Questões - IME 2006 | Gabarito e resoluções

(IME - 2006/2007 - 2 FASE) Considere as matrizes e , e seja P uma matriz inversvel tal que . Sendo n um nmero natural, calcule o determinante da matriz .

(IME - 2006/2007 - 2 FASE) Considere o conjunto formado por m bolas pretas e n bolas brancas. Determine o nmero de seqncias simtricas que podem ser formadas utilizando-se todas as m+n bolas. Observao: uma seqncia dita simtrica quando ela possui a mesma ordem de cores ao ser percorrida da direita para a esquerda e da esquerda para a direita.

(IME - 2006/2007 - 2 FASE) Sejam x1 e x2 as razes da equao. Sabendo que x1 e x2 so nmeros inteiros, determine o conjunto de valores possveis para m.

(IME - 2016 - 1a FASE) Um trapzio ABCD, de base menorAB e base maior CD, possui base mdia MN. Os pontos M e N dividem a base mdia em trs segmentos iguais, na ordem MMNN. Ao se traar as retas AM e BN, verificou-se que as mesmas se encontraram sobre o lado CD no ponto P. Calcule a rea do trapzio MNCD em funo da rea de ABCD.

[IME- 2006/2007] Sejam C e C* dois crculos tangentes exteriores de raios r e r* e centros O e O*, respectivamente, e seja t uma reta tangente comum a C e C* nos pontos no coincidentes A e A*. Considere o slido de revoluo gerado a partir da rotao do segmento AA* em torno do eixo OO*, e seja S a sua correspondente rea lateral. Determine S em funo de r e r*.

(IME 2006) Leia o texto “New planet definition sparks furore” e responda, EM PORTUGUÊS, as perguntas que se seguem.  New planet definition sparks furore The new planet definition that relegates Pluto to "dwarf planet" status is drawing intense criticism from astronomers. It appears likely that the definition will not be widely adopted by astronomers for everyday use, even though it is the International Astronomical Union's (IAU) official position. On Thursday, astronomers at the IAU meeting in Prague approved a resolution that says the solar system has only eight planets, with Pluto excluded. Pluto is considered a "dwarf planet" instead. But the new definition has provoked a backlash. Alan Stern, who heads NASA's New Horizons mission to Pluto and works at the Southwest Research Institute in Boulder, Colorado, US, says the new definition is "awful". "The definition introduced is fundamentally flawed," he told New Scientist. "As a scientist, I'm embarrassed." Four planets He says only four of the eight objects mentioned in the IAU definition actually meet the definition's criteria – and Earth, Mars, Jupiter and Neptune do not. That is because the definition stipulates that to be a planet, an object must have cleared the neighbourhood around its orbit. But Earth's orbital neighbourhood is filled with thousands of near-Earth asteroids, Stern says. And Mars, Jupiter and Neptune have so-called "Trojan" asteroids sharing their orbits. "This is a half-baked criterion for planethood," he says. He says the new definition was pushed by people who are unhappy with having large numbers of planets (an earlier proposal, which was abandoned after heavy criticism at the meeting, would have potentially allowed hundreds of new planets into the fold). "It's just people that say things like, 'School kids will have to memorise too many names.' Do we limit the number of stars because children have to think of too many names? Or rivers on the Earth? It's just crazy," Stern told New Scientist. Minority report Stern is also critical of the fact that only astronomers present for the vote, which occurred at the end of the two-week meeting, were allowed to have their say on the matter. No email voting was allowed for the decision – it was made by a show of hands – and that meant that less than 5% of the nearly 9000 IAU members actually voted. "You're going to see a real backlash in the coming weeks," he says. "I know there is a petition among planetary scientists that's getting a lot of support." In any case, he says, astronomers are not obligated to use the new definition, since the IAU does not have the power to enforce it. "I don't think it's going to be very widely followed," he says. David Weintraub, author of the upcoming book Is Pluto a Planet? and a researcher at Vanderbilt University in Nashville, Tennessee, US, says he thinks Pluto is still a planet – even under the new definition.  Grammar issue "As best I can tell, 'dwarf' is an adjective and 'planet' is a noun," he told New Scientist. "I think the IAU thinks they defined Pluto to not be a planet. But they in fact have defined Pluto to be a planet – a particular kind of planet." Astronomers who proposed the new definition respond that the term "dwarf planet" is meant to be thought of as a single concept. And others point out that "minor planets" – asteroids and other small bodies – are not considered planets. But he agrees with Stern that the stipulation that a planet clears out its neighbourhood is flawed. A better definition would say a planet is an object that orbits a star and is large enough to be spherical, but is not large enough to be a brown dwarf – a "failed" star with between about 13 and 75 times the mass of Jupiter – or a star, he says.  'This is crazy'  "Everyone agrees on those criteria," he says. "The question is, can we agree on additional criteria to refine the definition further? I think the answer is no." "Everybody who has communicated with me is saying, 'This is crazy and we don't agree with it,'" he adds. "I'm not convinced that the folks who were at the meeting represented well the larger community." But not everyone is unhappy with the decision. Richard Conn Henry of Johns Hopkins University in Baltimore, Maryland, US, says he is pleased with the outcome. "As far as I'm concerned, the right decision was made," he told New Scientist. "I know a planet when I see it and there are eight of them." He says it makes no sense to call Pluto a planet because it is just one of huge numbers of objects in the Kuiper Belt beyond Neptune. (Adaptado de New Scientist Space, August 2006) __________________________________________________________________________________________________________________________________________________________________________________________ Richard Conn Henry, of Johns Hopkins University, says he is pleased with the outcome. Find in the text one argument that supports his position.

