Quiz — تصحيح إختبار مادة الرياضيات شعبة الرياضيات دورة جوان 2018.pdf

Question 1 sur 10 20:00

[{"id":29560,"question":"Quelle est la limite de la suite définie par u\u003Csub\u003En\u003C\/sub\u003E = (3n² + 2n - 1)\/(2n² + 5) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients des termes de plus haut degré. Ici, 3n²\/2n² = 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\": \"A. 0\", \"b\": \"B. 3\/2\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":29561,"question":"La fonction f(x) = x³ - 3x² + 4 est convexe sur l'intervalle [-1, 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La convexité est déterminée par la dérivée seconde : f''(x) = 6x - 6. Sur [-1, 2], f''(x) change de signe (négative puis positive), donc la fonction n'est pas convexe sur tout l'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":29562,"question":"Quelle est la primitive de la fonction f(x) = 2x * e^(x²) ?","option_a":"A. e^(x²) + C","option_b":"B. x² * e^(x²) + C","option_c":"C. e^(x²) * (x² - 1) + C","option_d":"D. x * e^(x²) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 2x * e^(x²) est e^(x²) + C, car la dérivée de e^(x²) est 2x * e^(x²).","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. e^(x²) + C\", \"b\": \"B. x² * e^(x²) + C\", \"c\": \"C. e^(x²) * ","_debug_options_count":4},{"id":29563,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale :","option_a":"A. y = C * e^(-2x)","option_b":"B. y = C * e^(2x)","option_c":"C. y = C * x * e^(-2x)","option_d":"D. y = C * ln(x) * e^(-2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle linéaire du premier ordre y' + 2y = 0 a pour solution générale y = C * e^(-2x), où C est une constante.","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 = C * e^(-2x)\", \"b\": \"B. y = C * e^(2x)\", \"c\": \"C. y = C * x","_debug_options_count":4},{"id":29564,"question":"Le produit scalaire de deux vecteurs u et v est nul si et seulement si u et v sont orthogonaux.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit scalaire de deux vecteurs est nul si et seulement si les vecteurs sont orthogonaux (ou si l'un des vecteurs est nul).","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":29565,"question":"Quelle est la valeur de l'intégrale ∫\u003Csub\u003E0\u003C\/sub\u003E\u003Csup\u003E1\u003C\/sup\u003E (3x² + 2x + 1) dx ?","option_a":"A. 2","option_b":"B. 3","option_c":"C. 4","option_d":"D. 5","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale se calcule comme suit : [x³ + x² + x] de 0 à 1 = (1 + 1 + 1) - (0 + 0 + 0) = 3.","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. 2\", \"b\": \"B. 3\", \"c\": \"C. 4\", \"d\": \"D. 5\"}}","_debug_options_count":4},{"id":29566,"question":"La probabilité conditionnelle P(A|B) est égale à :","option_a":"A. P(A) * P(B)","option_b":"B. P(A ∩ B) \/ P(B)","option_c":"C. P(A) + P(B)","option_d":"D. P(A) \/ P(B)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité conditionnelle P(A|B) est définie par P(A ∩ B) \/ P(B), à condition que P(B) ≠ 0.","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. P(A) * P(B)\", \"b\": \"B. P(A ∩ B) \/ P(B)\", \"c\": \"C. P(A) + P(B","_debug_options_count":4},{"id":29567,"question":"L'équation de la tangente à la courbe y = x² + 3x - 2 au point d'abscisse x = 1 est :","option_a":"A. y = 5x - 3","option_b":"B. y = 5x + 3","option_c":"C. y = 3x - 1","option_d":"D. y = x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente a pour équation y = f'(1)(x - 1) + f(1). Ici, f'(x) = 2x + 3, donc f'(1) = 5 et f(1) = 2. D'où y = 5(x - 1) + 2 = 5x - 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. y = 5x - 3\", \"b\": \"B. y = 5x + 3\", \"c\": \"C. y = 3x - 1\", \"d\": ","_debug_options_count":4},{"id":29568,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","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":29569,"question":"Quelle est la valeur de la somme S = Σ\u003Csub\u003Ek=1\u003C\/sub\u003E\u003Csup\u003En\u003C\/sup\u003E k² ?","option_a":"A. n(n+1)\/2","option_b":"B. n(n+1)(2n+1)\/6","option_c":"C. n²(n+1)²\/4","option_d":"D. n(n-1)\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des carrés des n premiers entiers naturels est donnée par la formule S = n(n+1)(2n+1)\/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. n(n+1)\/2\", \"b\": \"B. n(n+1)(2n+1)\/6\", \"c\": \"C. n²(n+1)²\/4\", \"","_debug_options_count":4}]