Quiz — Copie de Série d'exercices N°9-MAS 3_.pdf

Question 1 sur 10 20:00

[{"id":79294,"question":"Soit la fonction f définie par f(x) = x² - 4x + 3. Quelle est la dérivée f'(x) ?","option_a":"A. f'(x) = 2x - 4","option_b":"B. f'(x) = x - 4","option_c":"C. f'(x) = 2x + 4","option_d":"D. f'(x) = x² - 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme f(x) = ax² + bx + c est f'(x) = 2ax + b. Ici, a=1 et b=-4, donc f'(x) = 2x - 4.","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. f'(x) = 2x - 4\", \"b\": \"B. f'(x) = x - 4\", \"c\": \"C. f'(x) = 2x ","_debug_options_count":4},{"id":79295,"question":"Une suite géométrique a pour premier terme u₀ = 2 et pour raison q = 3. Quel est le terme u₃ ?","option_a":"A. u₃ = 6","option_b":"B. u₃ = 18","option_c":"C. u₃ = 54","option_d":"D. u₃ = 27","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le terme général d'une suite géométrique est uₙ = u₀ × qⁿ. Donc u₃ = 2 × 3³ = 2 × 27 = 54.","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. u₃ = 6\", \"b\": \"B. u₃ = 18\", \"c\": \"C. u₃ = 54\", \"d\": \"D. ","_debug_options_count":4},{"id":79296,"question":"Vrai ou Faux ? La limite en +∞ de la fonction f(x) = (3x² + 2x - 1)\/(x² - 5) est égale à 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver la limite en +∞ d'un quotient de polynômes, on compare les degrés. Ici, les deux polynômes sont de degré 2, donc la limite est égale au 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":79297,"question":"Soit un vecteur u⃗(2 ; -1 ; 3) dans l'espace. Quelle est la norme de u⃗ ?","option_a":"A. √14","option_b":"B. √10","option_c":"C. 6","option_d":"D. √6","option_e":"","option_f":"","bonne_reponse":"a","explication":"La norme d'un vecteur u⃗(x ; y ; z) est donnée par √(x² + y² + z²). Donc ||u⃗|| = √(2² + (-1)² + 3²) = √(4 + 1 + 9) = √14.","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. √14\", \"b\": \"B. √10\", \"c\": \"C. 6\", \"d\": \"D. √6\"}}","_debug_options_count":4},{"id":79298,"question":"Vrai ou Faux ? Toute suite croissante est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite croissante peut être convergente (ex. suite constante) ou divergente (ex. suite uₙ = n). La convergence dépend de la majoration ou non de la suite.","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":79300,"question":"Quelle est la solution de l'équation complexe z² = -4 dans ℂ ?","option_a":"A. z = 2i ou z = -2i","option_b":"B. z = 4i ou z = -4i","option_c":"C. z = 2 ou z = -2","option_d":"D. z = i ou z = -i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de z² = -4 sont les racines carrées de -4, soit z = 2i ou z = -2i, car (2i)² = -4 et (-2i)² = -4.","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. z = 2i ou z = -2i\", \"b\": \"B. z = 4i ou z = -4i\", \"c\": \"C. z = ","_debug_options_count":4},{"id":79302,"question":"Soit une variable aléatoire X suivant une loi normale N(μ=5 ; σ=2). Quelle est la probabilité P(X ≤ 7) ?","option_a":"A. P(X ≤ 7) ≈ 0,8413","option_b":"B. P(X ≤ 7) ≈ 0,6915","option_c":"C. P(X ≤ 7) ≈ 0,5","option_d":"D. P(X ≤ 7) ≈ 0,3085","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale N(μ; σ), on calcule le z-score : z = (7 - 5)\/2 = 1. En utilisant la table de la loi normale, P(Z ≤ 1) ≈ 0,8413.","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. P(X ≤ 7) ≈ 0,8413\", \"b\": \"B. P(X ≤ 7) ≈ 0,6915\", \"c\": ","_debug_options_count":4},{"id":79304,"question":"Vrai ou Faux ? La fonction f(x) = e^x est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f'(x) = e^x et f''(x) = e^x \u003E 0 pour tout x ∈ ℝ, donc f est convexe.","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":79306,"question":"Soit une suite définie par uₙ₊₁ = 0,5uₙ + 3 avec u₀ = 1. Quelle est la limite de cette suite ?","option_a":"A. 6","option_b":"B. 3","option_c":"C. 0","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la suite converge vers L, alors L = 0,5L + 3 ⇒ 0,5L = 3 ⇒ L = 6. La limite est donc 6.","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. 6\", \"b\": \"B. 3\", \"c\": \"C. 0\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":79307,"question":"Quelle est l'équation cartésienne du plan passant par les points A(1;0;0), B(0;1;0) et C(0;0;1) ?","option_a":"A. x + y + z = 1","option_b":"B. x + y - z = 1","option_c":"C. 2x + 3y + z = 1","option_d":"D. x - y + z = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le plan passant par A, B et C a pour équation x + y + z = 1, car chaque point vérifie cette équation (ex. A(1;0;0) : 1+0+0=1).","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 + y + z = 1\", \"b\": \"B. x + y - z = 1\", \"c\": \"C. 2x + 3y + z ","_debug_options_count":4}]