Quiz — spe-mathematiques-2022-metropole-1-remplacement-sujet-officiel (2).pdf

Question 1 sur 10 20:00

[{"id":5516,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x - 5","option_d":"D. 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d’un polynôme s’obtient en appliquant la règle de dérivation : f'(x) = 2x + 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. 2x + 3\", \"b\": \"B. x² + 3\", \"c\": \"C. 2x - 5\", \"d\": \"D. 3x + 2\"","_debug_options_count":4},{"id":5517,"question":"La suite définie par uₙ = 3n + 2 est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite est arithmétique si la différence entre deux termes consécutifs est constante. Ici, uₙ₊₁ - uₙ = 3, donc elle est arithmétique.","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":5518,"question":"Quel est le résultat de l’intégrale ∫(2x + 1) dx ?","option_a":"A. x² + x + C","option_b":"B. x² + 2x + C","option_c":"C. 2x² + x + C","option_d":"D. x² + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L’intégrale d’une fonction linéaire s’obtient en appliquant la formule ∫(ax + b) dx = (a\/2)x² + bx + 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. x² + x + C\", \"b\": \"B. x² + 2x + C\", \"c\": \"C. 2x² + x + C\", ","_debug_options_count":4},{"id":5519,"question":"La matrice A = [[1, 2], [3, 4]] est-elle inversible ?","option_a":"A. Oui, car son déterminant est non nul","option_b":"B. Non, car elle n’est pas carrée","option_c":"C. Oui, car elle est symétrique","option_d":"D. Non, car son déterminant est nul","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une matrice carrée est inversible si et seulement si son déterminant est non nul. Ici, det(A) = (1×4) - (2×3) = -2 ≠ 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. Oui, car son déterminant est non nul\", \"b\": \"B. Non, car elle","_debug_options_count":4},{"id":5520,"question":"La probabilité conditionnelle P(A|B) est-elle toujours inférieure à P(A) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(A|B) peut être supérieure à P(A) si les événements A et B sont positivement corrélés. Par exemple, si B implique A.","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":5521,"question":"Quel est le module du nombre complexe z = 3 + 4i ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. √7","option_d":"D. 12","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d’un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |z| = √(3² + 4²) = 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\": \"A. 5\", \"b\": \"B. 7\", \"c\": \"C. √7\", \"d\": \"D. 12\"}}","_debug_options_count":4},{"id":5522,"question":"La fonction f(x) = eˣ est-elle solution de l’équation différentielle y' = y ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de eˣ est eˣ, donc f(x) = eˣ vérifie bien y' = y.","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":5523,"question":"Quelle est la valeur de la limite lim (x→∞) (3x² + 2x - 1)\/(x² + 5) ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. ∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour x→∞, on divise numérateur et dénominateur par x² : lim (3 + 2\/x - 1\/x²)\/(1 + 5\/x²) = 3\/1 = 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. 0\", \"b\": \"B. 3\", \"c\": \"C. ∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":5524,"question":"Un événement certain a une probabilité de 0,5 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un événement certain a une probabilité de 1. Un événement de probabilité 0,5 est simplement un événement équilibré.","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":5525,"question":"Quelle est la solution générale de l’équation différentielle y'' + y = 0 ?","option_a":"A. y = C₁eˣ + C₂e⁻ˣ","option_b":"B. y = C₁cos(x) + C₂sin(x)","option_c":"C. y = C₁x + C₂","option_d":"D. y = C₁e²ˣ + C₂e⁻²ˣ","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’équation caractéristique associée est r² + 1 = 0, dont les racines sont r = ±i. La solution générale est donc y = C₁cos(x) + C₂sin(x).","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. y = C₁eˣ + C₂e⁻ˣ\", \"b\": \"B. y = C₁cos(x) + C₂sin(x","_debug_options_count":4}]