[ { "year": 2006, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $g$ a fun\u00e7\u00e3o definida em $\\mathbb{R}$ por $g(x) = \\frac{e^x + 5}{2 + \\cos x}$\nConsidere a sucess\u00e3o de termo geral $u_n = \\frac{n + 1}{n^2}$\nIndique o valor de $\\lim_{n \\to +\\infty} g(u_n)$", "answer": 2, "choices": [ "4", "3", "2", "1" ], "is_completion": false }, { "year": 2006, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $h$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}$, definida por\n$h(x) = \\frac{\\ln \\sqrt{e^x}}{2}$ ($\\ln$ designa logaritmo de base $e$)\nQual das seguintes express\u00f5es pode tamb\u00e9m definir $h$?", "answer": 2, "choices": [ "$\\sqrt{x}$", "$\\frac{x}{2}$", "$\\frac{x}{4}$", "$\\frac{\\sqrt{x}}{2}$" ], "is_completion": true }, { "year": 2006, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que $P(A) = 0.3$\nApenas um dos acontecimentos seguintes pode ter probabilidade inferior a 0.3. Qual deles?", "answer": 2, "choices": [ "$A \\cup B$", "$\\overline{A} \\cup B$", "$A \\cap B$", "$\\overline{A\\cap B}$" ], "is_completion": true }, { "year": 2006, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "Documento table1:\n{'headers': ['$x_i$', '0', '1'], 'rows': [{'$x_i$': '$P (X = x_i)$', '0': '$\\\\frac{{^{2005}}C_{99}}{{^{2006}}C_{100}}$', '1': '$\\\\frac{a}{{^{2006}}C_{100}}$'}]}\n\nUma vari\u00e1vel aleat\u00f3ria $X$ tem a seguinte distribui\u00e7\u00e3o de probabilidades:\n\nIndique o valor de $a$.", "answer": 1, "choices": [ "${^{2005}}C_{99}$", "${^{2005}}C_{100}$", "${^{2006}}C_{99}$", "${^{2006}}C_{100}$" ], "is_completion": true }, { "year": 2006, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "Documento table1:\n{'header': ['$x_i$', '0', '1', '2'], 'data': [['$P(X = x_i)$', 'a', 'a', '0,4']]}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria $X$ \u00e9\n($a$ designa um n\u00famero real).\nQual \u00e9 o valor m\u00e9dio desta vari\u00e1vel aleat\u00f3ria?", "answer": 0, "choices": [ "1,1", "1,2", "1,3", "1,4" ], "is_completion": true }, { "year": 2006, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nQuatro raparigas e quatro rapazes entram num autocarro, no qual existem seis lugares sentados, ainda n\u00e3o ocupados.\nDe quantas maneiras diferentes podem ficar ocupados esses seis lugares, supondo que ficam dois rapazes em p\u00e9?", "answer": 3, "choices": [ "3560", "3840", "4180", "4320" ], "is_completion": true }, { "year": 2007, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nIdentifique o valor de $\\lim_{x \\to 2^{+}} \\frac{1}{4-x^2}$", "answer": 3, "choices": [ "0", "1", "$+\\infty$", "$-\\infty$" ], "is_completion": false }, { "year": 2007, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSabendo que:\n$\\ln(x) - \\ln(e^{1/3}) > 0$ ($\\ln$ designa logaritmo na base $e$ ),\num valor poss\u00edvel para $x$ \u00e9:", "answer": 3, "choices": [ "0", "-1", "1", "2" ], "is_completion": true }, { "year": 2007, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}$.\nSabe-se que 3 \u00e9 um zero da fun\u00e7\u00e3o $f$.\nSeja $g$ a fun\u00e7\u00e3o definida por $g(x) = f(x \u2212 1) + 4$, para qualquer n\u00famero real $x$.\nQual dos seguintes pontos pertence garantidamente ao gr\u00e1fico da fun\u00e7\u00e3o $g$?", "answer": 1, "choices": [ "(2,4)", "(4,4)", "(4,8)", "(1,7)" ], "is_completion": true }, { "year": 2007, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nEscolhem-se, ao acaso, dois v\u00e9rtices diferentes de um paralelep\u00edpedo rect\u00e2ngulo.\nQual \u00e9 a probabilidade de que esses dois v\u00e9rtices sejam extremos de uma aresta?", "answer": 0, "choices": [ "$\\frac{12}{^8C_2}$", "$\\frac{12}{8^2}$", "$\\frac{8}{^8C_2}$", "$\\frac{8}{^8A_2}$" ], "is_completion": true }, { "year": 2007, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nAs cinco letras da palavra TIMOR foram pintadas, cada uma em sua bola.\nAs cinco bolas, indistingu\u00edveis ao tacto, foram introduzidas num saco.\nExtraem-se, aleatoriamente, as bolas do saco, sem reposi\u00e7\u00e3o, e colocam-se em fila, da esquerda para a direita.\nQual \u00e9 a probabilidade de que, no final do processo, fique formada a palavra TIMOR, sabendo-se que, ao fim da terceira extrac\u00e7\u00e3o, estava formada a sucess\u00e3o de letras TIM?", "answer": 2, "choices": [ "0", "$\\frac{1}{3}$", "$\\frac{1}{2}$", "1" ], "is_completion": true }, { "year": 2007, "phase": "1.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nQual das op\u00e7\u00f5es seguintes apresenta duas ra\u00edzes quadradas de um mesmo n\u00famero complexo?", "answer": 3, "choices": [ "1 e $i$", "-1 e $i$", "1-$i$ e 1+$i$", "1-$i$ e -1+$i$" ], "is_completion": true }, { "year": 2007, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nConsidere um rect\u00e2ngulo cuja \u00e1rea \u00e9 igual a 5.\nQual das seguintes express\u00f5es representa o per\u00edmetro deste rect\u00e2ngulo, em fun\u00e7\u00e3o do comprimento, $x$, de um dos seus lados?", "answer": 0, "choices": [ "$2x + \\frac{10}{x}$", "$2x + \\frac{2x}{5}$", "$2x + \\frac{5}{x}$", "$x + \\frac{5}{x}$" ], "is_completion": true }, { "year": 2007, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $f: [0, 2\\pi] \\rightarrow \\mathbb{R}$ a fun\u00e7\u00e3o definida por $f(x) = 3 - 2 cos x$.\nIndique o valor de $x$ para o qual $f(x)$ \u00e9 m\u00e1ximo.", "answer": 2, "choices": [ "0", "$\\frac{\\pi}{2}$", "$\\pi$", "$\\frac{3\\pi}{2}$" ], "is_completion": true }, { "year": 2007, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nDois cientistas, que v\u00e3o participar num congresso no estrangeiro, mandam reservar hotel na mesma cidade, cada um sem conhecimento da marca\u00e7\u00e3o feita pelo outro.\nSabendo que nessa cidade existem sete hot\u00e9is, todos com igual probabilidade de serem escolhidos, qual \u00e9 a probabilidade de os dois cientistas ficarem no mesmo hotel?", "answer": 0, "choices": [ "$\\frac{1}{7}$", "$\\frac{2}{7}$", "$\\frac{5}{7}$", "$\\frac{6}{7}$" ], "is_completion": true }, { "year": 2007, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nLan\u00e7aram-se dois dados, ambos com as faces numeradas de um a seis. Sabe-se que a soma dos n\u00fameros sa\u00eddos foi quatro.\nQual \u00e9 a probabilidade de ter sa\u00eddo o mesmo n\u00famero, em ambos os dados?", "answer": 2, "choices": [ "$\\frac{1}{5}$", "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$" ], "is_completion": true }, { "year": 2007, "phase": "2.\u00aa FASE", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, seja $i$ a unidade imagin\u00e1ria.\nSeja n um n\u00famero natural tal que $i^n = -i$.\nIndique qual dos seguintes \u00e9 o valor de $i^{n+1}$.", "answer": 0, "choices": [ "1", "$i$", "-1", "$-i$" ], "is_completion": true }, { "year": 2008, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nO Jo\u00e3o e a Maria convidaram tr\u00eas amigos para irem, com eles, ao cinema. Compraram cinco bilhetes com numera\u00e7\u00e3o seguida, numa determinada fila, e distribu\u00edram-nos ao acaso.\nQual \u00e9 a probabilidade de o Jo\u00e3o e a Maria ficarem sentados um ao lado do outro?", "answer": 1, "choices": [ "$\\frac{1}{5}$", "$\\frac{2}{5}$", "$\\frac{3}{5}$", "$\\frac{4}{5}$" ], "is_completion": true }, { "year": 2008, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria, e sejam $A$ e $B$ dois acontecimentos ($A\\subset\\Omega$ e $B\\subset\\Omega$). Sabe-se que:\n$P(A\\cup B) = 80\\%$\n$P(B) = 60\\%$\n$P(A \\cap B) = 10\\%$\nQual \u00e9 o valor de $P(A)$?", "answer": 2, "choices": [ "10%", "20%", "30%", "40%" ], "is_completion": true }, { "year": 2008, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nAdmita que a vari\u00e1vel peso, expressa em gramas, das ma\u00e7\u00e3s de um pomar \u00e9 bem modelada por uma distribui\u00e7\u00e3o normal $\\mathcal{N} (60; 5)$, em que 60 \u00e9 o valor m\u00e9dio e 5 \u00e9 o valor do desvio-padr\u00e3o da distribui\u00e7\u00e3o.\nRetira-se, ao acaso, uma dessas ma\u00e7\u00e3s.\nConsidere os acontecimentos:\nA: \u00abo peso da ma\u00e7\u00e3 retirada \u00e9 superior a 66 gramas\u00bb\nB: \u00abo peso da ma\u00e7\u00e3 retirada \u00e9 inferior a 48 gramas\u00bb\nQual das seguintes afirma\u00e7\u00f5es \u00e9 verdadeira?", "answer": 2, "choices": [ "$P(A) = P(B)$", "$P(A) < P(B)$", "$P(B) < P(A)$", "$P(A) + P(B) = 1$" ], "is_completion": true }, { "year": 2008, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $a$ um n\u00famero real maior do que 1.\nQual dos seguintes valores \u00e9 igual a $2\\log_a(a^{1/3})$?", "answer": 3, "choices": [ "$-\\frac{2}{3}$", "$-\\frac{1}{3}$", "$\\frac{1}{3}$", "$\\frac{2}{3}$" ], "is_completion": true }, { "year": 2008, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSeja $z = 3i$ um n\u00famero complexo.\nQual dos seguintes valores \u00e9 um argumento de $z$?", "answer": 1, "choices": [ "0", "$\\frac{1}{2} \\pi$", "$\\pi$", "$\\frac{3}{2} \\pi$" ], "is_completion": true }, { "year": 2008, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nAo disputar um torneio de tiro ao alvo, o Jo\u00e3o tem de atirar sobre o alvo quatro vezes. Sabe-se que, em cada tiro, a probabilidade de o Jo\u00e3o acertar no alvo \u00e9 $0,8$.\nQual \u00e9 a probabilidade de o Jo\u00e3o acertar sempre no alvo, nas quatro vezes em que tem de atirar?", "answer": 3, "choices": [ "$0,0016$", "$0,0064$", "$0,0819$", "$0,4096$" ], "is_completion": true }, { "year": 2008, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nUma caixa A cont\u00e9m duas bolas verdes e uma bola amarela. Outra caixa B cont\u00e9m uma bola verde e tr\u00eas bolas amarelas. As bolas colocadas nas caixas A e B s\u00e3o indistingu\u00edveis ao tacto.\nLan\u00e7a-se um dado c\u00fabico perfeito, com as faces numeradas de 1 a 6. Se sair o n\u00famero 5, tira-se uma bola da caixa A; caso contr\u00e1rio, tira-se uma bola da caixa B.\nQual \u00e9 a probabilidade de a bola retirada ser verde, sabendo que saiu o n\u00famero 5 no lan\u00e7amento do dado?", "answer": 3, "choices": [ "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{3}{7}$", "$\\frac{2}{3}$" ], "is_completion": true }, { "year": 2008, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nUma linha do Tri\u00e2ngulo de Pascal tem quinze elementos.\nQuantos elementos dessa linha s\u00e3o inferiores a 100?", "answer": 2, "choices": [ "3", "4", "6", "8" ], "is_completion": true }, { "year": 2008, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSabe-se que o ponto $P(1, 3)$ pertence ao gr\u00e1fico da fun\u00e7\u00e3o $f(x) = 2^{ax} - 1$, com $a \\in \\mathbb{R}$.\nQual \u00e9 o valor de $a$?", "answer": 0, "choices": [ "2", "1", "0", "-2" ], "is_completion": true }, { "year": 2008, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSeja $z$ um n\u00famero complexo de argumento $\\frac{\\pi}{6}$.\nQual dos seguintes valores \u00e9 um argumento de $(-z)$?", "answer": 3, "choices": [ "$\\frac{-\\pi}{6}$", "$\\frac{5\\pi}{6}$", "$\\pi$", "$\\frac{7\\pi}{6}$" ], "is_completion": true }, { "year": 2009, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nDe um baralho com 40 cartas, repartidas por quatro naipes (Copas, Ouros, Espadas e Paus), em que cada naipe cont\u00e9m um \u00c1s, uma Dama, um Valete, um Rei e seis cartas (do Dois ao Sete), foram dadas sucessivamente, ao acaso, seis cartas a um jogador, que as coloca na m\u00e3o, pela ordem que as recebe.\nQual \u00e9 a probabilidade de o jogador obter a sequ\u00eancia 2 \u2013 4 \u2013 6 \u2013 7 \u2013 Dama \u2013 Rei, nas cartas recebidas?", "answer": 0, "choices": [ "$\\frac{4^6}{^{40} A_6}$", "$\\frac{4^6}{^{40} C_6}$", "$\\frac{1}{^{40} A_6}$", "$\\frac{1}{^{40} C_6}$" ], "is_completion": true }, { "year": 2009, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria. Sejam A e B dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n\u2022 $P(A) = 0,3$\n\u2022 $P(B) = 0,4$\n\u2022 $P(A \\cup B) = 0,5$\n(P designa probabilidade.)\nQual \u00e9 a probabilidade de se realizar A, sabendo que B se realiza?", "answer": 3, "choices": [ "$\\frac{1}{6}$", "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$" ], "is_completion": true }, { "year": 2009, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "Documento tabela_1:\n{'headers': ['$x_i$', '4', '5', '6'], 'rows': [['$P(X = x_i)$', '$\\\\frac{k}{8}$', '$\\\\frac{1}{4}$', '$\\\\frac{k}{4}$']]}\n\nConsidere uma vari\u00e1vel aleat\u00f3ria $X$, cuja distribui\u00e7\u00e3o de probabilidades \u00e9 dada pela tabela seguinte.\nQual \u00e9 o valor de $k$?", "answer": 1, "choices": [ "1", "2", "3", "4" ], "is_completion": true }, { "year": 2009, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $x$ um n\u00famero real positivo.\nQual das express\u00f5es seguintes \u00e9 igual a $e^{4 \\ln x}-10^{2 \\log x}$?