Quiz — test-2-corrige-4.pdf

Question 1 sur 10 20:00

[{"id":76276,"question":"Quelle est la solution de l'équation x² - 5x + 6 = 0 ?","option_a":"A) x = 2 ou x = 3","option_b":"B) x = -2 ou x = -3","option_c":"C) x = 1 ou x = 6","option_d":"D) x = -1 ou x = -6","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation se factorise en (x-2)(x-3)=0, donc les solutions sont x=2 et x=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) x = 2 ou x = 3\", \"b\": \"B) x = -2 ou x = -3\", \"c\": \"C) x = 1 ou","_debug_options_count":4},{"id":76277,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle dérivable en x=1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ. Sa dérivée f'(x)=3x²-6x+2 existe donc en x=1.","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":76278,"question":"Quelle est la limite de (2x² + 3x - 5)\/(x² - 1) lorsque x tend vers l'infini ?","option_a":"A) 0","option_b":"B) 2","option_c":"C) +∞","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient (2 + 3\/x - 5\/x²)\/(1 - 1\/x²) → 2\/1 = 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) 2\", \"c\": \"C) +∞\", \"d\": \"D) -∞\"}}","_debug_options_count":4},{"id":76279,"question":"Le théorème de Thalès s'applique-t-il dans un triangle rectangle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Thalès s'applique à tout triangle (quelconque ou rectangle) dès que deux droites sont 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":76280,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"A) cos(2x)","option_b":"B) 2cos(2x)","option_c":"C) 2sin(x)","option_d":"D) -2cos(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u) est u'cos(u). Ici u=2x donc u'=2, d'où f'(x)=2cos(2x).","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) cos(2x)\", \"b\": \"B) 2cos(2x)\", \"c\": \"C) 2sin(x)\", \"d\": \"D) -2co","_debug_options_count":4},{"id":76281,"question":"L'équation x³ + x + 1 = 0 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x)=x³+x+1 est strictement croissante (f'(x)=3x²+1\u003E0) et f(-1)=-1, f(0)=1, donc elle s'annule une fois sur 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":76282,"question":"Quelle est l'image de 2 par la fonction f définie par f(x) = √(x² - 4) ?","option_a":"A) 0","option_b":"B) 2","option_c":"C) √2","option_d":"D) 2√2","option_e":"","option_f":"","bonne_reponse":"a","explication":"f(2) = √(2² - 4) = √(4-4) = √0 = 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\": \"A) 0\", \"b\": \"B) 2\", \"c\": \"C) √2\", \"d\": \"D) 2√2\"}}","_debug_options_count":4},{"id":76283,"question":"Le produit de deux nombres complexes conjugués est toujours réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si z = a+ib, alors z̅ = a-ib et z×z̅ = a² + b² ∈ 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":76284,"question":"Quelle est la solution de l'inéquation x² - 4x + 3 \u003E 0 ?","option_a":"A) x ∈ ]1; 3[","option_b":"B) x ∈ ]-∞; 1[ ∪ ]3; +∞[","option_c":"C) x ∈ ]-∞; 1[","option_d":"D) x ∈ ]3; +∞[","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'inéquation se factorise en (x-1)(x-3)\u003E0. Le produit est positif à l'extérieur des racines, donc x\u003C1 ou x\u003E3.","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) x ∈ ]1; 3[\", \"b\": \"B) x ∈ ]-∞; 1[ ∪ ]3; +∞[\", \"c\": \"","_debug_options_count":4},{"id":76285,"question":"La fonction exponentielle est-elle toujours croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle f(x)=eˣ a pour dérivée f'(x)=eˣ \u003E 0 pour tout x ∈ ℝ, donc elle 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}]