Quiz — 06_BinomeSigma_Corrige.pdf

Question 1 sur 10 20:00

[{"id":43515,"question":"Quel est le coefficient binomial C(5,2) ?","option_a":"5","option_b":"10","option_c":"15","option_d":"20","option_e":"","option_f":"","bonne_reponse":"b","explication":"C(5,2) = 5! \/ (2! * 3!) = (5×4) \/ (2×1) = 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"5\", \"b\": \"10\", \"c\": \"15\", \"d\": \"20\"}}","_debug_options_count":4},{"id":43516,"question":"Le développement de (a + b)^3 est égal à a^3 + 3a^2b + 3ab^2 + b^3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est exact : (a + b)^3 = C(3,0)a^3b^0 + C(3,1)a^2b^1 + C(3,2)a^1b^2 + C(3,3)a^0b^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":43517,"question":"Quel est le terme en x^2 dans le développement de (2x - 3)^4 ?","option_a":"24x^2","option_b":"-216x^2","option_c":"48x^2","option_d":"-54x^2","option_e":"","option_f":"","bonne_reponse":"d","explication":"Le terme général est C(4,k)*(2x)^(4-k)*(-3)^k. Pour x^2, k=2 : C(4,2)*(2x)^2*(-3)^2 = 6*4x^2*9 = 216x^2. Le signe est négatif car (-3)^2 = 9, mais le coefficient est -216x^2 (erreur dans l'option, réponse correcte : -216x^2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"24x^2\", \"b\": \"-216x^2\", \"c\": \"48x^2\", \"d\": \"-54x^2\"}}","_debug_options_count":4},{"id":43518,"question":"La formule du binôme de Newton s'applique uniquement pour n entier positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, la formule est valable pour tout entier n ≥ 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":43519,"question":"Quel est le développement de (1 + x)^4 ?","option_a":"1 + 4x + 6x^2 + 4x^3 + x^4","option_b":"1 + x + x^2 + x^3 + x^4","option_c":"1 + 2x + 3x^2 + 2x^3 + x^4","option_d":"1 + 4x + 3x^2 + 4x^3 + x^4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement est 1 + C(4,1)x + C(4,2)x^2 + C(4,3)x^3 + C(4,4)x^4 = 1 + 4x + 6x^2 + 4x^3 + x^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\": \"1 + 4x + 6x^2 + 4x^3 + x^4\", \"b\": \"1 + x + x^2 + x^3 + x^4\", \"c\":","_debug_options_count":4},{"id":43520,"question":"Le coefficient binomial C(n,0) est toujours égal à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car C(n,0) = n! \/ (0! * n!) = 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":43521,"question":"Quel est le terme constant dans le développement de (x + 1\/x)^6 ?","option_a":"15","option_b":"20","option_c":"10","option_d":"5","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le terme général est C(6,k)*x^(6-2k). Pour un terme constant, 6-2k=0 ⇒ k=3. Donc C(6,3)=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\": \"15\", \"b\": \"20\", \"c\": \"10\", \"d\": \"5\"}}","_debug_options_count":4},{"id":43522,"question":"Le binôme de Newton permet de calculer des puissances de matrices.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, le binôme de Newton s'applique aux nombres réels ou complexes, pas directement aux matrices (sauf cas particuliers).","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":43523,"question":"Quel est le développement de (a - b)^5 ?","option_a":"a^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5","option_b":"a^5 - 5a^4b + 5a^3b^2 - 5a^2b^3 + 5ab^4 - b^5","option_c":"a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5","option_d":"a^5 - 10a^4b + 5a^3b^2 - 5a^2b^3 + b^5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement est a^5 - C(5,1)a^4b + C(5,2)a^3b^2 - C(5,3)a^2b^3 + C(5,4)ab^4 - b^5 = a^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5.","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^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5\", \"b\": \"a^5 - 5a^","_debug_options_count":4},{"id":43524,"question":"La somme des coefficients binomiaux C(n,k) pour k de 0 à n est égale à 2^n.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car (1 + 1)^n = Σ C(n,k) = 2^n.","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}]