\n($\\ln$ designa logaritmo de base $e$; $\\log$ designa logaritmo de base 10.)", "answer": 2, "choices": [ "$\\ln x^4 - \\log x^2$", "$x^4 + x^2$", "$x^4 - x^2$", "$\\frac{\\ln x^4}{\\log x^2}$" ], "is_completion": true }, { "year": 2009, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "Documento funcao_f:\n$f(x) = \\begin{cases} log_2(k+x) &\\text{se } x\\geq 0 \\\\ \\frac{sen(2x)}{x} &\\text{se } x < 0 \\end{cases}$\n\nPara um certo n\u00famero real positivo $k$, \u00e9 cont\u00ednua a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}$, definida por: [Ver documento]\nQual \u00e9 o valor de $k$?", "answer": 3, "choices": [ "1", "2", "3", "4" ], "is_completion": true }, { "year": 2009, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSeja $z$ um n\u00famero complexo, em que um dos argumentos \u00e9 $\\frac{\\pi}{3}$.\nQual dos valores seguintes \u00e9 um argumento de $\\frac{2i}{\\overline{z}}$, sendo $\\overline{z}$ o conjugado de $z$?", "answer": 2, "choices": [ "$\\frac{\\pi}{6}$", "$\\frac{2}{3}\\pi$", "$\\frac{5}{6}\\pi$", "$\\frac{7}{6}\\pi$" ], "is_completion": true }, { "year": 2009, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nA Maria gravou nove CD, sete com m\u00fasica rock e dois com m\u00fasica popular, mas esqueceu-se de identificar cada um deles.\nQual \u00e9 a probabilidade de, ao escolher dois CD ao acaso, um ser de m\u00fasica rock e o outro ser de m\u00fasica popular?", "answer": 3, "choices": [ "$\\frac{7}{36}$", "$\\frac{1}{4}$", "$\\frac{2}{9}$", "$\\frac{7}{18}$" ], "is_completion": true }, { "year": 2009, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nAdmita que um estudante tem de realizar dois testes no mesmo dia. A probabilidade de ter classifica\u00e7\u00e3o positiva no primeiro teste \u00e9 0,7, a de ter classifica\u00e7\u00e3o positiva no segundo teste \u00e9 0,8, e a de ter classifica\u00e7\u00e3o negativa em ambos os testes \u00e9 0,1.\nQual \u00e9 a probabilidade de o estudante ter negativa no segundo teste, sabendo que teve negativa no primeiro teste?", "answer": 2, "choices": [ "$\\frac{1}{8}$", "$\\frac{1}{7}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$" ], "is_completion": true }, { "year": 2009, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nUma certa linha do Tri\u00e2ngulo de Pascal \u00e9 constitu\u00edda por todos os elementos da forma $^{14}C_p$.\nEscolhido, ao acaso, um elemento dessa linha, qual \u00e9 a probabilidade de ele ser o n\u00famero 14?", "answer": 2, "choices": [ "$\\frac{1}{15}$", "$\\frac{1}{14}$", "$\\frac{2}{15}$", "$\\frac{4}{15}$" ], "is_completion": true }, { "year": 2009, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}$, definida por $f(x) = e^{x+1}$.\nQual dos pontos seguintes pertence ao gr\u00e1fico de $f$?\n($\\ln$ designa logaritmo de base $e$.)", "answer": 1, "choices": [ "$(-1, 0)$", "$(\\ln 2, 2e)$", "$(\\ln 5, 6)$", "$(-2, e)$" ], "is_completion": true }, { "year": 2009, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSeja $k$ um n\u00famero real, e $z_1 = (k \u2013 i) (3 \u2013 2i)$ um n\u00famero complexo.\nQual \u00e9 o valor de $k$, para que $z_1$ seja um n\u00famero imagin\u00e1rio puro?", "answer": 2, "choices": [ "$-\\frac{3}{2}$", "$-\\frac{2}{3}$", "$\\frac{2}{3}$", "$\\frac{3}{2}$" ], "is_completion": true }, { "year": 2010, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria, e sejam A e B dois acontecimentos ($A \\subset \\Omega \\text{ e } B \\subset \\Omega$).\nSabe-se que:\n* $P(A) = 30\\%$;\n* $P(A \\cup B) = 70\\%$;\n* A e B s\u00e3o incompat\u00edveis.\nQual \u00e9 o valor de P(B)?", "answer": 1, "choices": [ "21%", "40%", "60%", "61%" ], "is_completion": true }, { "year": 2010, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nNum grupo de dez trabalhadores de uma f\u00e1brica, v\u00e3o ser escolhidos tr\u00eas, ao acaso, para frequentarem uma ac\u00e7\u00e3o de forma\u00e7\u00e3o. Nesse grupo de dez trabalhadores, h\u00e1 tr\u00eas amigos, o Jo\u00e3o, o Ant\u00f3nio e o Manuel, que gostariam de frequentar essa ac\u00e7\u00e3o.\nQual \u00e9 a probabilidade de serem escolhidos, exactamente, os tr\u00eas amigos?", "answer": 2, "choices": [ "$\\frac{1}{^{10}A_3}$", "$\\frac{3}{^{10}A_3}$", "$\\frac{1}{^{10}C_3}$", "$\\frac{3}{^{10}C_3}$" ], "is_completion": true }, { "year": 2010, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "Documento tabela_1:\n{'headers': ['$x_i$', '0', '1', '2', '3'], 'rows': [['$P(X = x_i)$', '$\\\\frac{1}{5}$', '$\\\\frac{1}{2}$', '$2a$', '$a$']]}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria X \u00e9 a seguinte.", "answer": 1, "choices": [ "$P(X = 0) = P(X > 1)$", "$P(X = 0) = P(X = 2)$", "$P(X = 0) = P(X = 3)$", "$P(X < 2) = P(X = 3)$" ], "is_completion": true }, { "year": 2010, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nSeja $g$ a fun\u00e7\u00e3o, de dom\u00ednio $]-2, +\\infty[$, definida por $g(x) = \\ln(x + 2)$.\nConsidere, num referencial o.n. $*xOy*$, um tri\u00e2ngulo $[OAB]$ tal que:\n* $O$ \u00e9 a origem do referencial;\n* $A$ \u00e9 um ponto de ordenada 5;\n* $B$ \u00e9 o ponto de intersec\u00e7\u00e3o do gr\u00e1fico da fun\u00e7\u00e3o $g$ com o eixo das abcissas.\nQual \u00e9 a \u00e1rea do tri\u00e2ngulo $[OAB]$?", "answer": 0, "choices": [ "$\\frac{5}{2}$", "$\\frac{1}{2}$", "$\\frac{5 \\ln 2}{2}$", "$\\frac{\\ln 2}{2}$" ], "is_completion": true }, { "year": 2010, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, considere $z = 3 cis(\\frac{\\pi}{8} - \\theta)$, com $\\theta \\in \\mathbb{R}$.\nPara qual dos valores seguintes de $\\theta$ podemos afirmar que $z$ \u00e9 um n\u00famero imagin\u00e1rio puro?", "answer": 3, "choices": [ "$\\frac{-\\pi}{2}$", "$\\frac{\\pi}{2}$", "$\\frac{\\pi}{8}$", "$\\frac{5\\pi}{8}$" ], "is_completion": true }, { "year": 2010, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nUma caixa cont\u00e9m bolas indistingu\u00edveis ao tacto e de duas cores diferentes: azul e roxo.\nSabe-se que:\n\u2022 o n\u00famero de bolas azuis \u00e9 8\n\u2022 extraindo-se, ao acaso, uma bola da caixa, a probabilidade de ela ser azul \u00e9 igual a $\\frac{1}{2}$\nQuantas bolas roxas h\u00e1 na caixa?", "answer": 2, "choices": [ "16", "12", "8", "4" ], "is_completion": true }, { "year": 2010, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nConsidere todos os n\u00fameros de cinco algarismos que se podem formar com os algarismos 5, 6, 7, 8 e 9.\nDe entre estes n\u00fameros, quantos t\u00eam, exactamente, tr\u00eas algarismos 5?", "answer": 1, "choices": [ "${}^5C_3 \\times {}^4A_2$", "${}^5C_3\\times 4^2$", "${}^5A_3 \\times 4^2$", "${}^5A_3 \\times {}^4C_2$" ], "is_completion": true }, { "year": 2010, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nNa sequ\u00eancia seguinte, reproduzem-se os tr\u00eas primeiros elementos e os tr\u00eas \u00faltimos elementos de uma linha do Tri\u00e2ngulo de Pascal.\n1 15 105 ... 105 15 1\nS\u00e3o escolhidos, ao acaso, dois elementos dessa linha.\nQual \u00e9 a probabilidade de a soma desses dois elementos ser igual a 105?", "answer": 3, "choices": [ "1", "$\\frac{1}{60}$", "$\\frac{1}{120}$", "0" ], "is_completion": true }, { "year": 2010, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nDe uma fun\u00e7\u00e3o $h$, de dom\u00ednio $\\mathbb{R}$, sabe-se que:\n\u2022 h\u00e9 uma fun\u00e7\u00e3o par;\n\u2022 $\\lim_{x\\to+\\infty} (h(x) - 2x) = 0$\nQual \u00e9 o valor de $\\lim_{x\\to-\\infty} h(x)$?", "answer": 0, "choices": [ "$+\\infty$", "$-2$", "$0$", "$-\\infty$" ], "is_completion": true }, { "year": 2010, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nConsidere a fun\u00e7\u00e3o $g$, de dom\u00ednio $\\mathbb{R}$, definida por\n$g(x) = \\begin{cases} e^x & \\text{se } x \\leq 0 \\\\ \\ln x & \\text{se } x > 0 \\end{cases}$\nConsidere a sucess\u00e3o de termo geral $u_n = \\frac{1}{n}$\nQual \u00e9 o valor de $\\lim_{n\\to+\\infty} g(u_n)$?", "answer": 3, "choices": [ "$+\\infty$", "$1$", "$0$", "$-\\infty$" ], "is_completion": true }, { "year": 2011, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam A e B dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$) independentes, com $P(A)\neq0$\nQual das afirma\u00e7\u00f5es seguintes \u00e9 necessariamente verdadeira?", "answer": 3, "choices": [ "$P(A) + P(B) = 1$", "$P(A\\cup B) = P(A) + P(B)$", "$P(A) \neq P(B)$", "$P(B|A) = P(B)$" ], "is_completion": true }, { "year": 2011, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nO c\u00f3digo de um auto-r\u00e1dio \u00e9 constitu\u00eddo por uma sequ\u00eancia de quatro algarismos. Por exemplo, 0137\nQuantos desses c\u00f3digos t\u00eam dois e s\u00f3 dois algarismos iguais a 7?", "answer": 0, "choices": [ "486", "810", "432", "600" ], "is_completion": true }, { "year": 2011, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $[0, +\\infty[$, definida por\n$f(x) = \\begin{cases} 2^x-9 & \\text{se } 0\\leq x<5 \\\\ \\frac{1-e^x}{x} & \\text{se } x\\geq5 \\end{cases}$\nEm qual dos intervalos seguintes o teorema de Bolzano permite garantir a exist\u00eancia de, pelo menos, um zero da fun\u00e7\u00e3o $f$?", "answer": 1, "choices": [ "$]0,1[$", "$]1,4[$", "$]4,6[$", "$]6,7[$" ], "is_completion": true }, { "year": 2011, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nQual \u00e9 o valor de $\\lim_{x \\to 0} (\\frac{1}{x^2} sen^2(\\frac{x}{2}))$?", "answer": 2, "choices": [ "4", "0", "$\\frac{1}{4}$", "$\\frac{1}{2}$" ], "is_completion": true }, { "year": 2011, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nOs medicamentos produzidos num laborat\u00f3rio s\u00e3o embalados em caixas de igual aspecto exterior e indistingu\u00edveis ao tacto. Um lote cont\u00e9m dez caixas de um medicamento X e vinte caixas de um medicamento Y. Desse lote, retiram-se, ao acaso, simultaneamente, quatro caixas para controlo de qualidade.\nQual \u00e9 a probabilidade de as caixas retiradas serem todas do medicamento Y?", "answer": 1, "choices": [ "$\\frac{{10} C_4}{{30} C_4}$", "$\\frac{{20} C_4}{{30} C_4}$", "$\\frac{4}{{30} C_4}$", "$\\binom{2}{3}^4$" ], "is_completion": true }, { "year": 2011, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "Documento tabela1:\n{'$x_i$': ['0', '1', '2', '3', '4', '5'], '$P(X = x_i$)': ['$2a$', '$a$', '$b$', '$b$', '$b$', '$\\\\frac{1}{10}$']}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria X \u00e9 a seguinte.\nSabe-se que:\n\u2022 $a$ e $b$ s\u00e3o n\u00fameros reais\n\u2022 $P(X \\le 1) = 3P(X = 5)$\nQual \u00e9 o valor de $b$?", "answer": 3, "choices": [ "$\\frac{1}{10}$", "$\\frac{4}{15}$", "$\\frac{7}{30}$", "$\\frac{1}{5}$" ], "is_completion": true }, { "year": 2011, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $a$ um n\u00famero real positivo e seja $X$ uma vari\u00e1vel aleat\u00f3ria com distribui\u00e7\u00e3o Normal $N(0, 1)$.\nQual das igualdades seguintes \u00e9 verdadeira?", "answer": 1, "choices": [ "$P(X \\le a) + P(X \\ge -a) = 0$", "$P(X \\le a) = P(X \\ge -a)$", "$P(X \\le a) + P(X \\ge -a) = 1$", "$P(X \\le a) = P(X > a)$" ], "is_completion": true }, { "year": 2011, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nPara um certo n\u00famero real positivo, $k$, a fun\u00e7\u00e3o $g$ definida em $\\mathbb{R}$ por\n$g(x) = \\begin{cases} \\frac{\\text{sen} x}{3x} & \\text{se } x>0 \\\\ \\ln(k-x) & \\text{se } x \\leq 0 \\end{cases}$ \u00e9 cont\u00ednua.\nQual \u00e9 o valor de $k$?", "answer": 0, "choices": [ "$^3\\sqrt{e}$", "$e^3$", "$\\frac{e}{3}$", "$3e$" ], "is_completion": true }, { "year": 2011, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nNum determinado clube desportivo praticam-se apenas dois desportos, futebol e andebol. Dos jovens inscritos nesse clube, 28 jogam apenas futebol, 12 jogam apenas andebol e 12 jogam futebol e andebol. Escolhe-se, ao acaso, um dos jovens inscritos.\nQual \u00e9 a probabilidade de o jovem escolhido jogar andebol sabendo que joga futebol?", "answer": 1, "choices": [ "$\\frac{1}{2}$", "$\\frac{3}{10}$", "$\\frac{7}{10}$", "$\\frac{3}{7}$" ], "is_completion": true }, { "year": 2011, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nLan\u00e7a-se cinco vezes consecutivas um dado equilibrado, com as faces numeradas de 1 a 6, e regista-se, em cada lan\u00e7amento, o n\u00famero inscrito na face voltada para cima.\nConsidere os acontecimentos seguintes.\n$I$: \u00absair face \u00edmpar em exactamente dois dos cinco lan\u00e7amentos\u00bb;\n$J$: \u00absair face 4 em exactamente dois dos cinco lan\u00e7amentos\u00bb.\nQual dos acontecimentos seguintes \u00e9 mais prov\u00e1vel?", "answer": 3, "choices": [ "acontecimento $I$", "acontecimento $\\overline{I}$", "acontecimento $J$", "acontecimento $\\overline{J}$" ], "is_completion": true }, { "year": 2011, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam A e B dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$) incompat\u00edveis.\nSabe-se que $P(\\overline{A} \\cap \\overline{B}) = 0,3$ e que $P(A) = 0,5$\nQual \u00e9 o valor de $P(B)$?", "answer": 0, "choices": [ "0,2", "0", "0,5", "0,4" ], "is_completion": true }, { "year": 2011, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nConsidere uma fun\u00e7\u00e3o f, de dom\u00ednio $\\mathbb{R}\\setminus\\{3\\}$, cont\u00ednua em todo o seu dom\u00ednio.\nSabe-se que:\n\u2022 $\\lim_{x \\to +\\infty} f(x) = 1$\n\u2022 $\\lim_{x \\to 3} f(x) = -2$\n\u2022 $\\lim_{x \\to -\\infty} (f(x) + 2x) = 0$\nEm qual das op\u00e7\u00f5es seguintes as equa\u00e7\u00f5es definem duas assimptotas do gr\u00e1fico de $f$?", "answer": 2, "choices": [ "$x = -2$ e$y = 1$", "$x = 3$ e $y = -2x$", "$y = -2x$ e $y = 1$", "$y = 2x$ e $y = -1$" ], "is_completion": true }, { "year": 2011, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSejam $k$ e $p$ dois n\u00fameros reais e sejam $z_1 = (3k+2) + p i$ e $z_2 = (3p-4) + (2-5k)i$ dois n\u00fameros complexos.