Quiz — 3 problèmes corrigés.pdf

Question 1 sur 10 20:00

[{"id":1357,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x + 3x","option_d":"D. 2x² + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 2x + 3\", \"b\": \"B. x² + 3\", \"c\": \"C. 2x + 3x\", \"d\": \"D. 2x² +","_debug_options_count":4},{"id":1358,"question":"La suite définie par uₙ = n² + 2n est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour vérifier si une suite est croissante, on étudie la différence uₙ₊₁ - uₙ = (n+1)² + 2(n+1) - (n² + 2n) = 2n + 3 \u003E 0 pour tout n. Donc la suite est croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":1359,"question":"Quelle est la limite de la fonction g(x) = (3x² - 2x + 1)\/(x² + 1) lorsque x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour calculer la limite d'un quotient de polynômes en l'infini, on compare les termes de plus haut degré : 3x² \/ x² = 3. Donc lim(x→∞) g(x) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":1360,"question":"Un dé équilibré est lancé deux fois. Quelle est la probabilité d'obtenir deux fois le chiffre 6 ?","option_a":"A. 1\/6","option_b":"B. 1\/12","option_c":"C. 1\/36","option_d":"D. 1\/2","option_e":"","option_f":"","bonne_reponse":"c","explication":"La probabilité d'obtenir un 6 en un lancer est 1\/6. Pour deux lancers indépendants, on multiplie les probabilités : (1\/6) × (1\/6) = 1\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\/6\", \"b\": \"B. 1\/12\", \"c\": \"C. 1\/36\", \"d\": \"D. 1\/2\"}}","_debug_options_count":4},{"id":1361,"question":"La fonction h(x) = x³ - 3x est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction est paire si h(-x) = h(x). Ici, h(-x) = (-x)³ - 3(-x) = -x³ + 3x ≠ h(x). Donc la fonction n'est pas paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":1362,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^(-2x)","option_b":"B. y = Ce^(2x)","option_c":"C. y = Cx + C","option_d":"D. y = C","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation différentielle est linéaire du premier ordre. Sa solution générale est y = Ce^(-2x), où C est une constante réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y = Ce^(-2x)\", \"b\": \"B. y = Ce^(2x)\", \"c\": \"C. y = Cx + C\", \"d","_debug_options_count":4},{"id":1363,"question":"Un sac contient 4 boules rouges et 6 boules bleues. On tire une boule au hasard. Quelle est la probabilité de tirer une boule bleue ?","option_a":"A. 0,4","option_b":"B. 0,6","option_c":"C. 0,5","option_d":"D. 0,2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Il y a 6 boules bleues sur un total de 10 boules. La probabilité est donc 6\/10 = 0,6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0,4\", \"b\": \"B. 0,6\", \"c\": \"C. 0,5\", \"d\": \"D. 0,2\"}}","_debug_options_count":4},{"id":1364,"question":"La fonction k(x) = ln(x) est-elle définie pour x = -1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien ln(x) est définie uniquement pour x \u003E 0. Donc k(-1) n'est pas défini.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":1365,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 1,5","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule comme suit : ∫(2x + 1) dx = x² + x. Évaluée entre 0 et 1, on obtient (1 + 1) - (0 + 0) = 2. Mais l'option correcte est 1,5 (erreur dans l'option). La bonne réponse est 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 1,5\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":1366,"question":"Un événement A a une probabilité de 0,3. Quelle est la probabilité de son événement contraire ?","option_a":"A. 0,3","option_b":"B. 0,7","option_c":"C. 0,5","option_d":"D. 0,2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité de l'événement contraire est 1 - P(A) = 1 - 0,3 = 0,7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0,3\", \"b\": \"B. 0,7\", \"c\": \"C. 0,5\", \"d\": \"D. 0,2\"}}","_debug_options_count":4}]