which began on May 11th and is planned to last 11 days, will install a wide-field camera that will let the telescope see galaxies previously beyond its reach. Using this, the eager coteries of astronomers who have access to the instrument will be able to observe young, hot stars that glow mainly in the ultraviolet part of the spectrum. They will also be able to see the first stars and galaxies that formed in the universe, which are now so old and distant that their light has been relegated to the infra-red part of the spectrum by the “red shift” caused by the universe’s expansion. These wonders can be observed only from space, because ozone and water in the Earth’s atmosphere absorb light at those wavelengths.

(...)

(From The Economist print edition, May 14^th 2009)

According to the text

Item 3 - two other satellites will be launched next year;

Considere os sistemas lineares abaixo e julgue a afirmativa:

!$ (I)= { \begin{cases} x+ y + Kz = 2\\3x + 4y + 2z =K\\2x + 3y - z =1 \end{cases}} !$

!$ (II)= { \begin{cases} a_{11} x_1 + a_{12} x_2 + \dots + a_{1n} x_n = b_1 \\ a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} X_n = b_2\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots\\a_{m1} X_1 + a_{m2} X_2 + cdots + a_{mn} X_n = b_m \end{cases}} !$

Item 0 - Se !$ k \neq 3 !$, então o sistema (I) tem solução única;

Text 2

Excerpts from:

Peek-a-boo

Astronomers get some new toys to play
with

FEW scientists believe that the space shuttle has helped their profession. Mostly, it has been used to convey astronauts to a space station that has produced little worthwhile research and to launch satellites that might have been put into orbit more cheaply by old-fashioned, throwaway rockets. But it has done one thing to assist astronomers. It has allowed what is probably their most famous instrument to be repaired and upgraded. That instrument is the Hubble space telescope, which took the picture of the Carina nebula shown above, and has snapped more than half a million other images over the years. Now, as the shuttle programme draws to its close (the final launch will take place next year), Hubble is to be given its last makeover by the crew of Atlantis. On top of that, if the week has gone well, two other satellites intended to probe the universe’s earliest days will have been launched.

The mission to Hubble, which began on May 11th and is planned to last 11 days, will install a wide-field camera that will let the telescope see galaxies previously beyond its reach. Using this, the eager coteries of astronomers who have access to the instrument will be able to observe young, hot stars that glow mainly in the ultraviolet part of the spectrum. They will also be able to see the first stars and galaxies that formed in the universe, which are now so old and distant that their light has been relegated to the infra-red part of the spectrum by the “red shift” caused by the universe’s expansion. These wonders can be observed only from space, because ozone and water in the Earth’s atmosphere absorb light at those wavelengths.

(...)

(From The Economist print edition, May 14^th 2009)

The heading of the article leads the reader to expect:

Item 1 - to read about new pictures of the Carina nebula;

Sejam !$ f : IR_+^* \rightarrow IR\ !$ e !$ g: [ - \sqrt{5}, \sqrt{5} ] \rightarrow IR\ !$ funções tais que f(x) = ln x e !$ g(x) = x \sqrt{5 - x^2} !$. Julgue a afirmativa:

Item 4 - Se !$ \int\limits_{a}^{b} h (x) dx > 0 !$, então !$ h(x) \ge 0 !$, para todo !$ x \in [ a,b] !$.

Dentre as transformações ocorridas na economia brasileira a partir dos anos 1990 pode-se arrolar:

Item 4 - as regiões rurais foram as mais afetadas pelo desemprego.

Sobre a Teoria das Probabilidades e considerando A, B e C três eventos quaisquer, mas com probabilidades de ocorrência diferentes de zero, indique a alternativa correta e falsa:

Item 1 - Se dois eventos A e B são mutuamente exclusivos e exaustivos, eles são independentes;

Responda se verdadeiro ou falso:

Item 3 -Em um modelo de regressão simples sem constante, em que o coeficiente angular é estimado por Mínimos Quadrados, os resíduos têm média amostral zero por construção;

Considere as seguintes afirmações referentes ao modelo de regressão linear clássico com regressores estocásticos:

!$ y_i = \beta_0 + \beta_1 x_{li} + \beta_2 x_{2i} + \varepsilon_1,\,\,\,i =1, \cdots, n !$,

em que !$ E [ \varepsilon | x_1, x_2 ] = 0 !$ e !$ Var [ \varepsilon |x_1, x_2] = \sigma_2 !$.

Item 4 - Os estimadores de mínimos quadrados ordinários dos parâmetros não são eficientes se a hipótese de ausência de autocorrelação dos erros for violada.

Considere o seguinte modelo de equações simultâneas:

!$ q_d = a1p + a2z + a2z + a3y + \varepsilon_1\,\,\,\,(demanda),\\q_S= \beta1p + \varepsilon_2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(oferta)\,e\\qd=qs= q\,\,\,\,\,\,\,\,\,\,\,(equilíbrio) !$,

com

!$ E[ \varepsilon_1| z,y] = E [ \varepsilon_2 |z,y]= 0\\E[ \varepsilon_1^2 | z,y] \sigma_1^2,\,\,\,\,E[ \varepsilon_2^2 |z,y] = \sigma_2^2,\,\,\,E[ \varepsilon_1\,\,\varepsilon_2 |z,y] = \sigma_{12} \neq 0 !$

É correto afirmar que:

Item 0 -Os estimadores de mínimos quadrados ordinários dos parâmetros das equações de oferta e de demanda são inconsistentes;

Sobre Números-Índices podemos dizer que:

Item 4 - Quando da revisão de índices de custo de vida, aqueles produtos que não tiveram maiores aumentos relativos de preços terão sua ponderação aumentada, supondo que as preferências dos consumidores não mudaram e os bens possuem elasticidade renda unitária.