\nQuais s\u00e3o os valores de $k$ e de $p$ para os quais $z_1$ \u00e9 igual ao conjugado de $z_2$ ?", "answer": 1, "choices": [ "$k = -1$ e $p = 3$", "$k = 1$ e $p = 3$", "$k = 0$ e $p = -2$", "$k = 1$ e $p = -3$" ], "is_completion": true }, { "year": 2012, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria, e sejam A e B dois acontecimentos (A$\\subset\\Omega$ e B$\\subset\\Omega$). Sabe-se que:\n\u2022 A e B s\u00e3o acontecimentos independentes;\n\u2022 $P(\\overline{A}) = \\frac{7}{10}$\n\u2022$P(A\\cup B)=\\frac{3}{4}$\nQual \u00e9 o valor de $P(B)$?", "answer": 1, "choices": [ "$\\frac{5}{14}$", "$\\frac{9}{14}$", "$\\frac{9}{20}$", "$\\frac{11}{20}$" ], "is_completion": true }, { "year": 2012, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nPara assistirem a um espet\u00e1culo, o Jo\u00e3o, a Margarida e cinco amigos sentam-se, ao acaso, numa fila com sete lugares. Qual \u00e9 a probabilidade de o Jo\u00e3o e a Margarida n\u00e3o ficarem sentados um ao lado do outro?", "answer": 3, "choices": [ "$\\frac{2 \\times 5!}{7!}$", "$\\frac{5!}{7!}$", "$\\frac{2}{7}$", "$\\frac{5}{7}$" ], "is_completion": true }, { "year": 2012, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nNuma caixa com 12 compartimentos, pretende-se arrumar 10 copos, com tamanho e forma iguais: sete brancos, um verde, um azul e um roxo. Em cada compartimento pode ser arrumado apenas um copo. De quantas maneiras diferentes se podem arrumar os 10 copos nessa caixa?", "answer": 2, "choices": [ "$^{12}A_7 \\times 3!$", "$^{12}A_7 \\times ^5C_3$", "$^{12}C_7 \\times ^5A_3$", "$^{12}C_7 \\times ^{12}A_3$" ], "is_completion": true }, { "year": 2012, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}$, definida por $f(x) = e^{x}-3$. Em qual dos intervalos seguintes o teorema de Bolzano permite afirmar que a equa\u00e7\u00e3o $f(x)=-x-\\frac{3}{2}$ tem, pelo menos, uma solu\u00e7\u00e3o?", "answer": 1, "choices": [ "$]0,\\frac{1}{5}[$", "$] \\frac{1}{5}, \\frac{1}{4} [$", "$]\\frac{1}{4}, \\frac{1}{3}[$", "$]\\frac{1}{3}, 1[$" ], "is_completion": true }, { "year": 2012, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nO c\u00f3digo de acesso a uma conta de e-mail \u00e9 constitu\u00eddo por quatro letras e tr\u00eas algarismos. Sabe-se que um c\u00f3digo tem quatro \u00aba\u00bb, dois \u00ab5\u00bb e um \u00ab2\u00bb, como, por exemplo, o c\u00f3digo 2aa5a5a\nQuantos c\u00f3digos diferentes existem nestas condi\u00e7\u00f5es?", "answer": 0, "choices": [ "105", "210", "5040", "39" ], "is_completion": true }, { "year": 2012, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "Documento table_1:\n{'headers': ['$x_i$', '0', '1', '2'], 'rows': [['$P(X = x_i)$', '$b^3$', 'a', '2a']]}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria $X$ \u00e9 a seguinte.\nSabe-se que:\n\u2022 $a$ e $b$ s\u00e3o n\u00fameros reais;\n\u2022 o valor m\u00e9dio da vari\u00e1vel aleat\u00f3ria $X$ \u00e9 $\\frac{35}{24}$\nQual \u00e9 o valor de $b$?", "answer": 2, "choices": [ "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$", "$\\frac{1}{5}$" ], "is_completion": true }, { "year": 2012, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nNuma certa linha do tri\u00e2ngulo de Pascal, o pen\u00faltimo elemento \u00e9 111\nEscolhe-se, ao acaso, um elemento dessa linha.\nQual \u00e9 a probabilidade de esse elemento ser maior do que $10^5$?", "answer": 1, "choices": [ "$\\frac{3}{56}$", "$\\frac{53}{56}$", "$\\frac{2}{37}$", "$\\frac{35}{37}$" ], "is_completion": true }, { "year": 2012, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}$\nSabe-se que:\n\u2022 $lim_{x \\to +\\infty} (f(x)-2x)=1$\n\u2022 $lim_{x \\to -\\infty} f(x) = 3$\n\u2022 $lim_{x \\to 1^+} f(x) = +\\infty$\n\u2022 $lim_{x \\to 1^-} f(x) = 2$\nEm qual das op\u00e7\u00f5es seguintes as duas equa\u00e7\u00f5es definem ass\u00edntotas do gr\u00e1fico da fun\u00e7\u00e3o $f$?", "answer": 1, "choices": [ "$x=1$ e $y = -2x+1$", "$x = 1$ e $y = 2x + 1$", "$y = 3$ e $y = -2x+1$", "$y = 2$ e $y = 2x + 1$" ], "is_completion": true }, { "year": 2012, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSeja $k$ um n\u00famero real, e sejam $z\u2081 = 2 + i$ e $z\u2082 =3-ki$ dois n\u00fameros complexos.\nQual \u00e9 o valor de $k$ para o qual $z_1\\times \\overline{z_2}$ \u00e9 um imagin\u00e1rio puro?", "answer": 3, "choices": [ "$\\frac{3}{2}$", "-$\\frac{3}{2}$", "1", "6" ], "is_completion": true }, { "year": 2012, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nUma sequ\u00eancia de algarismos cuja leitura da direita para a esquerda ou da esquerda para a direita d\u00e1 o mesmo n\u00famero designa-se por capicua. Por exemplo, 103301 \u00e9 capicua.\nQuantos n\u00fameros com seis algarismos s\u00e3o capicuas?", "answer": 1, "choices": [ "729", "900", "810 000", "900 000" ], "is_completion": true }, { "year": 2012, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "Documento table_1:\n{'headers': ['$x_i$', '-1', '0', '1'], 'rows': [['$P(X = x_i)$', '1 - 3$a$', '2$a$', '$a$']]}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria $X$ \u00e9 a seguinte.\nSabe-se que $P(X=0 \\vee X=1) = 0,81$\nQual \u00e9 o valor m\u00e9dio de $X$?", "answer": 2, "choices": [ "0,46", "0,27", "0,08", "0" ], "is_completion": true }, { "year": 2012, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nConsidere um dado c\u00fabico, com as faces numeradas de 1 a 6, e um saco que cont\u00e9m cinco bolas, indistingu\u00edveis ao tato, cada uma delas numerada com um n\u00famero diferente: 0, 1, 2, 3 e 4.\nLan\u00e7a-se o dado uma vez e retira-se, ao acaso, uma bola do saco, registando-se os n\u00fameros que sa\u00edram.\nQual \u00e9 a probabilidade de o produto desses n\u00fameros ser igual a zero?", "answer": 3, "choices": [ "0", "$\\frac{1}{15}$", "$\\frac{1}{30}$", "$\\frac{1}{5}$" ], "is_completion": true }, { "year": 2012, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSejam $f$ e $g$ fun\u00e7\u00f5es de dom\u00ednio $]0, +\\infty[$.\nSabe-se que:\n\u2022 a reta de equa\u00e7\u00e3o $y = 3$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $f$\n\u2022 $f$ n\u00e3o tem zeros;\n\u2022 $g(x) = \\frac{e^{-x}-3}{f(x)}$\nQual das op\u00e7\u00f5es seguintes define uma ass\u00edntota horizontal do gr\u00e1fico de $g$?", "answer": 3, "choices": [ "$y = 3$", "$y = e$", "$y = 0$", "$y = -1$" ], "is_completion": true }, { "year": 2012, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nSejam $a$, $b$ e $c$ tr\u00eas n\u00fameros tais que $a \\in ]1, +\\infty[$, $b \\in R^+$ e $c \\in R^+$.\nSabe-se que $\\log_a b = c$ e que $\\log_a \\sqrt{c} = 3$\nQual das express\u00f5es seguintes \u00e9 equivalente a $\\log_a \\sqrt{b \\times c}$?", "answer": 2, "choices": [ "$c + 3$", "$c - 3$", "$\\frac{c}{2} + 3$", "$\\frac{c}{2} - 3$" ], "is_completion": true }, { "year": 2012, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nSejam $k$ e $p$ dois n\u00fameros reais tais que os n\u00fameros complexos $z = 1 + i$ e $w = (k - 1) + 2pi$ sejam inversos um do outro.\nQual \u00e9 o valor de $k + p$?", "answer": 3, "choices": [ "$-\\frac{1}{4}$", "$\\frac{1}{2}$", "$\\frac{5}{4}$", "$\\frac{7}{4}$" ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nNum grupo de nove pessoas, constitu\u00eddo por seis homens e tr\u00eas mulheres, v\u00e3o ser escolhidos tr\u00eas elementos para formarem uma comiss\u00e3o.\nQuantas comiss\u00f5es diferentes se podem formar com exatamente duas mulheres?", "answer": 1, "choices": [ "$^3C_2$", "$6 \\times ^3C_2$", "$^9A_3$", "$6 \\times ^3A_2$" ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria X \u00e9 a seguinte.\nSabe-se que:\n\u2022 $a$ e $b$ s\u00e3o n\u00fameros reais;\n\u2022 $P(X > 1) = P(X < 2)$\nQual \u00e9 o valor m\u00e9dio da vari\u00e1vel aleat\u00f3ria $X$?", "answer": 3, "choices": [ "$\\frac{3}{2}$", "$\\frac{7}{5}$", "$\\frac{17}{9}$", "$\\frac{19}{12}$" ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nConsidere uma vari\u00e1vel aleat\u00f3ria $X$ com distribui\u00e7\u00e3o normal de valor m\u00e9dio 11 e desvio padr\u00e3o $\\sigma$\nSabe-se que $\\sigma$ \u00e9 um n\u00famero natural e que $P(X > 23) \\approx 0,02275$\nQual \u00e9 o valor de $\\sigma$?", "answer": 2, "choices": [ "12", "11", "6", "4" ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R} \\setminus \\{0\\}$, definida por $f(x) = \\frac{\\operatorname{sen}(-x)}{x}$\nConsidere a sucess\u00e3o de n\u00fameros reais $(x_n)$ tal que $x_n = \\frac{1}{n}$\nQual \u00e9 o valor de $\\lim f(x_n)$?", "answer": 0, "choices": [ "-1", "0", "1", "+$\\infty$" ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}^+$\nSabe-se que $\\lim_{x \\to +\\infty} \\frac{\\ln x + f(x)}{3x} = 1$\nQual das equa\u00e7\u00f5es seguintes pode definir uma ass\u00edntota do gr\u00e1fico da fun\u00e7\u00e3o $f$?", "answer": 3, "choices": [ "$y = \\frac{1}{3}x$", "$y = \\frac{2}{3}x$", "$y = x$", "$y = 3x$" ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nConsidere, para um certo n\u00famero real $a$ superior a 1, as fun\u00e7\u00f5es $f$ e $g$, de dom\u00ednio $\\mathbb{R}$, definidas por $f(x) = a^x$ e $g(x) = a^{-x}$\nConsidere as afirma\u00e7\u00f5es seguintes.\nI) Os gr\u00e1ficos das fun\u00e7\u00f5es $f$ e $g$ n\u00e3o se intersectam.\nII) As fun\u00e7\u00f5es $f$ e $g$ s\u00e3o mon\u00f3tonas crescentes.\nIII) $f'(-1) - g'(1) = \\frac{2 \\ln a}{a}$\nQual das op\u00e7\u00f5es seguintes \u00e9 a correta?", "answer": 1, "choices": [ "II e III s\u00e3o verdadeiras.", "I \u00e9 falsa e III \u00e9 verdadeira.", "I \u00e9 verdadeira e III \u00e9 falsa.", "II e III s\u00e3o falsas." ], "is_completion": true }, { "year": 2013, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, considere $z = -8 + 6i$ e $w = \\frac{-i \\times z^2}{\\overline{z}}$\nSeja $\\alpha$ um argumento do n\u00famero complexo $z$\nQual das op\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 0, "choices": [ "$w = 10 \\operatorname{cis}(3\\alpha - \\frac{\\pi}{2})$", "$w = 2 \\operatorname{cis}(3\\alpha - \\frac{\\pi}{2})$", "$w = 10 \\operatorname{cis}(\\alpha - \\frac{\\pi}{2})$", "$w = 2 \\operatorname{cis}(\\alpha - \\frac{\\pi}{2})$" ], "is_completion": true }, { "year": 2013, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nConsidere a linha do tri\u00e2ngulo de Pascal em que o produto do segundo elemento pelo pen\u00faltimo elemento\n\u00e9 484.\nQual \u00e9 a probabilidade de escolher, ao acaso, um elemento dessa linha que seja superior a 1000?", "answer": 2, "choices": [ "$\\frac{15}{23}$", "$\\frac{6}{11}$", "$\\frac{17}{23}$", "$\\frac{8}{11}$" ], "is_completion": true }, { "year": 2013, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSejam $a$ e $b$ dois n\u00fameros reais tais que $1|z-1|\\right\\}$", "$\\left\\{z\\in \\mathbb{C}: |z|\\leq \\sqrt{2}\\right\\}$", "$\\left\\{z\\in \\mathbb{C}: z = \\overline{z}\\right\\}$", "$\\left\\{z\\in \\mathbb{C}: Re(z) = Im (z)\\right\\}$" ], "is_completion": true }, { "year": 2014, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n\u2022 $P(A) = 0,4$\n\u2022 $P(A \\cap B) = 0,2$\n\u2022 $P(B|\\overline{A}) = 0,8$\nQual \u00e9 o valor de $P(B)$?", "answer": 2, "choices": [ "0,28", "0,52", "0,68", "0,80" ], "is_completion": true }, { "year": 2014, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nConsidere todos os n\u00fameros naturais de dez algarismos que se podem escrever com os algarismos de 1 a 9.\nQuantos desses n\u00fameros t\u00eam exatamente seis algarismos 2?", "answer": 0, "choices": [ "${}^{10}C_6 \u00d7 8^4$", "${}^{10}C_6 \u00d7 {}^8A_4$", "${}^{10}A_6 \u00d7 {}^8A_4$", "${}^{10}A_6 \u00d7 8^4$" ], "is_completion": true }, { "year": 2014, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja \u0192 a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}^+$, definida por $f(x) = e^{\\frac{1}{x}} - 3$.\nConsidere a sucess\u00e3o de n\u00fameros reais $(x_n)$ tal que $x_n = \\frac{1}{\\sqrt{n}}$\nQual \u00e9 o valor de $lim \\frac{2}{f(x_n)}$ ?", "answer": 2, "choices": [ "$-\u221e$", "$-e$", "$0$", "$+\u221e$" ], "is_completion": true }, { "year": 2014, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nConsidere, para um certo n\u00famero real $k$, a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}$, definida por $f(x) = k e^x + x$\nO teorema de Bolzano garante que a fun\u00e7\u00e3o $f$ tem, pelo menos, um zero no intervalo $]0,1[$\nA qual dos intervalos seguintes pode pertencer $k$?", "answer": 1, "choices": [ "$\\left]-e, -\\frac{1}{e}\\right[$", "$\\left]-\\frac{1}{e},0\\right[$", "$\\left]0,\\frac{1}{e}\\right[$", "$\\left]\\frac{1}{e},1\\right[$" ], "is_completion": true }, { "year": 2014, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nConsidere, num referencial o.n. $Oxyz$, o plano $\\alpha$, definido por $4x \u2212 z + 1 = 0$\nSeja $r$ uma reta perpendicular ao plano $\\alpha$\nQual das condi\u00e7\u00f5es seguintes pode definir a reta $r$?", "answer": 3, "choices": [ "$\\frac{x}{4}=y \\wedge z = -1$", "$x = 4 \\wedge z = \u22121$", "$x - 3 = \\frac{z}{4} \\wedge y = 0$", "$\\frac{x-3}{4}=-z \\wedge y = 1$" ], "is_completion": true }, { "year": 2014, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o de resultados associado a uma experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n\u2022 $A$ e $B$ s\u00e3o acontecimentos independentes;\n\u2022 $P(A) = 0,4$\n\u2022 $P(\\overline{A} \\cap \\overline{B}) = 0,48$\nQual \u00e9 o valor de $P(B)$?", "answer": 2, "choices": [ "0,08", "0,12", "0,2", "0,6" ], "is_completion": true }, { "year": 2014, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nUm dos termos do desenvolvimento de $\\left( \\frac{2}{x} + x \\right)^{10}$, com $x \\neq 0$, n\u00e3o depende da vari\u00e1vel $x$.