Quiz — Examen 2011 + Corrigé acad a.pdf

Question 1 sur 10 20:00

[{"id":29070,"question":"Quelle est la dérivée de la fonction f(x) = x² * ln(x) ?","option_a":"A. 2x * ln(x) + x","option_b":"B. 2x * ln(x) + 1","option_c":"C. x * ln(x) + x","option_d":"D. 2x * ln(x) + x²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x et celle de ln(x) est 1\/x. En appliquant la règle du produit, on obtient : f'(x) = 2x * ln(x) + x² * (1\/x) = 2x * ln(x) + x.","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 * ln(x) + x\", \"b\": \"B. 2x * ln(x) + 1\", \"c\": \"C. x * ln(x) ","_debug_options_count":4},{"id":29071,"question":"Soit une fonction f continue sur [a, b] et dérivable sur ]a, b[. Si f'(x) \u003E 0 pour tout x dans ]a, b[, alors f est strictement croissante sur [a, b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental du calcul différentiel : si la dérivée est strictement positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","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":29072,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à 1) e^(-x) dx ?","option_a":"A. 1 - 1\/e","option_b":"B. e - 1","option_c":"C. 1\/e","option_d":"D. e","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de e^(-x) est -e^(-x). En évaluant entre 0 et 1, on obtient : [-e^(-1)] - [-e^(0)] = -1\/e + 1 = 1 - 1\/e.","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. 1 - 1\/e\", \"b\": \"B. e - 1\", \"c\": \"C. 1\/e\", \"d\": \"D. e\"}}","_debug_options_count":4},{"id":29073,"question":"Dans un espace vectoriel, si deux vecteurs sont colinéaires, alors ils sont parallèles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux vecteurs colinéaires ont la même direction ou des directions opposées, donc ils sont toujours parallèles.","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":29074,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A. 1\/6","option_b":"B. 1\/12","option_c":"C. 1\/36","option_d":"D. 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 36 issues possibles (6 faces × 6 faces). Le double 6 correspond à une seule issue, donc la probabilité est 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\/18\"}}","_debug_options_count":4},{"id":29075,"question":"La fonction f(x) = x^3 - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1. En étudiant le signe de f' autour de 1, on voit que la fonction change de variation, donc c'est un extremum local.","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":29076,"question":"Quelle est la primitive de la fonction f(x) = 1\/(x² + 1) ?","option_a":"A. arctan(x) + C","option_b":"B. ln(x² + 1) + C","option_c":"C. arccos(x) + C","option_d":"D. x\/(x² + 1) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arctan(x) est 1\/(x² + 1), donc la primitive est arctan(x) + C.","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. arctan(x) + C\", \"b\": \"B. ln(x² + 1) + C\", \"c\": \"C. arccos(x) ","_debug_options_count":4},{"id":29077,"question":"Dans une loi binomiale B(n, p), l'espérance est égale à n * p.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, l'espérance d'une loi binomiale B(n, p) est E(X) = n * p.","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":29078,"question":"Quelle est la limite de la fonction f(x) = (x² - 4)\/(x - 2) lorsque x tend vers 2 ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. -4","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"c","explication":"En simplifiant f(x) = (x - 2)(x + 2)\/(x - 2), on obtient f(x) = x + 2 pour x ≠ 2. Donc la limite est 2 + 2 = 4.","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. 4\", \"c\": \"C. -4\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":29079,"question":"L'équation de la tangente à la courbe de f(x) = x² au point d'abscisse a est y = 2a(x - a) + a².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en a a pour équation y = f'(a)(x - a) + f(a). Ici, f'(a) = 2a et f(a) = a², donc l'équation est correcte.","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}]