Quiz — COMA 2019.ppt

Question 1 sur 10 20:00

[{"id":137078,"question":"Quelle est la dérivée de la fonction f(x) = x² * ln(x) ?","option_a":"A) 2x * ln(x) + x","option_b":"B) 2x * ln(x) + 1","option_c":"C) x * ln(x) + 2x","option_d":"D) 2x * (ln(x) + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un produit u*v est u'v + uv'. Ici, u = x² (u' = 2x) et v = ln(x) (v' = 1\/x). Ainsi, f'(x) = 2x * ln(x) + x² * (1\/x) = 2x * ln(x) + x.","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 * ln(x) + x\", \"b\": \"B) 2x * ln(x) + 1\", \"c\": \"C) x * ln(x) ","_debug_options_count":4},{"id":137079,"question":"Soit une matrice A de taille 2x2. Si det(A) = 5, alors det(2A) est égal à :","option_a":"A) 5","option_b":"B) 10","option_c":"C) 20","option_d":"D) 25","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour une matrice carrée A de taille n, det(kA) = kⁿ * det(A). Ici, n=2, donc det(2A) = 2² * det(A) = 4 * 5 = 20.","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) 5\", \"b\": \"B) 10\", \"c\": \"C) 20\", \"d\": \"D) 25\"}}","_debug_options_count":4},{"id":137080,"question":"L'intégrale ∫(0 à 1) e^(-x) dx est égale à :","option_a":"A) 1 - 1\/e","option_b":"B) e - 1","option_c":"C) 1\/e","option_d":"D) 1 - e","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de e^(-x) est -e^(-x). Évaluée entre 0 et 1, on obtient [-e^(-1)] - [-e^(0)] = -1\/e + 1 = 1 - 1\/e.","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) 1 - 1\/e\", \"b\": \"B) e - 1\", \"c\": \"C) 1\/e\", \"d\": \"D) 1 - e\"}}","_debug_options_count":4},{"id":137081,"question":"Vrai ou Faux ? Toute matrice carrée inversible admet une unique inverse.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, une matrice inversible admet une unique matrice inverse qui vérifie A * A⁻¹ = I (matrice identité).","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":137082,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"A) 0,3413","option_b":"B) 0,6826","option_c":"C) 0,9544","option_d":"D) 0,9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, environ 68,26% des valeurs sont dans l'intervalle [-1, 1]. Ainsi, P(-1 ≤ X ≤ 1) ≈ 0,6826.","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) 0,3413\", \"b\": \"B) 0,6826\", \"c\": \"C) 0,9544\", \"d\": \"D) 0,9974\"}","_debug_options_count":4},{"id":137083,"question":"L'équation différentielle y' + 2y = 0 admet pour solution générale :","option_a":"A) y = C * e^(-2x)","option_b":"B) y = C * e^(2x)","option_c":"C) y = C * x * e^(-2x)","option_d":"D) y = C * e^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est y = C * e^(-2x), où C est une constante.","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 = C * e^(-2x)\", \"b\": \"B) y = C * e^(2x)\", \"c\": \"C) y = C * x","_debug_options_count":4},{"id":137084,"question":"Vrai ou Faux ? Le produit de deux matrices symétriques est toujours une matrice symétrique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit de deux matrices symétriques n'est pas nécessairement symétrique. Par exemple, si A et B sont symétriques, (AB)ᵀ = BᵀAᵀ = BA ≠ AB en général.","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":137085,"question":"Quelle est la limite de la suite uₙ = (n² + 1)\/(2n² - 3) quand n tend vers l'infini ?","option_a":"A) 0","option_b":"B) 1\/2","option_c":"C) 1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant le numérateur et le dénominateur par n², on obtient uₙ = (1 + 1\/n²)\/(2 - 3\/n²). Quand n → ∞, 1\/n² → 0, donc uₙ → 1\/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) 1\/2\", \"c\": \"C) 1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":137086,"question":"Soit A une matrice de rotation de 90° dans le plan. Quelle est sa matrice ?","option_a":"A) [[0, -1], [1, 0]]","option_b":"B) [[0, 1], [-1, 0]]","option_c":"C) [[1, 0], [0, -1]]","option_d":"D) [[-1, 0], [0, 1]]","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une rotation de 90° dans le sens direct a pour matrice [[cos(90°), -sin(90°)], [sin(90°), cos(90°)]] = [[0, -1], [1, 0]].","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) [[0, -1], [1, 0]]\", \"b\": \"B) [[0, 1], [-1, 0]]\", \"c\": \"C) [[1,","_debug_options_count":4},{"id":137087,"question":"Vrai ou Faux ? Si deux suites uₙ et vₙ convergent respectivement vers L et M, alors la suite uₙ + vₙ converge vers L + M.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des suites convergentes : la somme de deux suites convergentes converge vers la somme de leurs limites.","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}]