\nQual \u00e9 esse termo?", "answer": 1, "choices": [ "10240", "8064", "1024", "252" ], "is_completion": true }, { "year": 2014, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $g$ uma fun\u00e7\u00e3o, de dom\u00ednio $]-\\infty, e[$, definida por $g(x) = \\ln(e - x)$.\nConsidere a sucess\u00e3o estritamente crescente de termo geral $x_n = \\left(1 + \\frac{1}{n}\\right)^n$\nQual \u00e9 o valor de $\\lim g(x_n)$?", "answer": 3, "choices": [ "$+\\infty$", "$e$", "$1$", "$-\\infty$" ], "is_completion": true }, { "year": 2014, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nConsidere, para um certo n\u00famero real $k$, a fun\u00e7\u00e3o $f$, cont\u00ednua em $\\left[\\frac{\\pi}{4}, \\frac{\\pi}{2}\\right]$, definida por\n$f(x) = \\begin{cases}\\frac{\\cos x}{x - \\frac{\\pi}{2}} &\\text{ se } \\frac{\\pi}{4} \\leq x < \\frac{\\pi}{2} \\\\ k - 3 &\\text{ se } x = \\frac{\\pi}{2} \\end{cases}$\nQual \u00e9 o valor de $k$?", "answer": 2, "choices": [ "0", "1", "2", "4" ], "is_completion": true }, { "year": 2014, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere, num referencial o.n. $Oxyz$, o ponto $A$, de coordenadas $(1, 0, 3)$, e o plano $\\alpha$, definido por $3x + 2y - 4 = 0$.\nSeja $\\beta$ um plano perpendicular ao plano $\\alpha$ e que passa pelo ponto $A$.\nQual das condi\u00e7\u00f5es seguintes pode definir o plano $\\beta$?", "answer": 1, "choices": [ "$3x + 2y \u2212 3 = 0$", "$2x - 3y - z + 1 = 0$", "$2x - 3y + z = 0$", "$3x + 2y = 0$" ], "is_completion": true }, { "year": 2014, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nConsidere todos os n\u00fameros \u00edmpares com cinco algarismos.\nQuantos desses n\u00fameros t\u00eam quatro algarismos pares e s\u00e3o superiores a 20 000 ?", "answer": 3, "choices": [ "$5^4$", "$5^5$", "$3 \\times 5^4$", "$4 \\times 5^4$" ], "is_completion": true }, { "year": 2014, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nConsidere a linha do tri\u00e2ngulo de Pascal em que a soma dos dois primeiros elementos com os dois \u00faltimos elementos \u00e9 igual a 20\nEscolhendo, ao acaso, um elemento dessa linha, qual \u00e9 a probabilidade de ele ser par?", "answer": 2, "choices": [ "$\\frac{1}{5}$", "$\\frac{2}{5}$", "$\\frac{3}{5}$", "$\\frac{4}{5}$" ], "is_completion": true }, { "year": 2014, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}\\setminus\\{0\\}$, definida por $f(x) = \\frac{x-1}{e^{x}-1}$\nConsidere a sucess\u00e3o de n\u00fameros reais $(x_n)$ tal que $x_n = - \\frac{1}{n}$\nQual \u00e9 o valor de $\\lim f (x_n)$?", "answer": 3, "choices": [ "$-\\infty$", "0", "1", "$+\\infty$" ], "is_completion": true }, { "year": 2014, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}^+$\nA reta de equa\u00e7\u00e3o $y=2x-5$ \u00e9 ass\u00edntota do gr\u00e1fico da fun\u00e7\u00e3o $f$\nQual \u00e9 o valor de $\\lim_{x \\to +\\infty} \\frac{6x-1}{f(x)}$ ?", "answer": 2, "choices": [ "0", "2", "3", "$+\\infty$" ], "is_completion": true }, { "year": 2014, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere, num referencial o.n. $Oxyz$, o ponto $A$, de coordenadas $(2,0,3)$, e o plano $\\alpha$, definido por $x-y-2z = 3$\nSeja $r$ a reta perpendicular ao plano $\\alpha$ que passa pelo ponto $A$\nQual das condi\u00e7\u00f5es seguintes pode definir a reta $r$?", "answer": 1, "choices": [ "$x + 2 = z +1 \\wedge y=0$", "$-x + 5 = y + 3 = \\frac{z + 3}{2}$", "$\\frac{x-1}{2} = \\frac{z+2}{3} \\wedge y = -1$", "$x - 2 = -y = z - 3$" ], "is_completion": true }, { "year": 2015, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nDois rapazes e quatro raparigas v\u00e3o sentar-se num banco corrido com seis lugares.\nDe quantas maneiras o podem fazer, de modo que fique um rapaz em cada extremidade do banco?", "answer": 2, "choices": [ "12", "24", "48", "60" ], "is_completion": true }, { "year": 2015, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n\u2022 $P(A) = 0,4$\n\u2022 $P(\\overline{B})=0,7$\n\u2022 $P(A \\cup B) = 0,5$\nQual \u00e9 o valor de $P(\\overline{A} \\cup \\overline{B})$?", "answer": 2, "choices": [ "0,6", "0,7", "0,8", "0,9" ], "is_completion": true }, { "year": 2015, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nQual das seguintes express\u00f5es \u00e9, para qualquer n\u00famero real $k$, igual a $log_3(\\frac{3^k}{9})$?", "answer": 1, "choices": [ "$\\frac{k}{2}$", "$k-2$", "$\\frac{k}{9}$", "$k-9$" ], "is_completion": true }, { "year": 2015, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nConsidere a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}^+$, definida por $f(x)=\\frac{1 + \\ln x}{x}$.\nConsidere a sucess\u00e3o de termo geral $u_n = n^2$\nQual \u00e9 o valor de $\\lim f(u_n)$?", "answer": 0, "choices": [ "0", "1", "$e$", "$+\\infty$" ], "is_completion": true }, { "year": 2015, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nConsidere em $\\mathbb{C}$, conjunto dos n\u00fameros complexos, a condi\u00e7\u00e3o\n$|z+4-4i|=3 \\land \\frac{\\pi}{2} \\leq arg(z) \\leq \\frac{3\\pi}{4}$\nNo plano complexo, esta condi\u00e7\u00e3o define uma linha.\nQual \u00e9 o comprimento dessa linha?", "answer": 2, "choices": [ "$\\pi$", "$2\\pi$", "$3\\pi$", "$4\\pi$" ], "is_completion": true }, { "year": 2015, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nSeja $a$ um n\u00famero real.\nConsidere a sucess\u00e3o $(u_n)$ definida por\n$\\begin{cases} u_1 = a \\\\ u_{n+1} = -3u_n +2, \\forall n \\in \\mathbb{N} \\end{cases}$\nQual \u00e9 o terceiro termo desta sucess\u00e3o?", "answer": 1, "choices": [ "$6a +4$", "$9a-4$", "$6a-4$", "$9a +4$" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "Documento table_1:\n{'headers': ['$x_i$', '1', '2', '3'], 'rows': [['$P(X = x_i)$', '$a$', '$2a$', '0,4']]}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma certa vari\u00e1vel aleat\u00f3ria $X$ \u00e9\n\nQual \u00e9 o valor m\u00e9dio desta vari\u00e1vel aleat\u00f3ria?", "answer": 1, "choices": [ "2,1", "2,2", "2,3", "2,4" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nUm saco cont\u00e9m nove bolas indistingu\u00edveis ao tato, numeradas de 1 a 9. As bolas numeradas de 1 a 5 s\u00e3o pretas e as restantes s\u00e3o brancas.\nRetira-se, ao acaso, uma bola do saco e observa-se a sua cor e o seu n\u00famero.\nConsidere os seguintes acontecimentos, associados a esta experi\u00eancia aleat\u00f3ria:\nA: \u00aba bola retirada \u00e9 preta\u00bb\nB: \u00abo n\u00famero da bola retirada \u00e9 um n\u00famero par\u00bb\nQual \u00e9 o valor da probabilidade condicionada $P(A|B)$?", "answer": 1, "choices": [ "$\\frac{2}{5}$", "$\\frac{1}{2}$", "$\\frac{3}{5}$", "$\\frac{3}{4}$" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nPara certos valores de a e de b ($a>1$ e $b>1$), tem-se $log_b a = \\frac{1}{3}$ \nQual \u00e9, para esses valores de $a$ e de $b$, o valor de $log_a (a^2 b)$?", "answer": 3, "choices": [ "$\\frac{2}{3}$", "$\\frac{5}{3}$", "$2$", "$5$" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nPara um certo n\u00famero real $k$, \u00e9 cont\u00ednua em $\\mathbb{R}$ a fun\u00e7\u00e3o $f$ definida por\n$f(x) = \\begin{cases} 2 + e^{x+k}, & \\text{se } x \\leq 0 \\\\ \\frac{2x + \\ln(x+1)}{x}, & \\text{se } x > 0 \\end{cases}$\nQual \u00e9 o valor de $k$ ?", "answer": 0, "choices": [ "$0$", "$1$", "$\\ln 2$", "$\\ln 3$" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}$, definida por $f(x) = 3sen^2(x)$\nQual das express\u00f5es seguintes define a fun\u00e7\u00e3o $f''$, segunda derivada de $f$?", "answer": 2, "choices": [ "$6 sen(2x) \\cos(x)$", "$6 sen(x) \\cos(2x)$", "$6 \\cos(2x)$", "$6sen(2x)$" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere, num referencial o.n. $xOy$, a circunfer\u00eancia definida pela equa\u00e7\u00e3o\n$x^2 + (y - 1)^2 = 2$\nEsta circunfer\u00eancia intersecta o eixo $Ox$ em dois pontos. Destes pontos, seja $A$ o que tem abcissa positiva.\nSeja $r$ a reta tangente \u00e0 circunfer\u00eancia no ponto $A$\nQual \u00e9 a equa\u00e7\u00e3o reduzida da reta $r$?", "answer": 1, "choices": [ "$y = x + 1$", "$y = x - 1$", "$y = 2x + 2$", "$y = 2x - 2$" ], "is_completion": true }, { "year": 2015, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nQual das express\u00f5es seguintes \u00e9 termo geral de uma sucess\u00e3o mon\u00f3tona e limitada?", "answer": 2, "choices": [ "$(-1)^n$", "$(-1)^n \\cdot n$", "$-\\frac{1}{n}$", "$1 + n^2$" ], "is_completion": true }, { "year": 2015, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n$\\bullet$ $P(A \\cup B) = 0,7$\n$\\bullet$ $P(B) = 0,4$\n$\\bullet$ $P(A \\cap B) = 0,2$\nQual \u00e9 o valor de $P(B|A)$?", "answer": 3, "choices": [ "0,25", "0,3", "0,35", "0,4" ], "is_completion": true }, { "year": 2015, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nNove jovens, tr\u00eas rapazes e seis raparigas, v\u00e3o dispor-se, lado a lado, para uma fotografia.\nDe quantas maneiras o podem fazer, de modo que os rapazes fiquem juntos?", "answer": 1, "choices": [ "40 140", "30 240", "20 340", "10 440" ], "is_completion": true }, { "year": 2015, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $a$ um n\u00famero real.\nSeja a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}^+$, definida por $f(x) = e^{a \\ln x}$\nConsidere, num referencial o.n. $xOy$, o ponto $P(2,8)$\nSabe-se que o ponto $P$ pertence ao gr\u00e1fico de $f$.\nQual \u00e9 o valor de $a$?", "answer": 2, "choices": [ "1", "2", "3", "4" ], "is_completion": true }, { "year": 2015, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}$.\nSabe-se que $f'(2) = 6$ ($f'$ designa a derivada de $f$).\nQual \u00e9 o valor de $\\lim_{x \\to 2} \\frac{f(x)-f(2)}{x^2-2x}$?", "answer": 0, "choices": [ "3", "4", "5", "6" ], "is_completion": true }, { "year": 2015, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nOs segmentos de reta $[AB]$ e $[BC]$ s\u00e3o lados consecutivos de um hex\u00e1gono regular de per\u00edmetro 12.\nQual \u00e9 o valor do produto escalar $\\vec{BA} . \\vec{BC}$?", "answer": 1, "choices": [ "-3", "-2", "2", "3" ], "is_completion": true }, { "year": 2015, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nDe uma progress\u00e3o geom\u00e9trica $(a_n)$, sabe-se que o terceiro termo \u00e9 igual a $\\frac{1}{4}$ e que o sexto termo \u00e9\nigual a 2.\nQual \u00e9 o valor do vig\u00e9simo termo?", "answer": 2, "choices": [ "8192", "16384", "32768", "65536" ], "is_completion": true }, { "year": 2016, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam A e B dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n$\\bullet$ $P(A) = \\frac{2}{5}$\n$\\bullet$ $P(B) = \\frac{3}{10}$\n$\\bullet$ $P(A|B) = \\frac{1}{6}$\nQual \u00e9 o valor de $P(A\\cup B)$?", "answer": 2, "choices": [ "$\\frac{4}{5}$", "$\\frac{7}{10}$", "$\\frac{13}{20}$", "$\\frac{19}{30}$" ], "is_completion": true }, { "year": 2016, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $X$ uma vari\u00e1vel aleat\u00f3ria com distribui\u00e7\u00e3o normal de valor m\u00e9dio 10\nSabe-se que $P(713)$?", "answer": 1, "choices": [ "0,1", "0,2", "0,3", "0,4" ], "is_completion": true }, { "year": 2016, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $a$ um n\u00famero real diferente de 0\nQual \u00e9 o valor de $lim_{x \\rightarrow a} \\frac{ae^{x-a} - a}{x^2 - a^2}$?", "answer": 1, "choices": [ "$\\frac{1}{4}$", "$\\frac{1}{2}$", "1", "2" ], "is_completion": true }, { "year": 2016, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}^-$\nSabe-se que:\n$\\bullet$ $lim_{x \\rightarrow +\\infty} \\frac{f(x) + e^x - x}{x} = 1$\n$\\bullet$ o gr\u00e1fico de $f$ tem uma ass\u00edntota obl\u00edqua.\nQual \u00e9 o declive dessa ass\u00edntota?", "answer": 3, "choices": [ "-2", "-1", "1", "2" ], "is_completion": true }, { "year": 2016, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nSeja $\\theta$ um n\u00famero real pertencente ao intervalo $]\\pi, \\frac{3\\pi}{2}[$\nConsidere o n\u00famero complexo $z = -3 cis \\theta$\nA que quadrante pertence a imagem geom\u00e9trica do complexo $z$?", "answer": 0, "choices": [ "Primeiro", "Segundo", "Terceiro", "Quarto" ], "is_completion": true }, { "year": 2016, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nConsidere as sucess\u00f5es ($u_n$) e ($v_n$) de termos gerais\n$u_n = \\frac{kn+3}{2n}$ ($k$ \u00e9 um n\u00famero real) e $v_n = \\ln(1 + \\frac{1}{n})^n$\nSabe-se que $lim(u_n) = lim(v_n)$\nQual \u00e9 o valor de $k$?", "answer": 1, "choices": [ "1", "2", "$e$", "$2e$" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega \\space e B \\subset \\Omega$).