Quiz — 683466e38ebce_corrigé sujet 3-exercices 2+3.pdf

Question 1 sur 10 20:00

[{"id":38070,"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. 2x² + 3x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc 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. 2x² + ","_debug_options_count":4},{"id":38071,"question":"La suite (uₙ) définie par uₙ = 2n + 1 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 = 2(n+1) + 1 - (2n + 1) = 2, donc la suite est bien 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":38072,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) 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":"Pour les fonctions rationnelles, lorsque x tend vers +∞, on compare les degrés du numérateur et du dénominateur. Ici, les deux sont de degré 2, donc la limite est le rapport des coefficients dominants : 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":38073,"question":"Dans un sac contenant 5 boules rouges et 3 boules bleues, quelle est la probabilité de tirer une boule rouge ?","option_a":"A. 5\/8","option_b":"B. 3\/8","option_c":"C. 5\/3","option_d":"D. 8\/5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité est le rapport du nombre de cas favorables sur le nombre total de cas. Ici, 5 boules rouges sur 8 boules au total, donc P(rouge) = 5\/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. 5\/8\", \"b\": \"B. 3\/8\", \"c\": \"C. 5\/3\", \"d\": \"D. 8\/5\"}}","_debug_options_count":4},{"id":38074,"question":"L'équation x² - 4x + 4 = 0 a-t-elle deux solutions réelles distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant Δ = b² - 4ac = (-4)² - 4*1*4 = 16 - 16 = 0. Un discriminant nul signifie une solution réelle double (x = 2), donc il n'y a pas deux solutions distinctes.","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":38075,"question":"Quelle est la primitive de la fonction f(x) = 4x³ - 2x + 1 ?","option_a":"A. x⁴ - x² + x + C","option_b":"B. 12x² - 2 + C","option_c":"C. 4x⁴ - 2x² + x + C","option_d":"D. x⁴ - x² + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive d'une fonction polynôme s'obtient en augmentant chaque exposant de 1 et en divisant par ce nouvel exposant. Ainsi, la primitive de 4x³ est x⁴, celle de -2x est -x², et celle de 1 est x. On ajoute la constante 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. x⁴ - x² + x + C\", \"b\": \"B. 12x² - 2 + C\", \"c\": \"C. 4x⁴ -","_debug_options_count":4},{"id":38076,"question":"Dans un repère orthonormé, les vecteurs u(2; -3) et v(-1; 4) sont-ils orthogonaux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux vecteurs sont orthogonaux si leur produit scalaire est nul. u · v = (2)(-1) + (-3)(4) = -2 - 12 = -14 ≠ 0. Donc les vecteurs ne sont pas orthogonaux.","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":38077,"question":"Quelle est la valeur de la somme S = 1 + 2 + 3 + ... + 100 ?","option_a":"A. 5050","option_b":"B. 10000","option_c":"C. 5000","option_d":"D. 5005","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers entiers naturels est donnée par la formule S = n(n+1)\/2. Pour n = 100, S = 100*101\/2 = 5050.","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. 5050\", \"b\": \"B. 10000\", \"c\": \"C. 5000\", \"d\": \"D. 5005\"}}","_debug_options_count":4},{"id":38078,"question":"La fonction f(x) = e^(2x) est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x). Comme e^(2x) \u003E 0 pour tout x ∈ ℝ, f'(x) \u003E 0, 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":38079,"question":"Quel est le centre de symétrie de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"A. (3; 2)","option_b":"B. (-3; 2)","option_c":"C. (3; -2)","option_d":"D. (-3; -2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une fonction homographique f(x) = (ax + b)\/(cx + d), le centre de symétrie est le point d'intersection des asymptotes verticale (x = -d\/c) et horizontale (y = a\/c). Ici, x = 3 et y = 2, donc le centre est (3; 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. (3; 2)\", \"b\": \"B. (-3; 2)\", \"c\": \"C. (3; -2)\", \"d\": \"D. (-3; -","_debug_options_count":4}]