Quiz — Examen d'A-H 2015-2016.PDF

Question 1 sur 10 20:00

[{"id":16789,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 5x","option_c":"x² - 2x + 5","option_d":"3x² - 4x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle (x^n)' = n*x^(n-1). Ainsi, f'(x) = 3x² - 4x + 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 5x\", \"c\": \"x² - 2x + 5\", \"d\": ","_debug_options_count":4},{"id":16790,"question":"La limite en +∞ de la fonction f(x) = (2x² + 3x) \/ (x² - 1) est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient (2 + 3\/x) \/ (1 - 1\/x²) → 2\/1 = 2 quand 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":16791,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = Cx + D","option_d":"y = C sin(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est y = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = Cx + D\", \"d\": \"y = C","_debug_options_count":4},{"id":16792,"question":"Dans une classe de 30 élèves, 12 ont les yeux bleus. Quelle est la probabilité qu'un élève choisi au hasard ait les yeux bleus ?","option_a":"0.4","option_b":"0.3","option_c":"0.25","option_d":"0.5","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est le rapport du nombre d'élèves aux yeux bleus sur le nombre total d'élèves : 12\/30 = 0.4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.4\", \"b\": \"0.3\", \"c\": \"0.25\", \"d\": \"0.5\"}}","_debug_options_count":4},{"id":16793,"question":"La fonction f(x) = x² est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = 2 \u003E 0 pour tout x ∈ ℝ.","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":16794,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) (3x² + 2x) dx ?","option_a":"2","option_b":"1","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale se calcule en trouvant une primitive : [x³ + x²] de 0 à 1 = (1 + 1) - (0 + 0) = 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\": \"2\", \"b\": \"1\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":16795,"question":"Un vecteur directeur de la droite d'équation 2x - 3y + 5 = 0 est :","option_a":"(3, 2)","option_b":"(2, -3)","option_c":"(-3, 2)","option_d":"(1, 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un vecteur directeur d'une droite ax + by + c = 0 est (-b, a). Ici, le vecteur est (3, 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"(3, 2)\", \"b\": \"(2, -3)\", \"c\": \"(-3, 2)\", \"d\": \"(1, 1)\"}}","_debug_options_count":4},{"id":16796,"question":"La suite définie par uₙ = n² \/ (n + 1) est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite n'est pas bornée car uₙ → +∞ 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":16797,"question":"Quelle est la solution de l'inéquation x² - 5x + 6 \u003E 0 ?","option_a":"x ∈ ]2, 3[","option_b":"x ∈ ]-∞, 2[ ∪ ]3, +∞[","option_c":"x ∈ [2, 3]","option_d":"x ∈ ]-∞, +∞[","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les racines de x² - 5x + 6 = 0 sont 2 et 3. L'inéquation est positive à l'extérieur des racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x ∈ ]2, 3[\", \"b\": \"x ∈ ]-∞, 2[ ∪ ]3, +∞[\", \"c\": \"x ∈ ","_debug_options_count":4},{"id":16798,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours nul.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux vecteurs orthogonaux ont un produit scalaire égal à 0.","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}]