Quiz — Mines-Ponts MP 2017 (Corrigé).doc

Question 1 sur 10 20:00

[{"id":160284,"question":"Soit f une application linéaire de ℝ³ dans ℝ². Quelle est la dimension maximale de Ker(f) ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"d","explication":"D'après le théorème du rang, dim(Ker(f)) + dim(Im(f)) = 3. Comme dim(Im(f)) ≤ 2, dim(Ker(f)) ≥ 1. La dimension maximale est donc 3 (si f est l'application nulle).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":160285,"question":"La suite (uₙ) définie par u₀ = 1 et uₙ₊₁ = √(2 + uₙ) est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est croissante et majorée par 2, donc elle converge vers l'unique solution de l ∈ [1,2] telle que l = √(2 + l).","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":160286,"question":"Quel est le développement limité à l'ordre 2 de sin(x) en 0 ?","option_a":"x - x²\/2 + o(x²)","option_b":"x + x³\/6 + o(x³)","option_c":"x - x³\/6 + o(x³)","option_d":"x + x²\/2 + o(x²)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de sin(x) à l'ordre 2 est x - x³\/6 + o(x³), mais à l'ordre 2, c'est x + o(x²). La réponse correcte est donc x - x²\/2 + o(x²) (erreur courante).","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 + x³\/6 + o(x³)\", \"c\": \"x - x³\/6 +","_debug_options_count":4},{"id":160287,"question":"Soit A une matrice carrée inversible. Laquelle de ces propriétés est toujours vraie ?","option_a":"det(A) = 0","option_b":"det(A) ≠ 0","option_c":"A est symétrique","option_d":"A est diagonalisable","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice inversible a un déterminant non nul. Les autres propriétés ne sont pas systématiques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"det(A) = 0\", \"b\": \"det(A) ≠ 0\", \"c\": \"A est symétrique\", \"d\": ","_debug_options_count":4},{"id":160288,"question":"L'intégrale ∫₀¹ x ln(x) dx est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale est impropre en 0, mais ∫₀¹ x ln(x) dx converge (elle vaut -1\/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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":160289,"question":"Soit P un polynôme de degré 3. Combien de racines réelles peut-il avoir au maximum ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un polynôme de degré 3 peut avoir 1 ou 3 racines réelles (en comptant les multiplicités). Le maximum est donc 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":160290,"question":"La série ∑ (1\/n²) est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série ∑ (1\/n²) est une série de Riemann convergente (p = 2 \u003E 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":160291,"question":"Soit f une fonction deux fois dérivable sur ℝ. Si f''(x) \u003E 0 pour tout x, alors f est :","option_a":"Décroissante","option_b":"Croissante","option_c":"Convexe","option_d":"Concave","option_e":"","option_f":"","bonne_reponse":"c","explication":"Si f''(x) \u003E 0, la fonction f est convexe (sa courbe est au-dessus de ses tangentes).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Décroissante\", \"b\": \"Croissante\", \"c\": \"Convexe\", \"d\": \"Concave\"","_debug_options_count":4},{"id":160292,"question":"L'équation différentielle y' + 2y = e⁻ˣ a pour solution générale :","option_a":"y = Ce⁻²ˣ + e⁻ˣ","option_b":"y = Ce⁻²ˣ - e⁻ˣ","option_c":"y = Ce⁻ˣ + e⁻²ˣ","option_d":"y = Ce⁻ˣ - e⁻²ˣ","option_e":"","option_f":"","bonne_reponse":"b","explication":"La solution générale est y = Ce⁻²ˣ - e⁻ˣ (solution particulière + solution de l'équation homogène).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = Ce⁻²ˣ + e⁻ˣ\", \"b\": \"y = Ce⁻²ˣ - e⁻ˣ\", \"c\": \"y =","_debug_options_count":4},{"id":160293,"question":"Soit A une matrice symétrique réelle. Ses valeurs propres sont toujours :","option_a":"Réelles","option_b":"Complexes","option_c":"Nulles","option_d":"Positives","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute matrice symétrique réelle est diagonalisable dans ℝ, donc ses valeurs propres sont réelles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Réelles\", \"b\": \"Complexes\", \"c\": \"Nulles\", \"d\": \"Positives\"}}","_debug_options_count":4}]