(IME 2006) Leia o texto “New planet definition sparks furore” e responda, EM PORTUGUÊS, as perguntas que se seguem.  New planet definition sparks furore The new planet definition that relegates Pluto to "dwarf planet" status is drawing intense criticism from astronomers. It appears likely that the definition will not be widely adopted by astronomers for everyday use, even though it is the International Astronomical Union's (IAU) official position. On Thursday, astronomers at the IAU meeting in Prague approved a resolution that says the solar system has only eight planets, with Pluto excluded. Pluto is considered a "dwarf planet" instead. But the new definition has provoked a backlash. Alan Stern, who heads NASA's New Horizons mission to Pluto and works at the Southwest Research Institute in Boulder, Colorado, US, says the new definition is "awful". "The definition introduced is fundamentally flawed," he told New Scientist. "As a scientist, I'm embarrassed." Four planets He says only four of the eight objects mentioned in the IAU definition actually meet the definition's criteria – and Earth, Mars, Jupiter and Neptune do not. That is because the definition stipulates that to be a planet, an object must have cleared the neighbourhood around its orbit. But Earth's orbital neighbourhood is filled with thousands of near-Earth asteroids, Stern says. And Mars, Jupiter and Neptune have so-called "Trojan" asteroids sharing their orbits. "This is a half-baked criterion for planethood," he says. He says the new definition was pushed by people who are unhappy with having large numbers of planets (an earlier proposal, which was abandoned after heavy criticism at the meeting, would have potentially allowed hundreds of new planets into the fold). "It's just people that say things like, 'School kids will have to memorise too many names.' Do we limit the number of stars because children have to think of too many names? Or rivers on the Earth? It's just crazy," Stern told New Scientist. Minority report Stern is also critical of the fact that only astronomers present for the vote, which occurred at the end of the two-week meeting, were allowed to have their say on the matter. No email voting was allowed for the decision – it was made by a show of hands – and that meant that less than 5% of the nearly 9000 IAU members actually voted. "You're going to see a real backlash in the coming weeks," he says. "I know there is a petition among planetary scientists that's getting a lot of support." In any case, he says, astronomers are not obligated to use the new definition, since the IAU does not have the power to enforce it. "I don't think it's going to be very widely followed," he says. David Weintraub, author of the upcoming book Is Pluto a Planet? and a researcher at Vanderbilt University in Nashville, Tennessee, US, says he thinks Pluto is still a planet – even under the new definition.  Grammar issue "As best I can tell, 'dwarf' is an adjective and 'planet' is a noun," he told New Scientist. "I think the IAU thinks they defined Pluto to not be a planet. But they in fact have defined Pluto to be a planet – a particular kind of planet." Astronomers who proposed the new definition respond that the term "dwarf planet" is meant to be thought of as a single concept. And others point out that "minor planets" – asteroids and other small bodies – are not considered planets. But he agrees with Stern that the stipulation that a planet clears out its neighbourhood is flawed. A better definition would say a planet is an object that orbits a star and is large enough to be spherical, but is not large enough to be a brown dwarf – a "failed" star with between about 13 and 75 times the mass of Jupiter – or a star, he says.  'This is crazy'  "Everyone agrees on those criteria," he says. "The question is, can we agree on additional criteria to refine the definition further? I think the answer is no." "Everybody who has communicated with me is saying, 'This is crazy and we don't agree with it,'" he adds. "I'm not convinced that the folks who were at the meeting represented well the larger community." But not everyone is unhappy with the decision. Richard Conn Henry of Johns Hopkins University in Baltimore, Maryland, US, says he is pleased with the outcome. "As far as I'm concerned, the right decision was made," he told New Scientist. "I know a planet when I see it and there are eight of them." He says it makes no sense to call Pluto a planet because it is just one of huge numbers of objects in the Kuiper Belt beyond Neptune. (Adaptado de New Scientist Space, August 2006) __________________________________________________________________________________________________________________________________________________________________________________________ Alan Stern, of Southwest Research Institute, states, “The definition is fundamentally flawed”. Find in the text one argument that can be used to support his opinion.

