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Ficha de avaliacao dominio 01 12 ano - calculo combinatorio (enunciado)
1.
DIMENSÕES • Matemática
• 12.º ano © Santillana 1 FICHA DE AVALIAÇÃO 1 Matemática 12.º Ano NOME: ______________________________________ N.o : ____ TURMA: ____ DATA: _____________ Cálculo combinatório Duração: 90 minutos Grupo I Este grupo é constituído por cinco (5) itens de seleção. Para cada um deles, são indicadas quatro alternativas, das quais apenas uma está correta. Deverá registar as suas respostas na folha de teste. Se apresentar mais do que uma resposta, a questão será anulada, o mesmo acontecendo se a letra transcrita for ilegível. Não apresente cálculos nem justificações. 1. Sejam 𝑨 e 𝑩 dois conjuntos tais que 𝑨 ⊂ 𝑼 e 𝑩 ⊂ 𝑼. O conjunto (𝑨 ∪ 𝑩) ∩ (𝑨 ∪ 𝑩 ̅) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ é igual a: (A) 𝐴 (B) 𝑈 (C) ∅ (D) 𝐴̅ 2. Oito amigos, quatro rapazes e quatro raparigas, vão restaurante e sentam-se numa mesa re- tangular, ficando quatro de cada lado. De quantas formas se podem sentar de forma que os rapazes não fiquem todos do mesmo lado? (A) 4! .4! .2 (B) 4! .4! (C) 8! − 4! .4! .2 (D) 8! − 4! .4! 3. Considere todos os números pares com cinco algarismos. Quantos destes números têm exa- tamente quatro algarismos ímpares. (A) 4 5 5 C (B) 5 5 (C) ! 5 (D) 4 5 5 A 4. No Triângulo de Pascal, considere a linha que contém os elementos da forma k C 2017 . Quantos elementos desta linha são maiores do que 4 2017 C ? (A) 2012 (B) 2008 (C) 2007 (D) 2013 5. O termo independente do desenvolvimento (𝟐𝒙 + 𝟏 √𝒙 ) 𝟏𝟓 é: (A) 𝐶5 15 (B) 210 × 𝐶10 15 (C) 210 × 𝐶5 15 (D) 25 × 𝐶5 15
2.
DIMENSÕES • Matemática
• 12.º ano © Santillana 2 Grupo II Este grupo é constituído por cinco (5) itens de construção, pelo que deverá justificar convenientemente as suas respostas. Deverá registar todos os cálculos que efetuar. Atenção: quando, para o resultado, não é pedida a aproximação, pretende-se sempre o valor exato. 1. Sejam 𝑨 e 𝑩 dois conjuntos tais que 𝑨 ⊂ 𝑼 e 𝑩 ⊂ 𝑼. Mostre que: 𝑨 ̅ ∪ (𝑨 ∩ 𝑩) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = 𝑨𝑩. 2. Numa prova olímpica de natação, participam na final oito atletas que disputam as medalhas de ouro, prata e bronze. De quantas formas diferentes se pode fazer a distribuição das medalhas? 3. No jogo do Euromilhões, quantas apostas simples se podem efetuar? 4. Dos 12 elementos de uma associação de estudantes, vão ser selecionados três para representar a escola num encontro com elementos da Divisão de Educação do concelho onde a escola se situa. Dos 12 elementos, 7 são rapazes, um dos quais é o presidente da Associação de Estudantes. 4.1 De quantas formas diferentes se pode efetuar esta seleção? 4.2 Se o presidente da Associação de Estudantes for um dos participantes, de quantas formas diferentes se pode efetuar esta seleção? 4.3 De quantas formas se pode efetuar esta seleção se os representantes não forem todos do mesmo sexo? 5. Pretende-se arrumar numa estante 3 CD dos Xutos & Pontapés, 4 CD da Deolinda e 2 CD do Jorge Palma (todos diferentes). De quantas formas diferentes é possível fazê-lo se: 5.1 os CD de cada artista ficarem juntos? 5.2 os CD da Deolinda ficarem juntos? 6. Sabe-se que a soma dos últimos quatro números de uma determinada linha do Triângulo de Pascal é 383439. Sabendo que o terceiro elemento da linha seguinte é 8778, determine os quatro primeiros elementos das duas linhas referidas. 7. Determine 𝒌 tal que 𝟓𝒌 seja o coeficiente do termo de grau 4 no desenvolvimento de (𝒙𝟐 — 𝟏 𝒙 ) 𝟓 . 8. Mostre que: 𝑪𝟎 𝒏 + 𝟐 𝑪𝟏 𝒏 + 𝟐𝟐 𝑪𝟐 𝒏 + ⋯ + 𝟐𝒏 𝑪𝒏 𝒏 = 𝟑𝒏 , ∀𝒏 ∈ ℕ.
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