Quiz — J'intègre, Les Méthodes et Exercices de Mathématiques PCSI-PTSI.pdf

Question 1 sur 10 20:00

[{"id":44300,"question":"Quelle est la limite de la suite définie par u_n = (3n² + 2n - 1)\/(2n² + 5) lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle s'obtient en comparant les termes de plus haut degré au numérateur et au dénominateur. Ici, 3n²\/2n² = 3\/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. 3\/2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":44301,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) \u003E 0 pour tout x ∈ ℝ. Alors f est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est dérivable et f'(x) \u003E 0 pour tout x, alors f est strictement croissante sur ℝ (théorème fondamental de l'analyse).","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":44302,"question":"Quelle est la valeur propre d'une matrice triangulaire supérieure dont les éléments diagonaux sont 2, 3 et 5 ?","option_a":"A. 2","option_b":"B. 3","option_c":"C. 5","option_d":"D. 2, 3 et 5","option_e":"","option_f":"","bonne_reponse":"d","explication":"","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. 2\", \"b\": \"B. 3\", \"c\": \"C. 5\", \"d\": \"D. 2, 3 et 5\"}}","_debug_options_count":4},{"id":44303,"question":"Soit (u_n) une suite géométrique de raison q = -2 et de premier terme u_0 = 1. Que vaut u_3 ?","option_a":"A. -8","option_b":"B. 8","option_c":"C. -6","option_d":"D. 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"u_3 = u_0 * q^3 = 1 * (-2)^3 = -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. -8\", \"b\": \"B. 8\", \"c\": \"C. -6\", \"d\": \"D. 6\"}}","_debug_options_count":4},{"id":44304,"question":"La fonction f(x) = x^3 - 3x + 1 est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"f''(x) = 6x, qui n'est pas toujours positive (ex: f''(-1) = -6 \u003C 0). Donc f n'est pas convexe sur ℝ.","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":44305,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y' + 4y = 0 ?","option_a":"A. y(x) = (A + Bx)e^{-2x}","option_b":"B. y(x) = Ae^{-2x} + Be^{2x}","option_c":"C. y(x) = A cos(2x) + B sin(2x)","option_d":"D. y(x) = Ae^{-x} + Be^{-3x}","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² + 4r + 4 = 0, de racine double r = -2. La solution générale est donc y(x) = (A + Bx)e^{-2x}.","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. y(x) = (A + Bx)e^{-2x}\", \"b\": \"B. y(x) = Ae^{-2x} + Be^{2x}\", ","_debug_options_count":4},{"id":44306,"question":"Soit A une matrice carrée d'ordre 2 telle que A² = A. Que peut-on dire de A ?","option_a":"A. A est inversible","option_b":"B. A est la matrice nulle","option_c":"C. A est une matrice de projection","option_d":"D. A est symétrique","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une matrice vérifiant A² = A est appelée matrice de projection (ou idempotente). Elle projette tout vecteur sur son image.","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. A est inversible\", \"b\": \"B. A est la matrice nulle\", \"c\": \"C. ","_debug_options_count":4},{"id":44307,"question":"La série de terme général u_n = 1\/n^2 converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de Riemann Σ 1\/n^α converge si et seulement si α \u003E 1. Ici, α = 2 \u003E 1, donc la série converge.","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":44308,"question":"Quelle est la dimension de l'espace vectoriel ℝ^3 ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"ℝ^3 est un espace vectoriel de dimension 3, car il est engendré par 3 vecteurs linéairement indépendants (ex: la base canonique).","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. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":44309,"question":"Soit f une fonction continue sur [a, b] et dérivable sur ]a, b[. Si f(a) = f(b), alors il existe c ∈ ]a, b[ tel que f'(c) = 0 (théorème de Rolle).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Rolle stipule que sous ces conditions, il existe au moins un point c où la dérivée s'annule.","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}]