(IME - 2005/2006 - 1 FASE) Sejame termos de uma sequncia. Determine, em funo de n, os valores de r e s que tornam esta sequncia em uma PA, tal que r, se.

(IME 2006) Os náilons são polímeros usualmente empregados na forma de fios, úteis na fabricação de cordas, tecidos, linhas de pesca, etc. Um dos mais comuns é o náilon-66, resultante da reação de polimerização entre a hexametilenodiamina(1,6-diamino-n-hexano) e o ácido adípico (ácido hexanodioico). Com base nesta informação, DETERMINE a fórmula mínima do náilon-66.

(IME 2006) Leia o texto “New planet definition sparks furore” e responda, EM PORTUGUÊS, as perguntas que se seguem.  New planet definition sparks furore The new planet definition that relegates Pluto to "dwarf planet" status is drawing intense criticism from astronomers. It appears likely that the definition will not be widely adopted by astronomers for everyday use, even though it is the International Astronomical Union's (IAU) official position. On Thursday, astronomers at the IAU meeting in Prague approved a resolution that says the solar system has only eight planets, with Pluto excluded. Pluto is considered a "dwarf planet" instead. But the new definition has provoked a backlash. Alan Stern, who heads NASA's New Horizons mission to Pluto and works at the Southwest Research Institute in Boulder, Colorado, US, says the new definition is "awful". "The definition introduced is fundamentally flawed," he told New Scientist. "As a scientist, I'm embarrassed." Four planets He says only four of the eight objects mentioned in the IAU definition actually meet the definition's criteria – and Earth, Mars, Jupiter and Neptune do not. That is because the definition stipulates that to be a planet, an object must have cleared the neighbourhood around its orbit. But Earth's orbital neighbourhood is filled with thousands of near-Earth asteroids, Stern says. And Mars, Jupiter and Neptune have so-called "Trojan" asteroids sharing their orbits. "This is a half-baked criterion for planethood," he says. He says the new definition was pushed by people who are unhappy with having large numbers of planets (an earlier proposal, which was abandoned after heavy criticism at the meeting, would have potentially allowed hundreds of new planets into the fold). "It's just people that say things like, 'School kids will have to memorise too many names.' Do we limit the number of stars because children have to think of too many names? Or rivers on the Earth? It's just crazy," Stern told New Scientist. Minority report Stern is also critical of the fact that only astronomers present for the vote, which occurred at the end of the two-week meeting, were allowed to have their say on the matter. No email voting was allowed for the decision – it was made by a show of hands – and that meant that less than 5% of the nearly 9000 IAU members actually voted. "You're going to see a real backlash in the coming weeks," he says. "I know there is a petition among planetary scientists that's getting a lot of support." In any case, he says, astronomers are not obligated to use the new definition, since the IAU does not have the power to enforce it. "I don't think it's going to be very widely followed," he says. David Weintraub, author of the upcoming book Is Pluto a Planet? and a researcher at Vanderbilt University in Nashville, Tennessee, US, says he thinks Pluto is still a planet – even under the new definition.  Grammar issue "As best I can tell, 'dwarf' is an adjective and 'planet' is a noun," he told New Scientist. "I think the IAU thinks they defined Pluto to not be a planet. But they in fact have defined Pluto to be a planet – a particular kind of planet." Astronomers who proposed the new definition respond that the term "dwarf planet" is meant to be thought of as a single concept. And others point out that "minor planets" – asteroids and other small bodies – are not considered planets. But he agrees with Stern that the stipulation that a planet clears out its neighbourhood is flawed. A better definition would say a planet is an object that orbits a star and is large enough to be spherical, but is not large enough to be a brown dwarf – a "failed" star with between about 13 and 75 times the mass of Jupiter – or a star, he says.  'This is crazy'  "Everyone agrees on those criteria," he says. "The question is, can we agree on additional criteria to refine the definition further? I think the answer is no." "Everybody who has communicated with me is saying, 'This is crazy and we don't agree with it,'" he adds. "I'm not convinced that the folks who were at the meeting represented well the larger community." But not everyone is unhappy with the decision. Richard Conn Henry of Johns Hopkins University in Baltimore, Maryland, US, says he is pleased with the outcome. "As far as I'm concerned, the right decision was made," he told New Scientist. "I know a planet when I see it and there are eight of them." He says it makes no sense to call Pluto a planet because it is just one of huge numbers of objects in the Kuiper Belt beyond Neptune. (Adaptado de New Scientist Space, August 2006) __________________________________________________________________________________________________________________________________________________________________________________________ The scientist Alan Stern states that according to the criterion established by the resolution, Earth shouldn’t be considered a planet. Why wouldn’t Earth be considered a planet?

