Quiz — 1998m2.pdf

Question 1 sur 10 20:00

[{"id":123833,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. Donc f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x - 5\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":123834,"question":"La suite définie par uₙ = (n² + 1)\/n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant uₙ = n + 1\/n, on voit que uₙ tend vers +∞ quand n tend vers +∞. La suite n'est donc pas convergente.","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":123835,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = Cx + 2","option_d":"y = C\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation est de la forme y' + ay = 0, sa solution générale est y = Ce^(-ax). Ici a = 2, donc y = Ce^(-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\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = Cx + 2\", \"d\": \"y = C","_debug_options_count":4},{"id":123836,"question":"Si X suit une loi binomiale B(n=10, p=0.3), quelle est P(X=3) ?","option_a":"C(10,3) * 0.3^3 * 0.7^7","option_b":"C(10,3) * 0.3^7 * 0.7^3","option_c":"10! \/ (3!7!)","option_d":"0.3^3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale, P(X=k) = C(n,k) * p^k * (1-p)^(n-k). Ici k=3, n=10, p=0.3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"C(10,3) * 0.3^3 * 0.7^7\", \"b\": \"C(10,3) * 0.3^7 * 0.7^3\", \"c\": \"1","_debug_options_count":4},{"id":123837,"question":"Le nombre complexe z = 3 - 4i a pour module |z| = 5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(a² + b²). Ici |z| = √(3² + (-4)²) = √(9 + 16) = √25 = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":123838,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n - 3) quand n tend vers +∞ ?","option_a":"2","option_b":"1","option_c":"0","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 - 3\/n). Quand n→+∞, uₙ tend vers 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":123839,"question":"La fonction f(x) = x³ - 3x² + 2 est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x = 3x(x-2). f'(x) est négative pour x ∈ ]0,2[, donc f n'est pas croissante 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":123840,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré ?","option_a":"1\/2","option_b":"1\/3","option_c":"1\/6","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 3 nombres pairs (2,4,6) sur 6 possibles. La probabilité est donc 3\/6 = 1\/2.","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\/2\", \"b\": \"1\/3\", \"c\": \"1\/6\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":123841,"question":"L'intégrale ∫(x² + 1) dx entre 0 et 1 vaut :","option_a":"4\/3","option_b":"1\/3","option_c":"2\/3","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de x² est x³\/3 et celle de 1 est x. Entre 0 et 1, cela donne (1\/3 + 1) - (0 + 0) = 4\/3. Mais l'intégrale de x² + 1 est x³\/3 + x, donc [1\/3 + 1] - [0 + 0] = 4\/3. Correction : la bonne réponse est 4\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"4\/3\", \"b\": \"1\/3\", \"c\": \"2\/3\", \"d\": \"1\"}}","_debug_options_count":4},{"id":123842,"question":"La fonction exponentielle est-elle toujours positive sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout x réel, e^x \u003E 0. La fonction exponentielle est donc toujours positive.","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}]