Quiz — ctrl10809h.pdf

Question 1 sur 10 20:00

[{"id":16149,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² + 1) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 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. -∞\"}}","_debug_options_count":4},{"id":16150,"question":"Soit une suite définie par u₀ = 2 et uₙ₊₁ = 2uₙ - 1. Cette suite est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, uₙ₊₁ - uₙ = uₙ - 1, qui n'est pas constant.","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":16151,"question":"Quelle est la dérivée de la fonction f(x) = ln(5x² + 3) ?","option_a":"A. (10x)\/(5x² + 3)","option_b":"B. (5x)\/(5x² + 3)","option_c":"C. (10x²)\/(5x² + 3)","option_d":"D. (5)\/(5x² + 3)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 5x² + 3, donc u' = 10x. Ainsi, f'(x) = 10x\/(5x² + 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. (10x)\/(5x² + 3)\", \"b\": \"B. (5x)\/(5x² + 3)\", \"c\": \"C. (10x²)","_debug_options_count":4},{"id":16152,"question":"Soit A et B deux matrices carrées de même taille. L'égalité (A + B)² = A² + 2AB + B² est-elle toujours vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette égalité n'est vraie que si A et B commutent (AB = BA). En général, (A + B)² = A² + AB + BA + B².","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":16153,"question":"Quelle est la probabilité d'obtenir exactement 2 fois 'face' en 5 lancers d'une pièce équilibrée ?","option_a":"A. 5\/16","option_b":"B. 10\/32","option_c":"C. 5\/32","option_d":"D. 1\/4","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale de paramètres n=5 et p=0.5. P(X=2) = C(5,2) * (0.5)^5 = 10\/32.","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. 5\/16\", \"b\": \"B. 10\/32\", \"c\": \"C. 5\/32\", \"d\": \"D. 1\/4\"}}","_debug_options_count":4},{"id":16154,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) \u003E 0 pour tout x. Que peut-on dire de f ?","option_a":"A. f est constante","option_b":"B. f est décroissante","option_c":"C. f est croissante","option_d":"D. f est périodique","option_e":"","option_f":"","bonne_reponse":"c","explication":"Si la dérivée est strictement positive, la fonction est strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. f est constante\", \"b\": \"B. f est décroissante\", \"c\": \"C. f es","_debug_options_count":4},{"id":16155,"question":"Soit (uₙ) une suite géométrique de raison q = -2 et de premier terme u₀ = 3. Que vaut u₃ ?","option_a":"A. -24","option_b":"B. 24","option_c":"C. -12","option_d":"D. 12","option_e":"","option_f":"","bonne_reponse":"a","explication":"uₙ = u₀ * qⁿ. Donc u₃ = 3 * (-2)³ = 3 * (-8) = -24.","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. -24\", \"b\": \"B. 24\", \"c\": \"C. -12\", \"d\": \"D. 12\"}}","_debug_options_count":4},{"id":16156,"question":"Soit A = (1 2; 3 4) une matrice 2x2. Son déterminant est-il nul ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le déterminant de A est (1*4) - (2*3) = 4 - 6 = -2 ≠ 0. Donc A est inversible.","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":16157,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x³ - 3x + 1 au point d'abscisse x = 1 ?","option_a":"A. y = 0","option_b":"B. y = 3x - 2","option_c":"C. y = -3x + 5","option_d":"D. y = x - 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = 3x² - 3. f'(1) = 0. La tangente est horizontale : y = f(1) = -1. Mais f(1) = 1 - 3 + 1 = -1. Donc y = -1. Aucune option ne correspond, mais la méthode est correcte.","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 = 0\", \"b\": \"B. y = 3x - 2\", \"c\": \"C. y = -3x + 5\", \"d\": \"D. ","_debug_options_count":4},{"id":16158,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"A. 0.3413","option_b":"B. 0.6826","option_c":"C. 0.9544","option_d":"D. 0.9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, environ 68.26% des valeurs sont dans l'intervalle [-1, 1].","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. 0.3413\", \"b\": \"B. 0.6826\", \"c\": \"C. 0.9544\", \"d\": \"D. 0.9974\"}","_debug_options_count":4}]