\nSabe-se que:\n$\\bullet P(A) = 0,2$\n$\\bullet P(B) = 0,3$\n$\\bullet P(\\overline{A} \\cap \\overline{B})=0,6$\nQual \u00e9 o valor de $P(A|B)$?", "answer": 0, "choices": [ "$\\frac{1}{3}$", "$\\frac{1}{2}$", "$\\frac{2}{3}$", "$\\frac{5}{6}$" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nO Carlos joga basquetebol na equipa da sua escola.\nAdmita que, em cada lance livre, a probabilidade de o Carlos encestar \u00e9 0,4\nNum treino, o Carlos vai executar uma s\u00e9rie de cinco lances livres.\nQual \u00e9 a probabilidade de o Carlos encestar exatamente quatro vezes?", "answer": 2, "choices": [ "0,01536", "0,05184", "0,0768", "0,2592" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nPara certos valores de a e de b ($a>1$ e $b>1$), tem-se $log_a(ab^3) = 5$\nQual \u00e9, para esses valores de $a$ e de $b$, o valor de $\\log_b a$?", "answer": 1, "choices": [ "$\\frac{5}{3}$", "$\\frac{3}{4}$", "$\\frac{3}{5}$", "$\\frac{1}{3}$" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nConsidere a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}^+$, definida por $f(x) = \\ln x$\nConsidere a sucess\u00e3o de termo geral $u_n = \\frac{n}{e^n}$\nQual \u00e9 o valor de $\\lim f (u_n)$?", "answer": 0, "choices": [ "$- \\infty$", "0", "e", "$+\\infty$" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, seja $z = 3 + 4i$\nSabe-se que $z$ \u00e9 uma das ra\u00edzes de \u00edndice 6 de um certo n\u00famero complexo $w$\nConsidere, no plano complexo, o pol\u00edgono cujos v\u00e9rtices s\u00e3o as imagens geom\u00e9tricas das ra\u00edzes de \u00edndice 6 desse n\u00famero complexo $w$\nQual \u00e9 o per\u00edmetro do pol\u00edgono?", "answer": 2, "choices": [ "42", "36", "30", "24" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere, num referencial o.n. $xOy$, o quadrado definido pela condi\u00e7\u00e3o\n$0\\le x \\le 4 \\space \\land \\space 1\\le y \\le 5$\nQual das condi\u00e7\u00f5es seguintes define a circunfer\u00eancia inscrita neste quadrado?", "answer": 2, "choices": [ "$(x - 4)^2 + (y - 5)^2 = 16$", "$(x - 4)^2 + (y - 5)^2 = 4$", "$(x \u2013 2)^2 + (y \u2013 3)^2 = 4$", "$(x \u2013 2)^2 + (y - 3)^2 = 16$" ], "is_completion": true }, { "year": 2016, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nDe uma progress\u00e3o geom\u00e9trica ($u_n$), mon\u00f3tona crescente, sabe-se que $u_4 = 32$ e que $u_8 = 8192$\nQual \u00e9 o quinto termo da sucess\u00e3o ($u_n$)?", "answer": 1, "choices": [ "64", "128", "256", "512" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nConsidere, num referencial o.n. $xOy$, os pontos $A(-1,3)$ e $B(2, 4)$. Qual das seguintes equa\u00e7\u00f5es define uma reta paralela \u00e0 reta $AB$?", "answer": 1, "choices": [ "$y = -\\frac{1}{3} x$", "$y = \\frac{1}{3} x$", "$y = 3x$", "$y = -3x$" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nUma pessoa lan\u00e7a um dado c\u00fabico, com as faces numeradas de 1 a 6, e regista o n\u00famero da face que ficou voltada para cima.\nUma outra pessoa lan\u00e7a um dado com a forma de um tetraedro regular, com as faces numeradas de 1 a 4, e regista o n\u00famero da face que ficou voltada para baixo.\nAdmita que ambos os dados s\u00e3o equilibrados.\nQual \u00e9 a probabilidade de, pelo menos, uma dessas pessoas registar o n\u00famero 4?", "answer": 0, "choices": [ "$\\frac{3}{8}$", "$\\frac{5}{8}$", "$\\frac{5}{12}$", "$\\frac{7}{12}$" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $X$ uma vari\u00e1vel aleat\u00f3ria com distribui\u00e7\u00e3o normal de valor m\u00e9dio 2 e desvio padr\u00e3o 0,5\nQual \u00e9 o valor, arredondado \u00e0s cent\u00e9simas, de $P(X > 2,5)$?", "answer": 3, "choices": [ "0,68", "0,34", "0,32", "0,16" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSejam $a$ e $b$ dois n\u00fameros reais superiores a 1, tais que $a = b^3$.\nQual dos valores seguintes \u00e9 igual a $log_a b + log_b a$?", "answer": 2, "choices": [ "$\\frac{4}{3}$", "1", "$\\frac{10}{3}$", "3" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nPara um certo n\u00famero real $k$, \u00e9 cont\u00ednua em $\\mathbb{R}$ a fun\u00e7\u00e3o $f$ definida por\n$f(x) = \\begin{cases} \\frac{sen(3x + 3)}{4x + 4} & \\text{se } x \\neq -1 \\\\ k+2 & \\text{se } x = -1 \\end{cases}$\nQual \u00e9 o valor de $k$?", "answer": 1, "choices": [ "$-\\frac{5}{3}$", "$-\\frac{5}{4}$", "$\\frac{5}{4}$", "$\\frac{5}{3}$" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere em $\\mathbb{C}$, conjunto dos n\u00fameros complexos, a condi\u00e7\u00e3o\n$0 \\leq \\text{arg} z \\leq \\frac{\\pi}{4} \\wedge 1 \\leq \\text{Re}z \\leq 5$\nEsta condi\u00e7\u00e3o define uma regi\u00e3o no plano complexo. Qual dos seguintes n\u00fameros complexos tem a sua imagem geom\u00e9trica nesta regi\u00e3o?", "answer": 2, "choices": [ "$3+4i$", "$6+2i$", "$2cis \\frac{13\\pi}{6}$", "$cis \\frac{\\pi}{6}$" ], "is_completion": true }, { "year": 2016, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nConsidere as sucess\u00f5es convergentes $(a_n)$ e $(b_n)$, de termos gerais\n$a_n = (1 + \\frac{1}{n})^{3n}$ e $b_n = \\ln(1 - 2e^{-n})$\nSejam $a$ e $b$ os n\u00fameros reais tais que $a = lim(a_n)$ e $b = lim(b_n)$. Qual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 1, "choices": [ "$a = 3e$ e $b=0$", "$a = e^3$ e $b=0$", "$a = 3e$ e $b=1$", "$a = e^3$ e $b=1$" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nConsidere todos os n\u00fameros naturais de quatro algarismos que se podem formar com os algarismos de 1 a 9.\nDestes n\u00fameros, quantos s\u00e3o m\u00faltiplos de 5?", "answer": 0, "choices": [ "729", "1458", "3645", "6561" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nUma turma \u00e9 constitu\u00edda por rapazes e por raparigas, num total de 20 alunos.\nSabe-se que:\n\u2022 $\\frac{1}{4}$ dos rapazes tem olhos verdes;\n\u2022 escolhido, ao acaso, um aluno da turma, a probabilidade de ele ser rapaz e de ter olhos verdes \u00e9 $\\frac{1}{10}$.\nQuantos rapazes tem a turma?", "answer": 1, "choices": [ "4", "8", "12", "16" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "4", "question": "\nSejam $f$ e $g$ duas fun\u00e7\u00f5es de dom\u00ednio $\\mathbb{R}^+$.\nSabe-se que a reta de equa\u00e7\u00e3o $y = -x$ \u00e9 ass\u00edntota obl\u00edqua do gr\u00e1fico de $f$ e do gr\u00e1fico de $g$.\nQual \u00e9 o valor de $\\lim_{x \\to +\\infty} \\frac{f(x) \\times g(x)}{x}$?", "answer": 0, "choices": [ "+$\\infty$", "1", "-1", "-$\\infty$" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $A$ e contradom\u00ednio $]-1, +\\infty[$, definida por $f'(x) = tg x$.\nQual dos conjuntos seguintes pode ser o conjunto $A$?", "answer": 1, "choices": [ "$]-\\frac{\\pi}{4}, \\frac{\\pi}{4}[$", "$]\\frac{3\\pi}{4}, \\frac{3\\pi}{2}[$", "$]\\frac{\\pi}{2}, \\frac{3\\pi}{4}[$", "$]\\frac{5\\pi}{4}, \\frac{3\\pi}{2}[$" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nConsidere, num referencial o.n. $xOy$, uma reta $r$ de inclina\u00e7\u00e3o $\\alpha$.\nSabe-se que $\\cos \\alpha = -\\frac{1}{\\sqrt{5}}$.\nQual pode ser a equa\u00e7\u00e3o reduzida da reta $r$?", "answer": 2, "choices": [ "$y = -5x$", "$y = 4x$", "$y = -2x$", "$y = 3x$" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere em $\\mathbb{C}$, conjunto dos n\u00fameros complexos, a condi\u00e7\u00e3o $\\frac{5\\pi}{4} \\leq arg(z) \\leq \\frac{7\\pi}{4} \\land Im(z) \\geq -1$.\nNo plano complexo, esta condi\u00e7\u00e3o define uma regi\u00e3o.\nQual \u00e9 a \u00e1rea dessa regi\u00e3o?", "answer": 3, "choices": [ "$\\frac{\\sqrt{2}}{2}$", "$\\frac{1}{2}$", "$\\sqrt{2}$", "1" ], "is_completion": true }, { "year": 2017, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nSeja $(u_n)$ a sucess\u00e3o definida por $u_n = \\begin{cases} n & \\text{se } n \\leq 20 \\\\ (-1)^n & \\text{se } n > 20 \\end{cases}$\nQual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 2, "choices": [ "A sucess\u00e3o $(u_n)$ \u00e9 mon\u00f3tona crescente.", "A sucess\u00e3o $(u_n)$ \u00e9 mon\u00f3tona decrescente.", "A sucess\u00e3o $(u_n)$ \u00e9 limitada.", "A sucess\u00e3o $(u_n)$ \u00e9 um infinitamente grande." ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nConsidere todos os n\u00fameros naturais de cinco algarismos diferentes que se podem formar com os algarismos 1,2,3,4 e 5\nDestes n\u00fameros, quantos t\u00eam os algarismos pares um a seguir ao outro?", "answer": 1, "choices": [ "24", "48", "72", "96" ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "Documento document_1:\n{'$x_i$': ['1', '2', '3', '4'], '$P(X = x_i)$': ['$\\\\frac{1}{3}$', '$\\\\frac{1}{4}$', '$\\\\frac{1}{6}$', '$\\\\frac{1}{4}$']}\n\nA tabela de distribui\u00e7\u00e3o de probabilidades de uma vari\u00e1vel aleat\u00f3ria X \u00e9 a seguinte.\nQual \u00e9 o valor da probabilidade condicionada $P(X>1| X \\le 3)$?", "answer": 3, "choices": [ "$\\frac{3}{4}$", "$\\frac{1}{4}$", "$\\frac{8}{9}$", "$\\frac{5}{9}$" ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nDe uma fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}$, com derivada finita em todos os pontos do seu dom\u00ednio, sabe-se que $lim_{x\\rightarrow 2} \\frac{x^2 - 2x}{f(x)-f(2)} = 4$\nQual \u00e9 o valor de $f'(2)$?", "answer": 2, "choices": [ "$-\\frac{1}{2}$", "$-\\frac{1}{4}$", "$\\frac{1}{2}$", "$\\frac{1}{4}$" ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "Documento document_2:\n{'$x$': ['$-\\\\infty$', '-10', '', '0', '', '10', '$+\\\\infty$'], \"$f''$\": ['-', '0', '+', '0', '-', '0', '+']}\n\nSeja $f$ uma fun\u00e7\u00e3o de dom\u00ednio $\\mathbb{R}$\nA tabela de varia\u00e7\u00e3o de sinal da fun\u00e7\u00e3o $f''$, segunda derivada de $f$, \u00e9 a seguinte.\nSeja $g$ a fun\u00e7\u00e3o definida por $g(x) = \u2212 f (x - 5)$\nEm qual dos intervalos seguintes o gr\u00e1fico de $g$ tem concavidade voltada para baixo?", "answer": 2, "choices": [ "$]-15, -5[$", "$]0,10[$", "$]-5, 5[$", "$]5,15[$" ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nSeja $z$ um n\u00famero complexo de argumento $\\frac{\\pi}{5}$\nQual dos seguintes valores \u00e9 um argumento do n\u00famero complexo $-5iz$?", "answer": 0, "choices": [ "$-\\frac{3\\pi}{10}$", "$-\\frac{4\\pi}{5}$", "$-\\frac{7\\pi}{5}$", "$-\\frac{13\\pi}{10}$" ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "7", "question": "\nConsidere, num referencial o.n. $xOy$, a regi\u00e3o definida pela condi\u00e7\u00e3o\n$(x + 1)^2 + (y + 1)^2\\le1 \\land x + y + 2\\ge0$\nQual \u00e9 o per\u00edmetro dessa regi\u00e3o?", "answer": 2, "choices": [ "$\\pi+1$", "$\\frac{\\pi}{2}+1$", "$\\pi + 2$", "$\\frac{\\pi}{2}+2$" ], "is_completion": true }, { "year": 2017, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nSeja $(u_n)$ a sucess\u00e3o definida por $u_n = (\\frac{1}{2})^{1-n}$\nQual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 1, "choices": [ "A sucess\u00e3o $(u_n)$ \u00e9 uma progress\u00e3o geom\u00e9trica de raz\u00e3o $\\frac{1}{2}$", "A sucess\u00e3o $(u_n)$ \u00e9 uma progress\u00e3o geom\u00e9trica de raz\u00e3o 2", "A sucess\u00e3o $(u_n)$ \u00e9 uma progress\u00e3o aritm\u00e9tica de raz\u00e3o $\\frac{1}{2}$", "A sucess\u00e3o $(u_n)$ \u00e9 uma progress\u00e3o aritm\u00e9tica de raz\u00e3o 2" ], "is_completion": true }, { "year": 2017, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "1", "question": "\nCom os algarismos 0, 1, 2, 3 e 4, quantos n\u00fameros naturais maiores do que 20 000 e com os cinco\nalgarismos todos diferentes \u00e9 poss\u00edvel formar?", "answer": 2, "choices": [ "24", "48", "72", "96" ], "is_completion": true }, { "year": 2017, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "2", "question": "\nSeja $X$ uma vari\u00e1vel aleat\u00f3ria com distribui\u00e7\u00e3o normal de valor m\u00e9dio 10\nSabe-se que $P(10 15)$ ?", "answer": 1, "choices": [ "0,1", "0,2", "0,4", "0,6" ], "is_completion": true }, { "year": 2017, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "3", "question": "\nSeja $a$ um n\u00famero real superior a 1\nQual \u00e9 o valor de $4+\\log_a (5^{\\ln a})$?", "answer": 1, "choices": [ "$\\ln (10e)$", "$\\ln (5e^4)$", "$\\ln (5e^2)$", "$\\ln (20e)$" ], "is_completion": true }, { "year": 2017, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "5", "question": "\nSejam $f$ e $g$ duas fun\u00e7\u00f5es de dom\u00ednio $\\mathbb{R}$, tais que a fun\u00e7\u00e3o $f-g$ admite inversa.\nSabe-se que $f(3)=4$ e que $(f-g)^{-1} (2) = 3$\nQual \u00e9 o valor de $g(3)$?", "answer": 1, "choices": [ "1", "2", "3", "4" ], "is_completion": true }, { "year": 2017, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "6", "question": "\nConsidere, num referencial o.n. $xOy$, dois pontos distintos, $R$ e $S$\nSeja $A$ o conjunto dos pontos $P$ desse plano que verificam a condi\u00e7\u00e3o $\\vec{PR}.\\vec{PS} = 0$\n($\\vec{PR}.\\vec{PS}$ designa o produto escalar de $\\vec{PR}$ por $\\vec{PS}$).