Quiz — dzexams-3as-mathematiques-as_t3-20160-4.pdf

Question 1 sur 10 20:00

[{"id":11463,"question":"Quelle est la solution de l'équation 2x² - 5x + 2 = 0 ?","option_a":"x = 1 ou x = 2","option_b":"x = 2 ou x = 0.5","option_c":"x = -1 ou x = -2","option_d":"x = 1 ou x = -2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Résolvons l'équation : 2x² - 5x + 2 = 0. Le discriminant Δ = (-5)² - 4*2*2 = 25 - 16 = 9. Les solutions sont x = (5 ± √9)\/4, soit x = 2 ou x = 0.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x = 1 ou x = 2\", \"b\": \"x = 2 ou x = 0.5\", \"c\": \"x = -1 ou x = -2\"","_debug_options_count":4},{"id":11464,"question":"La fonction f(x) = x³ - 3x est croissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [-1, 1] (car 3x² ≤ 3). La fonction est donc décroissante sur cet intervalle.","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":11465,"question":"Quelle est la limite de (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un résultat fondamental en analyse : lim(x→0) (sin x)\/x = 1, utilisé dans de nombreux calculs de limites.","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\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":11466,"question":"Dans un triangle ABC, si AB = 5, AC = 3 et BC = 4, alors le triangle est rectangle en A.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Vérifions avec le théorème de Pythagore : 3² + 4² = 9 + 16 = 25 = 5². Le triangle est bien rectangle en A.","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":11467,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"f'(x) = e^(2x)","option_b":"f'(x) = 2e^(2x)","option_c":"f'(x) = e^(x)","option_d":"f'(x) = 2e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x)*e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f'(x) = e^(2x)\", \"b\": \"f'(x) = 2e^(2x)\", \"c\": \"f'(x) = e^(x)\", \"d","_debug_options_count":4},{"id":11468,"question":"L'intégrale de 0 à π de sin(x) dx vaut 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de sin(x) est -cos(x). Évaluée entre 0 et π : -cos(π) - (-cos(0)) = -(-1) - (-1) = 1 + 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":11469,"question":"Quelle est la solution de l'inéquation x² - 4x + 3 \u003E 0 ?","option_a":"x ∈ ]1, 3[","option_b":"x ∈ ]-∞, 1[ ∪ ]3, +∞[","option_c":"x ∈ ]-∞, 3[","option_d":"x ∈ ]1, +∞[","option_e":"","option_f":"","bonne_reponse":"b","explication":"Résolvons x² - 4x + 3 \u003E 0. Les racines sont x = 1 et x = 3. Le polynôme est positif à l'extérieur des racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x ∈ ]1, 3[\", \"b\": \"x ∈ ]-∞, 1[ ∪ ]3, +∞[\", \"c\": \"x ∈ ","_debug_options_count":4},{"id":11470,"question":"La probabilité d'obtenir un nombre pair en lançant un dé équilibré est de 1\/2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un dé a 6 faces : 3 nombres pairs (2, 4, 6) et 3 nombres impairs. 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\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":11471,"question":"Quelle est l'équation de la tangente à la courbe y = x² + 1 au point d'abscisse x = 1 ?","option_a":"y = 2x + 1","option_b":"y = 2x","option_c":"y = x + 2","option_d":"y = 3x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f(1) = 2 et f'(x) = 2x, donc f'(1) = 2. L'équation est y = 2(x - 1) + 2 = 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 = 2x + 1\", \"b\": \"y = 2x\", \"c\": \"y = x + 2\", \"d\": \"y = 3x - 1\"}}","_debug_options_count":4},{"id":11472,"question":"Le vecteur u(2, -3) et le vecteur v(-4, 6) sont colinéaires.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Deux vecteurs sont colinéaires si l'un est un multiple de l'autre. Ici, v = -2u, donc ils sont colinéaires.","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}]