Quiz — fiche_TD_2011.pdf

Question 1 sur 10 20:00

[{"id":75315,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"6x² + 2","option_c":"3x + 2","option_d":"6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6x + 2\", \"b\": \"6x² + 2\", \"c\": \"3x + 2\", \"d\": \"6x - 5\"}}","_debug_options_count":4},{"id":75316,"question":"La suite (u_n) définie par u_n = 2n - 1 est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite est arithmétique si la différence entre deux termes consécutifs est constante. Ici, u_{n+1} - u_n = 2, donc la suite est arithmétique.","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":75317,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"e²","option_b":"2e","option_c":"ln(2)","option_d":"e\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation ln(x) = a a pour solution x = e^a. Ici, x = e².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"e²\", \"b\": \"2e\", \"c\": \"ln(2)\", \"d\": \"e\/2\"}}","_debug_options_count":4},{"id":75318,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à :","option_a":"1","option_b":"0","option_c":"-1","option_d":"impossible à déterminer","option_e":"","option_f":"","bonne_reponse":"b","explication":"Deux vecteurs orthogonaux ont un produit scalaire nul par définition.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\", \"b\": \"0\", \"c\": \"-1\", \"d\": \"impossible à déterminer\"}}","_debug_options_count":4},{"id":75319,"question":"La fonction f(x) = x³ est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = 6x, qui n'est pas toujours positive sur ℝ.","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":75320,"question":"Quelle est la limite de (3n² + 2n - 1)\/(n² + 5) quand n tend vers +∞ ?","option_a":"0","option_b":"3","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n², on obtient (3 + 2\/n - 1\/n²)\/(1 + 5\/n²) → 3\/1 = 3 quand n → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"3\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":75321,"question":"Dans un triangle rectangle, le carré de l'hypoténuse est égal à :","option_a":"La somme des carrés des deux autres côtés","option_b":"Le produit des deux autres côtés","option_c":"La différence des carrés des deux autres côtés","option_d":"Le double de la somme des deux autres côtés","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de Pythagore : dans un triangle rectangle, le carré de l'hypoténuse est égal à la somme des carrés des deux autres côtés.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"La somme des carrés des deux autres côtés\", \"b\": \"Le produit d","_debug_options_count":4},{"id":75322,"question":"La fonction f(x) = e^x est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^x est f'(x) = e^x, qui est toujours positive sur ℝ. Donc f est strictement 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":75323,"question":"Quelle est la valeur de l'intégrale ∫(2x + 1) dx entre 0 et 1 ?","option_a":"2","option_b":"1","option_c":"3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule comme [x² + x] de 0 à 1 = (1 + 1) - (0 + 0) = 2. Cependant, l'option correcte est 3 si l'intégrale est ∫(2x + 1) dx entre -1 et 1. Ici, correction : 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\": \"2\", \"b\": \"1\", \"c\": \"3\", \"d\": \"0\"}}","_debug_options_count":4},{"id":75324,"question":"Le discriminant d'une équation du second degré ax² + bx + c = 0 est donné par :","option_a":"b² - 4ac","option_b":"a² - 4bc","option_c":"4ac - b²","option_d":"b² + 4ac","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = b² - 4ac détermine le nombre et la nature des solutions de l'équation du second degré.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"b² - 4ac\", \"b\": \"a² - 4bc\", \"c\": \"4ac - b²\", \"d\": \"b² + 4ac\"}","_debug_options_count":4}]