\nQual das seguintes afirma\u00e7\u00f5es \u00e9 verdadeira?", "answer": 3, "choices": [ "O conjunto $A$ \u00e9 a mediatriz do segmento de reta $[RS]$", "O conjunto $A$ \u00e9 o segmento de reta $[RS]$", "O conjunto $A$ \u00e9 o tri\u00e2ngulo $[ROS]$", "O conjunto $A$ \u00e9 a circunfer\u00eancia de di\u00e2metro $[RS]$" ], "is_completion": true }, { "year": 2017, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "I", "question_number": "8", "question": "\nSeja $(u_n)$ uma sucess\u00e3o real em que todos os termos s\u00e3o positivos.\nSabe-se que, para todo o n\u00famero natural $n$, $\\frac{u_{n+1}}{u_n} <1$\nQual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 0, "choices": [ "A sucess\u00e3o $(u_n)$ \u00e9 limitada.", "A sucess\u00e3o $(u_n)$ \u00e9 uma progress\u00e3o aritm\u00e9tica.", "A sucess\u00e3o $(u_n)$ \u00e9 crescente.", "A sucess\u00e3o $(u_n)$ \u00e9 um infinitamente grande." ], "is_completion": true }, { "year": 2018, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "1.2", "question": "\nSeja $f$ uma fun\u00e7\u00e3o diferenci\u00e1vel no intervalo $[0,2]$ tal que:\n\u2022 $f(0)=1$\n\u2022 $\u2200x\u2208 [0,2], 0 \\mu - 2\\sigma)$?", "answer": 2, "choices": [ "0,926", "0,982", "0,977", "0,943" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "3.1", "question": "\nDisp\u00f5e-se de catorze caracteres (a saber: os algarismos 1, 2, 3, 4, 5, 6, 7, 8, 9 e as vogais a, e, i, o, u) para formar c\u00f3digos de quatro caracteres.\nQuantos c\u00f3digos iniciados por uma vogal seguida de tr\u00eas algarismos diferentes se podem formar?", "answer": 3, "choices": [ "420", "504", "1840", "2520" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "6", "question": "\nA primeira derivada de uma fun\u00e7\u00e3o $f$, de dom\u00ednio $]0, \\frac{\\pi}{2}[$, \u00e9 dada por $f'(x) = 3x - \\text{tg} x$.\nSabe-se que o gr\u00e1fico de $f$ tem um \u00fanico ponto de inflex\u00e3o.\nQual \u00e9 a abcissa desse ponto, arredondada \u00e0s cent\u00e9simas?", "answer": 3, "choices": [ "0,84", "0,88", "0,92", "0,96" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "9.1", "question": "\nNum dado problema de Programa\u00e7\u00e3o Linear, pretende-se determinar o valor m\u00e1ximo que a fun\u00e7\u00e3o objetivo, definida por $L = 3x + 5y$, pode alcan\u00e7ar.\nSabe-se que a regi\u00e3o admiss\u00edvel \u00e9 definida pelo seguinte sistema.\n$\\begin{cases} x \\geq 0 \\\\ y \\geq 0 \\\\ x + y \\leq 10 \\\\ x \\leq 5 \\end{cases}$\nQual \u00e9 esse valor m\u00e1ximo?", "answer": 1, "choices": [ "56", "50", "40", "36" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "9.2", "question": "\nConsidere, num referencial o.n. $xOy$, uma elipse centrada na origem do referencial e de focos $F_1$ e $F_2$ pertencentes ao eixo $Ox$.\nSabe-se que:\n* $\\overline{F_1F_2} = 12$\n* sendo $P$ um ponto qualquer da elipse, tem-se $\\overline{PF_1} + \\overline{PF_2} = 20$\nQual \u00e9 a equa\u00e7\u00e3o reduzida desta elipse?", "answer": 1, "choices": [ "$\\frac{x^2}{64} + \\frac{y^2}{100} = 1$", "$\\frac{x^2}{100} + \\frac{y^2}{64} = 1$", "$\\frac{x^2}{100} + \\frac{y^2}{36} = 1$", "$\\frac{x^2}{36} + \\frac{y^2}{100} = 1$" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "11", "question": "\nConsidere, num referencial o.n. $xOy$, a circunfer\u00eancia centrada na origem do referencial e que passa no ponto $A(2,1)$.\nSeja $r$ a reta tangente \u00e0 circunfer\u00eancia no ponto $A$.\nQual \u00e9 a ordenada na origem da reta $r$?", "answer": 1, "choices": [ "4", "5", "6", "7" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "12.1", "question": "\nConsidere, num referencial o.n. $Oxyz$, os planos $\\alpha$, $\\beta$ e $\\gamma$ definidos pelas equa\u00e7\u00f5es $y = -x, y = z$ e $2x + 3y - z - 1 = 0$, respetivamente.\nA intersec\u00e7\u00e3o dos planos $\\alpha$, $\\beta$ e $\\gamma$ \u00e9:", "answer": 3, "choices": [ "um ponto.", "uma reta.", "um plano.", "o conjunto vazio." ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "12.2", "question": "\nQual \u00e9 o valor do limite da sucess\u00e3o de termo geral $\\left( \\frac{n + 5}{n + 1} \\right)^{\\frac{n}{2}}$?", "answer": 3, "choices": [ "$+\\infty$", "1", "$e^4$", "$e^2$" ], "is_completion": true }, { "year": 2018, "phase": "2.\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "14.3", "question": "\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}$, definida por\n$f(x) = \\begin{cases} 3+\\frac{e^x}{1 - x} & \\text{se } x < 1 \\\\ \\frac{\\ln(x^2) + 2}{x} & \\text{se } x \\geq 1 \\end{cases}$\nSeja $h$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}$, definida por $h(x) = x + 1$.\nQual \u00e9 o valor de $(f \\circ h^{-1})(2)$?\n(o s\u00edmbolo $\\circ$ designa a composi\u00e7\u00e3o de fun\u00e7\u00f5es)", "answer": 2, "choices": [ "0", "1", "2", "3" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "1.1", "question": "\nSeja $\\Omega$ o espa\u00e7o amostral (espa\u00e7o de resultados) associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n$\\bullet$ $A$ e $B$ s\u00e3o acontecimentos independentes e equiprov\u00e1veis;\n$\\bullet$ $P(A \\cup B) = 0,64$\nQual \u00e9 o valor de $P(A)$?", "answer": 1, "choices": [ "0,42", "0,40", "0,38", "0,36" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "1.2", "question": "\nUm ponto $P$ desloca-se numa reta num\u00e9rica, no intervalo de tempo $I = [0,4]$ (medido em segundos), de tal forma que a respetiva abcissa \u00e9 dada por\n$x(t) = sen(\\pi t) + cos (\\pi t)$, com $t \\in I$\nQual \u00e9 a amplitude deste oscilador harm\u00f3nico?", "answer": 1, "choices": [ "1", "$\\sqrt{2}$", "$\\sqrt{3}$", "2" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nCom cinco pessoas, quantos conjuntos com, pelo menos, tr\u00eas pessoas \u00e9 poss\u00edvel formar?", "answer": 3, "choices": [ "60", "81", "10", "16" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "6", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, a express\u00e3o $i^0 + i^1 + i^2 + ... + i^{2018}$ \u00e9 igual a", "answer": 0, "choices": [ "$i$", "$-i$", "$-1 + i$", "$1 + i$" ], "is_completion": false }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "9", "question": "\nPara um certo n\u00famero real $a$, diferente de zero, s\u00e3o paralelas as retas $r$ e $s$, definidas, num referencial o.n. $xOy$, pelas condi\u00e7\u00f5es $r : ax + 2y + 1 = 0$ e $s : (x, y) = (1,1) + k(a,2a), k \\in \\mathbb{R}$.\nQual \u00e9 o valor de $a$?", "answer": 0, "choices": [ "-4", "2", "-2", "4" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "10.1", "question": "\nConsidere, num referencial o.n. $Oxyz$,\n$\\bullet$ a reta $r$, definida pela condi\u00e7\u00e3o $\\frac{x-1}{2} = \\frac{3-y}{5} = \\frac{z}{3}$\n$\\bullet$ o plano $\\alpha$, definido pela equa\u00e7\u00e3o $3x - 2z - 3 = 0$\nQual \u00e9 a posi\u00e7\u00e3o relativa da reta $r$ e do plano $\\alpha$?", "answer": 3, "choices": [ "$r$ \u00e9 estritamente paralela a $\\alpha$", "$r$ e $\\alpha$ s\u00e3o concorrentes, mas n\u00e3o perpendiculares.", "$r$ \u00e9 perpendicular a $\\alpha$", "$r$ est\u00e1 contida em $\\alpha$" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "10.2", "question": "\nSeja $g$ uma fun\u00e7\u00e3o real, de dom\u00ednio $[0,1]$\nSabe-se que a fun\u00e7\u00e3o $g$ n\u00e3o tem m\u00ednimo.\nQual das afirma\u00e7\u00f5es seguintes \u00e9 necessariamente verdadeira?", "answer": 3, "choices": [ "A fun\u00e7\u00e3o $g$ n\u00e3o tem zeros.", "A fun\u00e7\u00e3o $g$ n\u00e3o \u00e9 limitada.", "A fun\u00e7\u00e3o $g$ n\u00e3o tem m\u00e1ximo.", "A fun\u00e7\u00e3o $g$ n\u00e3o \u00e9 cont\u00ednua." ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "12.1", "question": "Documento tabela1:\n{'$x_i$': '0, 1', '$P(X = x_i)$': '$\\\\frac{19}{27}$, $\\\\frac{8}{27}$'}\n\nUm dado c\u00fabico equilibrado tem todas as faces numeradas, umas com o n\u00famero 0 e as restantes com o n\u00famero 1.\nLan\u00e7a-se o dado tr\u00eas vezes e, em cada lan\u00e7amento, regista-se o n\u00famero da face que fica voltada para cima.\nSeja X a vari\u00e1vel aleat\u00f3ria \u00abproduto dos n\u00fameros sa\u00eddos nos tr\u00eas lan\u00e7amentos\u00bb.\nA tabela de distribui\u00e7\u00e3o de probabilidades da vari\u00e1vel X \u00e9 a seguinte.\nQuantas faces est\u00e3o numeradas com o n\u00famero 1?", "answer": 2, "choices": [ "Duas.", "Tr\u00eas.", "Quatro.", "Cinco." ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "12.2", "question": "\nQual \u00e9 o valor do limite da sucess\u00e3o $(\\frac{n+2}{n+1})^{2n}$?", "answer": 2, "choices": [ "1", "$e$", "$e^2$", "$+\\infty$" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "14.3", "question": "\nSeja $h$ a fun\u00e7\u00e3o, de dom\u00ednio $[-\\frac{\\pi}{3}, +\\infty[$, definida por: $h(x) = \\begin{cases} \\frac{sen^2(x)}{sen(x^2)} & \\text{se } -\\frac{\\pi}{3} \\leq x < 0 \\\\ \\frac{e^x}{x+1} & \\text{se } x \\geq 0 \\end{cases}$. Qual das seguintes afirma\u00e7\u00f5es \u00e9 verdadeira?", "answer": 1, "choices": [ "A fun\u00e7\u00e3o $h$ \u00e9 estritamente decrescente em $[0, +\\infty[$", "A fun\u00e7\u00e3o $h$ \u00e9 estritamente crescente em $[0, +\\infty[$", "A fun\u00e7\u00e3o $h$ tem um m\u00e1ximo para $x = 1$", "A fun\u00e7\u00e3o $h$ tem um m\u00ednimo para $x = 1$" ], "is_completion": true }, { "year": 2018, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "15", "question": "\nConsidere a fun\u00e7\u00e3o $f$, definida em $[-\\frac{\\pi}{6}, \\frac{\\pi}{3}]$ por $f(x) = cos x$\nQual dos seguintes conjuntos \u00e9 o contradom\u00ednio da fun\u00e7\u00e3o $f$?", "answer": 2, "choices": [ "$[\\frac{1}{2}, \\frac{\\sqrt{3}}{2}]$", "$[\\frac{\\sqrt{3}}{2}, 1]$", "$[\\frac{1}{2}, 1]$", "$[0, \\frac{1}{2}]$" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "2.1", "question": "\nSeja $X$ uma vari\u00e1vel aleat\u00f3ria com distribui\u00e7\u00e3o normal de valor m\u00e9dio 5 e desvio padr\u00e3o $\\frac{1}{2}$. Qual \u00e9 o valor, arredondado \u00e0s mil\u00e9simas, de $P(X > 6)$?", "answer": 2, "choices": [ "0,046", "0,042", "0,023", "0,021" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "2.2", "question": "\nQual \u00e9 o limite da sucess\u00e3o de termo geral $(\\frac{n-2}{n})^{3n}$?", "answer": 2, "choices": [ "$\\frac{1}{e^3}$", "$e^3$", "$\\frac{1}{e^6}$", "$e^6$" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "3.2", "question": "\nUma caixa cont\u00e9m bolas de v\u00e1rias cores, indistingu\u00edveis ao tato, umas com um logotipo desenhado e outras n\u00e3o. Das bolas existentes na caixa, dez s\u00e3o amarelas. Dessas dez bolas, tr\u00eas t\u00eam o logotipo desenhado.\nDisp\u00f5em-se, ao acaso, as dez bolas amarelas, lado a lado, em linha reta. Qual \u00e9 a probabilidade de as tr\u00eas bolas com o logotipo desenhado ficarem juntas?", "answer": 1, "choices": [ "$\\frac{1}{16}$", "$\\frac{1}{15}$", "$\\frac{1}{14}$", "$\\frac{1}{13}$" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "1", "question_number": "6", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, seja $z = -1+ 2i$. Seja $\\theta$ o menor argumento positivo do n\u00famero complexo $\\overline{z}$ (conjugado de $z$). A qual dos intervalos seguintes pertence $\\theta$?", "answer": 3, "choices": [ "$]0, \\frac{\\pi}{4}[$", "$]\\frac{\\pi}{4}, \\frac{\\pi}{2}[$", "$]\\pi, \\frac{5\\pi}{4}[$", "$]\\frac{5\\pi}{4}, \\frac{3\\pi}{2}[$" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "9.1", "question": "\nConsidere, num referencial o.n. $Oxyz$, os planos $\\alpha$, $\\beta$ e $\\gamma$, definidos pelas equa\u00e7\u00f5es $x + y + z = 1$, $2x + 2y + 2z =1$ e $x+y= 0$, respetivamente. A intersec\u00e7\u00e3o dos planos $\\alpha$, $\\beta$ e $\\gamma$ \u00e9", "answer": 0, "choices": [ "o conjunto vazio.", "um ponto.", "uma reta.", "um plano." ], "is_completion": false }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "11", "question": "\nQual \u00e9 a solu\u00e7\u00e3o da equa\u00e7\u00e3o $2cos x+1=0$ no intervalo $[-\\pi,0]$?", "answer": 1, "choices": [ "$\\frac{-5\\pi}{6}$", "$\\frac{-2\\pi}{3}$", "$\\frac{-\\pi}{3}$", "$\\frac{-\\pi}{6}$" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "12.1", "question": "\nUm dado c\u00fabico equilibrado tem uma face numerada com o n\u00famero -1 e cinco faces numeradas com o n\u00famero 1. Lan\u00e7a-se este dado duas vezes. Seja $X$ a vari\u00e1vel aleat\u00f3ria \u00absoma dos n\u00fameros sa\u00eddos nos dois lan\u00e7amentos\u00bb. Qual \u00e9 o valor de $k$ para o qual $P(X = k) = \\frac{5}{18}$ ?", "answer": 0, "choices": [ "0", "2", "-2", "-1" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "12.2", "question": "\nUm ponto $P$ desloca-se numa reta num\u00e9rica, no intervalo de tempo $I = [0,10]$ (medido em segundos), de tal forma que a respetiva abcissa \u00e9 dada por $x(t) = 3 \\cos(\\pi t + \\frac{2\\pi}{3})$ com $t \\in I$. Qual \u00e9 o per\u00edodo, em segundos, deste oscilador harm\u00f3nico?", "answer": 0, "choices": [ "2", "3", "$2\\pi$", "$3\\pi$" ], "is_completion": true }, { "year": 2019, "phase": "1\u00aa", "subject": "Mathematics A", "question_group": "2", "question_number": "14.2", "question": "\nSeja $g$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R} \\setminus \\{0\\}$, definida por $g(x) = \\frac{e^{-x}}{x}$\nSeja $h$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}^+$, definida por $h(x) = g(x) + 2x - \\frac{1}{\\sqrt{x}}$. Sabe-se que o gr\u00e1fico da fun\u00e7\u00e3o $h$ tem uma ass\u00edntota obl\u00edqua. Qual \u00e9 o declive dessa ass\u00edntota?", "answer": 1, "choices": [ "1", "2", "$e$", "$e^2$" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "2.2", "question": "\nUma escola secund\u00e1ria tem apenas turmas de 10.