Quiz — 663b84a71abd8_Corrigé-Sujet N°01.pdf

Question 1 sur 10 20:00

[{"id":82427,"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 = 0 ou x = 5","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":82428,"question":"La dérivée de la fonction f(x) = x³ - 2x² + 5x est f'(x) = 3x² - 4x + 5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée est bien f'(x) = 3x² - 4x + 5, car la dérivée de x³ est 3x², celle de -2x² est -4x, et celle de 5x est 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":82429,"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), 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":82430,"question":"Dans une loi binomiale de paramètres n=10 et p=0.3, la probabilité P(X=3) est égale à :","option_a":"A) C(10,3) * 0.3³ * 0.7⁷","option_b":"B) C(10,3) * 0.7³ * 0.3⁷","option_c":"C) 0.3³ * 0.7⁷","option_d":"D) 10 * 0.3 * 0.7","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité P(X=k) dans une loi binomiale est donnée par C(n,k) * p^k * (1-p)^(n-k). Ici, k=3, n=10, p=0.3.","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) C(10,3) * 0.3³ * 0.7⁷\", \"b\": \"B) C(10,3) * 0.7³ * 0.3⁷\",","_debug_options_count":4},{"id":82431,"question":"Une suite géométrique de premier terme u₀=5 et de raison q=2 est définie par uₙ = 5 * 2ⁿ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule générale d'une suite géométrique est uₙ = u₀ * qⁿ. Avec u₀=5 et q=2, on obtient bien uₙ = 5 * 2ⁿ.","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":82432,"question":"Quelle est la solution de l'inéquation 2x - 7 \u003E 3x + 1 ?","option_a":"A) x \u003C -8","option_b":"B) x \u003E -8","option_c":"C) x \u003C 8","option_d":"D) x \u003E 8","option_e":"","option_f":"","bonne_reponse":"a","explication":"En résolvant 2x - 7 \u003E 3x + 1, on obtient -x \u003E 8, donc x \u003C -8.","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 \u003C -8\", \"b\": \"B) x \u003E -8\", \"c\": \"C) x \u003C 8\", \"d\": \"D) x \u003E 8\"}}","_debug_options_count":4},{"id":82433,"question":"L'espérance d'une variable aléatoire suivant une loi binomiale B(n,p) est égale à :","option_a":"A) n * p","option_b":"B) n + p","option_c":"C) p \/ n","option_d":"D) n² * p","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n,p) est E(X) = n * p.","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) n * p\", \"b\": \"B) n + p\", \"c\": \"C) p \/ n\", \"d\": \"D) n² * p\"}}","_debug_options_count":4},{"id":82434,"question":"La fonction f(x) = x⁴ - 4x³ + 6x² est convexe sur tout son domaine de définition.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée seconde f''(x) = 12x² - 24x + 12 = 12(x² - 2x + 1) = 12(x-1)² ≥ 0. La fonction est convexe partout sauf en x=1 où elle est linéaire.","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":82435,"question":"Quelle est la somme des 5 premiers termes d'une suite arithmétique de premier terme u₀=3 et de raison r=2 ?","option_a":"A) 25","option_b":"B) 35","option_c":"C) 45","option_d":"D) 55","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des n premiers termes d'une suite arithmétique est Sₙ = n\/2 * (2u₀ + (n-1)r). Pour n=5, u₀=3, r=2, on obtient S₅ = 5\/2 * (6 + 8) = 5\/2 * 14 = 35.","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) 25\", \"b\": \"B) 35\", \"c\": \"C) 45\", \"d\": \"D) 55\"}}","_debug_options_count":4},{"id":82436,"question":"La probabilité d'obtenir exactement 2 fois le chiffre 6 en lançant 5 fois un dé équilibré est :","option_a":"A) C(5,2) * (1\/6)² * (5\/6)³","option_b":"B) C(5,2) * (1\/6)³ * (5\/6)²","option_c":"C) (1\/6)² * (5\/6)³","option_d":"D) 5 * (1\/6) * (5\/6)","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une loi binomiale avec n=5, p=1\/6, k=2. La probabilité est C(5,2) * (1\/6)² * (5\/6)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) C(5,2) * (1\/6)² * (5\/6)³\", \"b\": \"B) C(5,2) * (1\/6)³ * (5\/6)","_debug_options_count":4}]