(IME 2006) Leia o texto “New planet definition sparks furore” e responda, EM PORTUGUÊS, as perguntas que se seguem.  New planet definition sparks furore The new planet definition that relegates Pluto to "dwarf planet" status is drawing intense criticism from astronomers. It appears likely that the definition will not be widely adopted by astronomers for everyday use, even though it is the International Astronomical Union's (IAU) official position. On Thursday, astronomers at the IAU meeting in Prague approved a resolution that says the solar system has only eight planets, with Pluto excluded. Pluto is considered a "dwarf planet" instead. But the new definition has provoked a backlash. Alan Stern, who heads NASA's New Horizons mission to Pluto and works at the Southwest Research Institute in Boulder, Colorado, US, says the new definition is "awful". "The definition introduced is fundamentally flawed," he told New Scientist. "As a scientist, I'm embarrassed." Four planets He says only four of the eight objects mentioned in the IAU definition actually meet the definition's criteria – and Earth, Mars, Jupiter and Neptune do not. That is because the definition stipulates that to be a planet, an object must have cleared the neighbourhood around its orbit. But Earth's orbital neighbourhood is filled with thousands of near-Earth asteroids, Stern says. And Mars, Jupiter and Neptune have so-called "Trojan" asteroids sharing their orbits. "This is a half-baked criterion for planethood," he says. He says the new definition was pushed by people who are unhappy with having large numbers of planets (an earlier proposal, which was abandoned after heavy criticism at the meeting, would have potentially allowed hundreds of new planets into the fold). "It's just people that say things like, 'School kids will have to memorise too many names.' Do we limit the number of stars because children have to think of too many names? Or rivers on the Earth? It's just crazy," Stern told New Scientist. Minority report Stern is also critical of the fact that only astronomers present for the vote, which occurred at the end of the two-week meeting, were allowed to have their say on the matter. No email voting was allowed for the decision – it was made by a show of hands – and that meant that less than 5% of the nearly 9000 IAU members actually voted. "You're going to see a real backlash in the coming weeks," he says. "I know there is a petition among planetary scientists that's getting a lot of support." In any case, he says, astronomers are not obligated to use the new definition, since the IAU does not have the power to enforce it. "I don't think it's going to be very widely followed," he says. David Weintraub, author of the upcoming book Is Pluto a Planet? and a researcher at Vanderbilt University in Nashville, Tennessee, US, says he thinks Pluto is still a planet – even under the new definition.  Grammar issue "As best I can tell, 'dwarf' is an adjective and 'planet' is a noun," he told New Scientist. "I think the IAU thinks they defined Pluto to not be a planet. But they in fact have defined Pluto to be a planet – a particular kind of planet." Astronomers who proposed the new definition respond that the term "dwarf planet" is meant to be thought of as a single concept. And others point out that "minor planets" – asteroids and other small bodies – are not considered planets. But he agrees with Stern that the stipulation that a planet clears out its neighbourhood is flawed. A better definition would say a planet is an object that orbits a star and is large enough to be spherical, but is not large enough to be a brown dwarf – a "failed" star with between about 13 and 75 times the mass of Jupiter – or a star, he says.  