\u00b0, 11.\u00ba e 12.\u00ba anos.\nUma turma dessa escola tem 26 alunos, dos quais 15 s\u00e3o raparigas. O delegado de turma \u00e9 um rapaz. Pretende-se formar uma comiss\u00e3o com tr\u00eas alunos desta turma, para organizar uma festa de fim de ano. Quantas comiss\u00f5es diferentes, que incluam rapazes e raparigas, se podem formar, sabendo-se que o delegado de turma tem de fazer parte da comiss\u00e3o?", "answer": 3, "choices": [ "195", "215", "235", "255" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3.1", "question": "\nUma caixa cont\u00e9m duas bolas brancas e tr\u00eas bolas pretas. Retiram-se, ao acaso e em simult\u00e2neo, duas bolas da caixa. Seja $X$ a vari\u00e1vel aleat\u00f3ria \u00abn\u00famero de bolas brancas retiradas\u00bb. Qual \u00e9 o valor m\u00e9dio da vari\u00e1vel aleat\u00f3ria $X$?", "answer": 1, "choices": [ "0,9", "0,8", "0,7", "0,6" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3.2", "question": "\nDe um tri\u00e2ngulo, sabe-se que os comprimentos dos seus lados s\u00e3o 4, 5 e 6. Seja $\\alpha$ a amplitude do maior \u00e2ngulo interno desse tri\u00e2ngulo. Qual \u00e9 o valor de $\\text{sen} \\alpha$, arredondado \u00e0s mil\u00e9simas?", "answer": 1, "choices": [ "0,989", "0,992", "0,995", "0,998" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "5", "question": "\nPara um certo n\u00famero real $k$, \u00e9 cont\u00ednua em $\\mathbb{R}$ a fun\u00e7\u00e3o $f$, definida por\n$f(x) = \\begin{cases} \\log_3 k & \\text{se } x = 1 \\\\ \\frac{x^2 - 1}{x - 1} & \\text{se } x \\neq 1 \\end{cases}$\nQual \u00e9 o valor de $k$?", "answer": 3, "choices": [ "5", "6", "8", "9" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "2", "question_number": "9.2", "question": "\nQual \u00e9 o valor de $sen(3 \\arccos(\\frac{1}{2}))$?", "answer": 2, "choices": [ "$\\frac{\\sqrt{3}}{2}$", "$\\frac{\\sqrt{2}}{2}$", "0", "1" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "2", "question_number": "11.1", "question": "\nConsidere, num referencial o.n. $Oxyz$, a reta $r$ definida por $\\frac{x-1}{2} = \\frac{3-y}{4} = z$. Qual dos seguintes vetores pode ser um vetor diretor de uma reta perpendicular \u00e0 reta $r$?", "answer": 2, "choices": [ "$\\vec{a} (2, 4, 1)$", "$\\vec{b} (-3, 1, 0)$", "$\\vec{c} (1, 1, 2)$", "$\\vec{d} (-4, 2, 0)$" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "2", "question_number": "11.2", "question": "\nQual \u00e9, para qualquer n\u00famero real positivo $a$, o limite da sucess\u00e3o $\\left(\\frac{n + \\ln a}{n}\\right)^{n+2}$ ?", "answer": 2, "choices": [ "$a^2$", "$2a$", "$a$", "$\\sqrt{a}$" ], "is_completion": true }, { "year": 2019, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "2", "question_number": "13.2", "question": "\nSeja $g$ a fun\u00e7\u00e3o definida em $]0,\\pi[$ por $g(x) = \\frac{1}{4} cos(2x) - cos x$\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $]-\\frac{\\pi}{2}, 0[$, definida por $f(x) = g(-x) + g(\\frac{\\pi}{2} - x)$. Qual das express\u00f5es seguintes tamb\u00e9m pode definir a fun\u00e7\u00e3o $f$?", "answer": 1, "choices": [ "$\\text{sen} x + \\cos x$", "$-\\text{sen} x - \\cos x$", "$\\text{sen} x - \\cos x$", "$-\\text{sen} x + \\cos x$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "2.1", "question": "\nLan\u00e7a-se cinco vezes um dado tetra\u00e9drico equilibrado, com as faces numeradas de 1 a 4, e regista-se o n\u00famero da face voltada para baixo.\nQual \u00e9 o valor, arredondado \u00e0s cent\u00e9simas, da probabilidade de, nos cinco lan\u00e7amentos, sair face 4 exatamente tr\u00eas vezes?", "answer": 3, "choices": [ "0,01", "0,03", "0,07", "0,09" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "2.2", "question": "\nDe um tri\u00e2ngulo, sabe-se que os comprimentos dos seus lados s\u00e3o 4, 5 e 8.\nSeja $\\alpha$ a amplitude, em graus, do maior \u00e2ngulo interno desse tri\u00e2ngulo.\nQual \u00e9 o valor de $\\alpha$, arredondado \u00e0s unidades?", "answer": 3, "choices": [ "75\u00b0", "100\u00b0", "120\u00b0", "125\u00b0" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "3.1", "question": "\nUm saco cont\u00e9m nove cart\u00f5es, indistingu\u00edveis ao tato, numerados de 1 a 9.\nRetiram-se, simultaneamente e ao acaso, quatro cart\u00f5es do saco.\nQual \u00e9 a probabilidade de o menor dos n\u00fameros sa\u00eddos ser 3 e o maior ser 8?", "answer": 1, "choices": [ "$\\frac{1}{18}$", "$\\frac{1}{21}$", "$\\frac{1}{6}$", "$\\frac{1}{7}$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "8", "question": "\nPara um certo n\u00famero real $k$, \u00e9 cont\u00ednua em $\\mathbb{R}$ a fun\u00e7\u00e3o $f$, definida por\n$f(x) = \\begin{cases} \\frac{x-1}{x^2 + x - 2} & \\text{se } x > 1 \\\\ k & \\text{se } x \\leq 1 \\end{cases}$\nQual \u00e9 o valor de $k$?", "answer": 2, "choices": [ "2", "3", "$\\frac{1}{3}$", "$\\frac{1}{2}$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "9.1", "question": "\nConsidere, num referencial o.n. $Oxyz$, a reta $r$, definida por $1-x = y \\wedge z=3$\nQual das equa\u00e7\u00f5es seguintes define um plano perpendicular \u00e0 reta $r$?", "answer": 1, "choices": [ "$x + y = 5$", "$x - y = 5$", "$x + y + 3z = 5$", "$x - y + 3z = 5$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "9.2", "question": "\nConsidere, num referencial o.n. $xOy$, a elipse que passa nos pontos de coordenadas (2,0) e (0,-1) e que \u00e9 sim\u00e9trica em rela\u00e7\u00e3o aos eixos coordenados.\nQual \u00e9 a dist\u00e2ncia focal desta elipse?", "answer": 1, "choices": [ "$2\\sqrt{2}$", "$2\\sqrt{3}$", "$2\\sqrt{5}$", "$2\\sqrt{6}$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "11", "question": "\nSejam $f$ e $g$ duas fun\u00e7\u00f5es de dom\u00ednio $\\mathbb{R}$.\nSabe-se que $f(x) = 2x + 1$ e que $(f \\circ g)(x) = 7$, para todo o valor real de $x$.\nQual das seguintes express\u00f5es define a fun\u00e7\u00e3o $g$?", "answer": 1, "choices": [ "-3", "3", "$x-3$", "$x+3$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "12.1", "question": "\nSeja $\\Omega$ o espa\u00e7o de resultados associado a uma certa experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos equiprov\u00e1veis e independentes.\nSabe-se que $P(\\overline{A} \\cup \\overline{B}) = \\frac{8}{9}$.\nQual \u00e9 o valor de $P(A)$?", "answer": 2, "choices": [ "$\\frac{1}{4}$", "$\\frac{3}{4}$", "$\\frac{1}{3}$", "$\\frac{2}{3}$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "12.2", "question": "\nConsidere a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\mathbb{R}^+$, definida por $f(x) = \\ln x$.\nSeja $(u_n)$ a sucess\u00e3o de termo geral $u_n = \\left(\\frac{n + 2}{n}\\right)^{\\frac{n}{4}}$.\nQual \u00e9 o valor de $\\lim f(u_n)$?", "answer": 2, "choices": [ "4", "2", "$\\frac{1}{2}$", "$\\frac{1}{4}$" ], "is_completion": true }, { "year": 2019, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "2", "question_number": "13.1", "question": "\nSeja $g$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}$, definida por\n$g(x) = \\begin{cases} x \\ln (1-x) & \\text{se } x \\leq 0 \\\\ \\frac{1 - 3x}{1 - e^{-x}} & \\text{se } x > 0 \\end{cases}$\nQual \u00e9 o declive da reta tangente ao gr\u00e1fico de $g$ no ponto de abcissa -1?", "answer": 0, "choices": [ "$0,5 + \\ln 2$", "$-0,5 + \\ln 2$", "$0,5 - \\ln 2$", "$-0,5 - \\ln 2$" ], "is_completion": true }, { "year": 2020, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "1.2", "question": "\nConsidere a sucess\u00e3o $(u_n)$ de termo geral $u_n = \\frac{8n-4}{n+1}$.\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $]-\\infty, 8[$, definida por $f(x) = log_2 (8 \u2013 x)$. A que \u00e9 igual $\\lim f (u_n)$?", "answer": 0, "choices": [ "-$\\infty$", "0", "1", "$+\\infty$" ], "is_completion": true }, { "year": 2020, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "2", "question": "\nQuatro pessoas v\u00e3o escolher, cada uma e em segredo, um dos seguintes n\u00fameros: 1, 2, 3, 4 e 5. Qual \u00e9 a probabilidade de exatamente duas delas escolherem o n\u00famero 5?", "answer": 3, "choices": [ "0,1530", "0,1532", "0,1534", "0,1536" ], "is_completion": true }, { "year": 2020, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3.2", "question": "\nUm saco cont\u00e9m bolas azuis e bolas brancas, indistingu\u00edveis ao tato. Cada bola tem uma \u00fanica cor e s\u00f3 existem bolas azuis e bolas brancas no saco.\nConsidere que se alterou a constitui\u00e7\u00e3o inicial do saco e que, neste, est\u00e3o agora oito bolas azuis e sete bolas brancas.\nPretende-se colocar todas estas bolas em dez caixas numeradas de 1 a 10, de tal forma que:\n\u2022 cada caixa com n\u00famero par tenha, pelo menos, uma bola azul;\n\u2022 cada caixa com n\u00famero \u00edmpar tenha, pelo menos, uma bola branca;\n\u2022 cada caixa tenha, no m\u00e1ximo, duas bolas.\nNestas condi\u00e7\u00f5es, de quantas maneiras diferentes podem ficar colocadas as bolas nas dez caixas?", "answer": 1, "choices": [ "1176", "2520", "28 016", "30 550" ], "is_completion": true }, { "year": 2020, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "9.3", "question": "\nSeja $f$ a fun\u00e7\u00e3o definida em]$-\\infty, 2]$ por $f(x) = x + \\ln (e^x + 1)$.\nSeja $h$ a fun\u00e7\u00e3o definida em $]-\\infty,2]$ por $h(x) = f(x) - x$.\nQual das express\u00f5es seguintes pode ser a express\u00e3o anal\u00edtica da fun\u00e7\u00e3o $h^{-1}$, fun\u00e7\u00e3o inversa de $h$?", "answer": 2, "choices": [ "$e^{x-1}$", "$1-e^{x}$", "$\\ln (e^{x}-1)$", "$\\ln (1-e^{x})$" ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "2", "question": "\nConsidere um cubo $[MNPQRSTU]$. Escolhem-se, ao acaso, tr\u00eas v\u00e9rtices distintos desse cubo. Qual \u00e9 a probabilidade de o plano por eles definido conter uma das faces do cubo?", "answer": 1, "choices": [ "$\\frac{1}{7}$", "$\\frac{3}{7}$", "$\\frac{1}{8}$", "$\\frac{3}{8}$" ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "4", "question": "\nConsidere todos os n\u00fameros naturais superiores a 9999 e inferiores a 22 000. Destes n\u00fameros, quantos se podem escrever com os algarismos 0, 1, 2 e 3?", "answer": 2, "choices": [ "192", "236", "384", "512" ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "5", "question": "\nDados dois n\u00fameros reais positivos, sabe-se que a soma dos seus logaritmos na base 8 \u00e9 igual a $\\frac{1}{3}$. A que \u00e9 igual o produto desses dois n\u00fameros?", "answer": 0, "choices": [ "2", "3", "8", "9" ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "7", "question": "\nSeja $(v_n)$ a sucess\u00e3o definida por:\n$v_n = \\begin{cases} n & \\text{se } n < 10 \\\\ 1 + \\frac{1}{n} & \\text{se } n \\geq 10 \\end{cases}$\nQual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 2, "choices": [ "A sucess\u00e3o $(v_n)$ tem limite nulo.", "A sucess\u00e3o $(v_n)$ \u00e9 divergente.", "A sucess\u00e3o $(v_n)$ \u00e9 limitada.", "A sucess\u00e3o $(v_n)$ \u00e9 mon\u00f3tona." ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "8.2", "question": "\nSeja $k$ um n\u00famero real. Sabe-se que $k + i$ \u00e9 uma das ra\u00edzes quadradas do n\u00famero complexo $3-4i$. Qual \u00e9 o valor de $k$?", "answer": 3, "choices": [ "2", "1", "-1", "-2" ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "10.1", "question": "\nSejam $f$ e $g$ as fun\u00e7\u00f5es, de dom\u00ednio $\\mathbb{R}$, definidas, respetivamente, por $f(x)= x^2$ e $g(x) = cos x$.\nQual \u00e9 o declive da reta tangente ao gr\u00e1fico da fun\u00e7\u00e3o $f \\circ g$ no ponto de abcissa $\\frac{\\pi}{4}$?", "answer": 1, "choices": [ "-2", "-1", "1", "2" ], "is_completion": true }, { "year": 2020, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "12.2", "question": "\nQual \u00e9 o valor de $\\lim_{x \\to 1} \\frac{f(x) - f(1)}{1 - x^2}$?", "answer": 1, "choices": [ "-2", "-1", "0", "2" ], "is_completion": true }, { "year": 2020, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nConsidere um dado c\u00fabico equilibrado, com as faces numeradas de 1 a 6.\nLan\u00e7a-se esse dado quatro vezes e escrevem-se, da esquerda para a direita, os algarismos sa\u00eddos, obtendo-se, assim, um n\u00famero com quatro algarismos.\nQual \u00e9 a probabilidade de esse n\u00famero ser par, menor do que 5000 e capicua (sequ\u00eancia de algarismos cuja leitura da direita para a esquerda ou da esquerda para a direita d\u00e1 o mesmo n\u00famero)?", "answer": 2, "choices": [ "$\\frac{1}{36}$", "$\\frac{5}{36}$", "$\\frac{1}{108}$", "$\\frac{5}{108}$" ], "is_completion": true }, { "year": 2020, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "4.2", "question": "\nUm hotel, que promove atividades ao ar livre, \u00e9 procurado por turistas de v\u00e1rias nacionalidades.