Quiz — Capes_2010_M1_Enonce.pdf

Question 1 sur 10 20:00

[{"id":161201,"question":"Soit une matrice carrée A d'ordre 2 telle que A² = I (matrice identité). Que peut-on dire de ses valeurs propres ?","option_a":"Elles sont toutes égales à 1","option_b":"Elles valent 1 ou -1","option_c":"Elles sont complexes conjuguées","option_d":"Elles sont toutes nulles","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si A² = I, alors les valeurs propres λ de A vérifient λ² = 1, donc λ = 1 ou λ = -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\": \"Elles sont toutes égales à 1\", \"b\": \"Elles valent 1 ou -1\", \"c\"","_debug_options_count":4},{"id":161202,"question":"La fonction f(x) = x³ - 3x + 2 est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1, donc la fonction n'est pas 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":161203,"question":"Quelle est la limite de la suite définie par uₙ = (1 + 1\/n)^n lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"e","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette suite converge vers le nombre d'Euler e ≈ 2,71828.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":161204,"question":"Soit une variable aléatoire X suivant une loi normale N(0,1). Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"≈ 0,34","option_b":"≈ 0,68","option_c":"≈ 0,95","option_d":"≈ 0,99","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, environ 68% 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\": \"≈ 0,34\", \"b\": \"≈ 0,68\", \"c\": \"≈ 0,95\", \"d\": \"≈ 0,99\"}}","_debug_options_count":4},{"id":161205,"question":"L'intégrale ∫₀^π sin(x) dx est égale à :","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) entre 0 et π vaut [-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":161206,"question":"Une application linéaire injective d'un espace vectoriel de dimension finie dans lui-même est bijective.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Dans un espace de dimension finie, injectif ⇒ surjectif ⇒ bijectif (théorème du rang).","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":161207,"question":"Quel est le développement limité à l'ordre 2 de la fonction f(x) = ln(1+x) au voisinage de 0 ?","option_a":"x - x²\/2 + o(x²)","option_b":"x + x²\/2 + o(x²)","option_c":"x - x² + o(x²)","option_d":"x + o(x²)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de ln(1+x) est x - x²\/2 + x³\/3 - ..., donc à l'ordre 2 : x - x²\/2 + o(x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x - x²\/2 + o(x²)\", \"b\": \"x + x²\/2 + o(x²)\", \"c\": \"x - x² + o","_debug_options_count":4},{"id":161208,"question":"Soit un triangle ABC avec AB = 5, AC = 7 et BC = 8. L'angle en A est aigu.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"D'après la loi des cosinus, cos(A) = (AB² + AC² - BC²)\/(2·AB·AC) = (25 + 49 - 64)\/70 = 10\/70 \u003E 0, donc A est aigu.","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":161209,"question":"La série ∑ (1\/n²) converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une série de Riemann avec α = 2 \u003E 1, donc elle converge (vers π²\/6).","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":161210,"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.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de Rolle, qui garantit l'existence d'un tel point 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]