'This is crazy'  "Everyone agrees on those criteria," he says. "The question is, can we agree on additional criteria to refine the definition further? I think the answer is no." "Everybody who has communicated with me is saying, 'This is crazy and we don't agree with it,'" he adds. "I'm not convinced that the folks who were at the meeting represented well the larger community." But not everyone is unhappy with the decision. Richard Conn Henry of Johns Hopkins University in Baltimore, Maryland, US, says he is pleased with the outcome. "As far as I'm concerned, the right decision was made," he told New Scientist. "I know a planet when I see it and there are eight of them." He says it makes no sense to call Pluto a planet because it is just one of huge numbers of objects in the Kuiper Belt beyond Neptune. (Adaptado de New Scientist Space, August 2006) __________________________________________________________________________________________________________________________________________________________________________________________ Considering the content of the text, why is the title “New planet definition sparks furore” appropriate?  

Considere o sistema de equações dado por : Sendovalores reais quaisquer, a condição para que o sistema possua infinitas soluçõesé :

(IME - 2006/2007) O elemento constituinte da substncia simples A possui um nome que em grego significa verde. Livre, como molcula, um gs venenoso. Na crosta terrestre, encontra-se combinado a outros elementos, como minerais em depsitos subterrneos e em oceanos. solvel em gua e tambm em ter. Quando A reage com hidrxido de sdio em soluo aquosa, produz a substncia composta B, usada como agente alvejante e bactericida. Quando A reage com sdio fundido, produz a substncia composta C, que essencial ao ser humano. A eletrlise de C em soluo aquosa, produz no ctodo de ferro a substncia simples D. A substncia simples E o produto gasoso da reao, sob aquecimento, entre o sdio metlico e nitrato de sdio. Ao reagir E com D, produz-se a substncia composta F, utilizada na fabricao de cido ntrico, corantes, explosivos, medicamentos, detergentes e ainda, na forma de seus sais, como fertilizante. Determine: as frmulas moleculares de B, C, E e F; as equaes qumicas das reaes de produo de B, E e F; o nome e a frmula do composto produzido pela reao de F com cido ntrico em soluo aquosa.

(IME - 2006/2007 - 2 FASE) O quadriltero BRAS, de coordenadas A(1,0), B(-2,0), e construdo tal que . Sabendo que o ponto R pertence reta t de equao y=x+1, determine a equao algbrica do lugar geomtrico descrito pelo ponto S ao se deslocar R sobre t.

(IME - 2006/2007) Para a reao hipottica A + B Produtos, tem-se os seguintes dados Considerando a mesma reao, verificou-se tambm a seguinte correlao: Onde e so, respectivamente, as ordens da reao em relao a A e a B. Sabendo que / = 10,0, determine: a) a constante de velocidade k; b) os valores numricos das ordens parciais e global da reao