\nTr\u00eas h\u00f3spedes suecos e quatro h\u00f3spedes dinamarqueses pretendem visitar os arredores do hotel. Para tal, o hotel disponibiliza quatro motos de dois lugares cada uma (uma preta, uma amarela, uma branca e uma verde).\nSabe-se que apenas os h\u00f3spedes dinamarqueses podem conduzir.\nDe quantas maneiras distintas se podem distribuir, deste modo, os sete h\u00f3spedes pelas quatro motos?", "answer": 3, "choices": [ "21", "35", "268", "576" ], "is_completion": true }, { "year": 2020, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "6", "question": "\nConsidere a sucess\u00e3o $(v_n)$ definida, por recorr\u00eancia, por\n$v_1 = 2$\n$v_{n+1} = \\frac{1}{v_n}$, para qualquer n\u00famero natural $n$\nQual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?", "answer": 3, "choices": [ "A sucess\u00e3o $(v_n)$ \u00e9 uma progress\u00e3o aritm\u00e9tica.", "A sucess\u00e3o $(v_n)$ \u00e9 uma progress\u00e3o geom\u00e9trica.", "A sucess\u00e3o $(v_n)$ \u00e9 mon\u00f3tona.", "A sucess\u00e3o $(v_n)$ \u00e9 limitada." ], "is_completion": true }, { "year": 2020, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "10.1", "question": "\nPara um certo n\u00famero real $k$, seja $g$ a fun\u00e7\u00e3o, de dom\u00ednio $\\mathbb{R}$, definida por\n$g(x) = \\begin{cases} \\frac{x^2 - x}{k - kx} & \\text{se } x < 1 \\\\ x^2 - 10 + 8\\ln x & \\text{se } x \\geq 1 \\end{cases}$\nSabe-se que $g$ \u00e9 cont\u00ednua no ponto 1.\nQual \u00e9 o valor de $k$?", "answer": 3, "choices": [ "$\\frac{1}{6}$", "$\\frac{1}{7}$", "$\\frac{1}{8}$", "$\\frac{1}{9}$" ], "is_completion": true }, { "year": 2020, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "11.1", "question": "\nConsidere a fun\u00e7\u00e3o $h$, de dom\u00ednio $\\mathbb{R}$, definida por $h(x) = \\frac{5}{4 + 3\\cos(2x)}$.\nQual \u00e9 a taxa m\u00e9dia de varia\u00e7\u00e3o da fun\u00e7\u00e3o $h$ entre $\\frac{\\pi}{6}$ e $\\frac{7\\pi}{6}$?", "answer": 2, "choices": [ "1", "$\\frac{1}{2}$", "0", "$-\\frac{1}{2}$" ], "is_completion": true }, { "year": 2021, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nNuma escola frequentada por estudantes portugueses e estrangeiros, 60% dos alunos s\u00e3o raparigas e 15% s\u00e3o rapazes estrangeiros.\nEscolheu-se, ao acaso, um aluno dessa escola e verificou-se que era um rapaz.\nQual \u00e9 a probabilidade de ele ser portugu\u00eas?", "answer": 3, "choices": [ "45%", "50%", "57,5%", "62,5%" ], "is_completion": true }, { "year": 2021, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "6", "question": "\nSeja $(V_n)$ uma progress\u00e3o geom\u00e9trica.\nSabe-se que $v_5 = 4$ e que $v_8 = 108$\nQual \u00e9 o valor de $v_6$?", "answer": 0, "choices": [ "12", "24", "48", "60" ], "is_completion": true }, { "year": 2021, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "8", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, considere $z\u2081 = 2e^{i\\frac{\\pi}{4}}$ e $z\u2082 = 2e^{i\\frac{3 \\pi}{28}}$\nSeja $w$ o n\u00famero complexo tal que $w = \\frac{z\u2081}{z\u2082}$\nSabe-se que, no plano complexo, o afixo do n\u00famero complexo $w$ \u00e9 um dos v\u00e9rtices de um pol\u00edgono regular com centro na origem do referencial e com outro v\u00e9rtice sobre o semieixo real positivo.\nQual \u00e9 o n\u00famero m\u00ednimo de v\u00e9rtices desse pol\u00edgono?", "answer": 1, "choices": [ "7", "14", "21", "28" ], "is_completion": true }, { "year": 2021, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3.1", "question": "\nNuma dada localidade, existe um clube onde se pratica badminton e t\u00e9nis.\nCom doze raquetes distintas, sendo seis de badminton e seis de t\u00e9nis, formam-se, ao acaso, dois conjuntos de seis raquetes cada um.\nQual \u00e9 o valor, arredondado \u00e0s cent\u00e9simas, da probabilidade de cada um dos dois conjuntos ficar com tr\u00eas raquetes de badminton e tr\u00eas raquetes de t\u00e9nis?", "answer": 1, "choices": [ "0,22", "0,43", "0,50", "0,87" ], "is_completion": true }, { "year": 2021, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "7", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, considere $z = 2e^{i \\frac{3\\pi}{5}}$.\nSeja $w$ o n\u00famero complexo tal que $z \\times w = i$.\nQual dos valores seguintes \u00e9 um argumento do n\u00famero complexo $w$?", "answer": 0, "choices": [ "$\\frac{19\\pi}{10}$", "$\\frac{2\\pi}{5}$", "$- \\frac{2\\pi}{5}$", "$- \\frac{19\\pi}{10}$" ], "is_completion": true }, { "year": 2021, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "11.1", "question": "\nNum laborat\u00f3rio cuja temperatura ambiente \u00e9 constante, aqueceu-se uma subst\u00e2ncia at\u00e9 atingir uma certa temperatura, superior \u00e0 temperatura ambiente, e, a seguir, deixou-se arrefecer essa subst\u00e2ncia durante uma hora.\nAdmita que a temperatura dessa subst\u00e2ncia, em graus Celsius, $t$ minutos ap\u00f3s o in\u00edcio do arrefecimento, \u00e9 dada por\n$T(t) = 20 + 100e^{-kt}, 0 \\leq t \\leq 60$\nem que $k$ \u00e9 uma constante real positiva.\nDurante o arrefecimento, houve um instante $t_1$ em que a temperatura da subst\u00e2ncia foi 30 \u00b0C.\nQual \u00e9 o valor de $k$?", "answer": 2, "choices": [ "$\\ln(\\frac{10}{t_1})$", "$t_1 - \\ln 10$", "$\\frac{\\ln10}{t_1}$", "$t_1 + \\ln 10$" ], "is_completion": true }, { "year": 2021, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "8", "question": "\nEm $\\mathbb{C}$, conjunto dos n\u00fameros complexos, considere os n\u00fameros complexos $z_1$ e $z_2$ tais que, para $\\theta \\in ]0, \\frac{\\pi}{2}[$, $z_1 = e^{i \\theta}$ e $z_2 = 2e^{i(\\theta + \\pi)}$.\nA qual dos quadrantes do plano complexo pertence o afixo do n\u00famero complexo $z_1 + z_2$?", "answer": 2, "choices": [ "Primeiro", "Segundo", "Terceiro", "Quarto" ], "is_completion": true }, { "year": 2021, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "12.1", "question": "\nPara conhecer a varia\u00e7\u00e3o do n\u00famero de bact\u00e9rias de uma determinada estirpe, colocou-se num tubo de ensaio fechado, com alguns nutrientes, um certo n\u00famero de bact\u00e9rias dessa estirpe.\nAdmita que, nessas condi\u00e7\u00f5es, o n\u00famero, $N$, em milhares, de bact\u00e9rias vivas existentes no tubo, $t$ horas ap\u00f3s a sua coloca\u00e7\u00e3o nesse tubo, \u00e9 dado por\n$N(t) = N_0 e^{1,08t - 0,3t^2}$\nem que $N_0$ representa a dimens\u00e3o, em milhares, da popula\u00e7\u00e3o inicial.\nCom o decorrer do tempo, para que tende o n\u00famero de bact\u00e9rias vivas existentes no tubo?", "answer": 3, "choices": [ "$+\\infty$", "0,78$N_0$", "$N_0$", "0" ], "is_completion": true }, { "year": 2022, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "1", "question": "\nQual das express\u00f5es seguintes \u00e9 termo geral de uma sucess\u00e3o convergente?", "answer": 1, "choices": [ "$(-1)^n \\times n$", "$\\frac{(-1)^n}{n}$", "$(-1)^n + n$", "$(-1)^n - n$" ], "is_completion": true }, { "year": 2022, "phase": "1.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nSeja $\\Omega$, conjunto finito, o espa\u00e7o amostral associado a uma dada experi\u00eancia aleat\u00f3ria.\nSejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n\u2022 $P(\\overline{B}) = 0,6$;\n\u2022 $P(A \\cup B) = 0,6$;\n\u2022 $A \\cap B = \\varnothing$.\nQual \u00e9 o valor de $P(\\overline{A})$?", "answer": 3, "choices": [ "0,2", "0,4", "0,6", "0,8" ], "is_completion": true }, { "year": 2022, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "2", "question": "\nA soma de todos os elementos de uma dada linha do tri\u00e2ngulo de Pascal \u00e9 igual a 16 384.\nQual \u00e9 o valor do quarto elemento da linha seguinte?", "answer": 1, "choices": [ "286", "455", "715", "1365" ], "is_completion": true }, { "year": 2022, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "11.1", "question": "\nUm tanque, que inicialmente tinha um certo volume de \u00e1gua salgada, disp\u00f5e de duas torneiras, uma de enchimento e outra de vazamento. As duas torneiras s\u00e3o abertas, em simult\u00e2neo, sendo vertida \u00e1gua do mar para o tanque at\u00e9 este estar cheio.\nAdmita que a massa de sal, $m$, em quilogramas, no tanque, $t$ minutos ap\u00f3s a abertura das torneiras, at\u00e9 o tanque estar cheio, \u00e9 dada por\n$m(t) = \\frac{30(1 + 0,006t)^3 - 29}{(1 + 0,006t)^2}$, com $t \\in [0, 250]$.\nQual \u00e9, com aproxima\u00e7\u00e3o \u00e0s unidades, a percentagem de aumento da massa de sal no tanque, no primeiro minuto ap\u00f3s a abertura das torneiras?", "answer": 1, "choices": [ "152%", "52%", "250%", "25%" ], "is_completion": true }, { "year": 2022, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nConsidere todos os n\u00fameros naturais de cinco algarismos diferentes que se podem formar com os algarismos de 0 a 5.\nDestes n\u00fameros, quantos t\u00eam o algarismo das unidades igual a 5?", "answer": 3, "choices": [ "625", "256", "128", "96" ], "is_completion": true }, { "year": 2022, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "6", "question": "\nSeja $k$ um n\u00famero natural.\nQual \u00e9 o limite da sucess\u00e3o $(u_n)$ definida por $u_n = \\left( \\frac{n+k}{n} \\right)^n$?", "answer": 2, "choices": [ "1", "$+\\infty$", "$e^{k}$", "$e^{-k}$" ], "is_completion": true }, { "year": 2023, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "1", "question": "\nQual \u00e9 o limite da sucess\u00e3o de termo geral $(1 + \\frac{1}{n})^{2n}$?", "answer": 2, "choices": [ "1", "$2e$", "$e^2$", "$+\\infty$" ], "is_completion": true }, { "year": 2023, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "5.1", "question": "\nUm grupo de jovens inscreveu-se num campo de f\u00e9rias que oferece as modalidades de surf e de skate.\nDez dos jovens do grupo v\u00e3o deslocar-se em fila, pela praia, para uma aula de surf.\nA Ana, o Diogo e o Francisco s\u00e3o tr\u00eas desses jovens.\nDe quantas formas diferentes se podem dispor os jovens na fila, ficando a Ana, o Diogo e o Francisco juntos?", "answer": 1, "choices": [ "483 840", "241 920", "60 480", "30 240" ], "is_completion": true }, { "year": 2023, "phase": "1\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "12.1", "question": "\nSeja $f$ a fun\u00e7\u00e3o, de dom\u00ednio $[0, \\pi]$, definida por $f(x) = sen (2x) +x$, e seja $r$ a reta de equa\u00e7\u00e3o $y = \u2212 x + 2$.\nQual das express\u00f5es seguintes pode definir a fun\u00e7\u00e3o derivada de $f$?", "answer": 3, "choices": [ "$2-2 cos^2x$", "$2-2 sen^2 x$", "$3-4 cos^2x$", "$3-4 sen^2x$" ], "is_completion": true }, { "year": 2023, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "1", "question": "\nSeja $(u_n)$ uma sucess\u00e3o tal que $\\lim u_n = 0$.\nQual das express\u00f5es seguintes pode ser termo geral de $(u_n)$?", "answer": 3, "choices": [ "$(1-\\frac{2}{n})^n$", "$-\\frac{n^2 + 1}{n}$", "$\\frac{4n+3}{3n+4}$", "$\\frac{(-1)^n}{n}$" ], "is_completion": true }, { "year": 2023, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nConsidere todos os n\u00fameros naturais de seis algarismos que \u00e9 poss\u00edvel formar com os algarismos de 1 a 9.\nDestes n\u00fameros, quantos t\u00eam exatamente dois cincos?", "answer": 1, "choices": [ "98 415", "61 440", "36 015", "25 200" ], "is_completion": true }, { "year": 2023, "phase": "2.\u00aa Fase", "subject": "Mathematics A", "question_group": "1", "question_number": "7", "question": "\nQual \u00e9 o valor de $\\lim_{x \\to 0} \\frac{\\text{sen}(2x)}{x}$?", "answer": 3, "choices": [ "0", "$\\frac{1}{2}$", "1", "2" ], "is_completion": true }, { "year": 2023, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "1", "question": "\nQual das express\u00f5es seguintes \u00e9 termo geral de uma sucess\u00e3o mon\u00f3tona?", "answer": 3, "choices": [ "$(n-5)^2$", "$\\frac{(-1)^n}{n+3}$", "$(-2)^n$", "$\\frac{1}{n}$" ], "is_completion": true }, { "year": 2023, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "3", "question": "\nSeja $\\Omega$ o espa\u00e7o amostral associado a uma certa experi\u00eancia aleat\u00f3ria, e sejam $A$ e $B$ dois acontecimentos ($A \\subset \\Omega$ e $B \\subset \\Omega$).\nSabe-se que:\n$\\cdot P(A \\cup B) = 0,8$;\n$\\cdot P(A \\cap \\overline{B}) = 0,5$.\nQual \u00e9 o valor de $P(B)$?", "answer": 1, "choices": [ "0,2", "0,3", "0,5", "0,6" ], "is_completion": true }, { "year": 2023, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "8", "question": "\nPara certos valores de $a$ e de $b$ ($a > 1$ e $b > 1$), tem-se $log_a \\left(\\frac{b}{a}\\right) = 2$.\nQual \u00e9 o valor de $log_a(\\sqrt{a^3} \\times b^2)$?", "answer": 1, "choices": [ "$\\frac{13}{2}$", "$\\frac{15}{2}$", "$\\frac{19}{2}$", "$\\frac{21}{2}$" ], "is_completion": true }, { "year": 2023, "phase": "\u00c9poca Especial", "subject": "Mathematics A", "question_group": "1", "question_number": "12", "question": "\nSeja $z$ um n\u00famero complexo de argumento $\\frac{\\pi}{7}$.\nQual dos seguintes valores \u00e9 um argumento de $2iz$?", "answer": 1, "choices": [ "$\\frac{5\\pi}{14}$", "$\\frac{9\\pi}{14}$", "$\\frac{6\\pi}{7}$", "$\\frac{8\\pi}{7}$" ], "is_completion": true } ]