Quiz — mp-2020-2021-dm07.pdf

Question 1 sur 10 20:00

[{"id":60312,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A) 6x + 2","option_b":"B) 3x + 2","option_c":"C) 6x² + 2","option_d":"D) 3x² + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. Donc 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\": \"A) 6x + 2\", \"b\": \"B) 3x + 2\", \"c\": \"C) 6x² + 2\", \"d\": \"D) 3x² +","_debug_options_count":4},{"id":60313,"question":"L'équation x² - 5x + 6 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = (-5)² - 4*1*6 = 25 - 24 = 1 \u003E 0, donc l'équation admet deux solutions réelles distinctes.","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":60314,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) 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 - 1\/x²)\/(1 - 4\/x²). Lorsque x → ∞, les termes en 1\/x et 1\/x² tendent vers 0, donc la limite est 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":60315,"question":"Le produit scalaire de deux vecteurs u et v est toujours positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit scalaire u·v = ||u||*||v||*cos(θ), où θ est l'angle entre les vecteurs. Si θ \u003E 90°, cos(θ) \u003C 0, donc le produit scalaire peut être négatif.","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":60316,"question":"Résolvez l'inéquation 2x - 3 \u003E 5. Quelle est la solution ?","option_a":"A) x \u003E 1","option_b":"B) x \u003E 4","option_c":"C) x \u003C 4","option_d":"D) x \u003E -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"2x - 3 \u003E 5 ⇒ 2x \u003E 8 ⇒ x \u003E 4. La solution est donc x \u003E 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\": \"A) x \u003E 1\", \"b\": \"B) x \u003E 4\", \"c\": \"C) x \u003C 4\", \"d\": \"D) x \u003E -1\"}}","_debug_options_count":4},{"id":60317,"question":"La fonction f(x) = x³ est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² est toujours positive (sauf en x=0 où elle est nulle), donc la fonction est strictement croissante 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":60318,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"A) x = e²","option_b":"B) x = 2","option_c":"C) x = 100","option_d":"D) x = ln(2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation ln(x) = 2 ⇒ x = e², car ln(e²) = 2 par définition du logarithme.","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 = e²\", \"b\": \"B) x = 2\", \"c\": \"C) x = 100\", \"d\": \"D) x = ln(","_debug_options_count":4},{"id":60319,"question":"La probabilité d'un événement impossible est égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, la probabilité d'un événement impossible est toujours égale 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},{"id":60320,"question":"Quelle est la valeur de l'intégrale ∫(3x² + 2x) dx ?","option_a":"A) x³ + x² + C","option_b":"B) 3x³ + 2x² + C","option_c":"C) x³ + x + C","option_d":"D) 6x + 2 + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de 3x² est x³, celle de 2x est x², donc ∫(3x² + 2x) dx = x³ + x² + 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) x³ + x² + C\", \"b\": \"B) 3x³ + 2x² + C\", \"c\": \"C) x³ + x + ","_debug_options_count":4},{"id":60321,"question":"L'équation de la tangente à la courbe y = x² au point d'abscisse x=1 est y = 2x - 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en x=1 a pour équation y = f'(1)(x-1) + f(1) = 2(x-1) + 1 = 2x - 1. L'affirmation est donc vraie.","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}]