Quiz — probabilité-et-géométrie-dans-lespace.pdf

Question 1 sur 10 20:00

[{"id":66931,"question":"Quelle est la probabilité de tirer un point aléatoire dans un cube de volume 8 cm³, sachant qu'il doit se situer à moins de 1 cm de la face avant ?","option_a":"1\/8","option_b":"1\/4","option_c":"1\/2","option_d":"3\/8","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le volume favorable est un cube de côté 2 cm (1 cm de chaque côté de la face avant), soit un volume de 8 cm³. La probabilité est donc 8\/8 = 1\/4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\/8\", \"b\": \"1\/4\", \"c\": \"1\/2\", \"d\": \"3\/8\"}}","_debug_options_count":4},{"id":66932,"question":"Dans un repère orthonormé de l'espace, les points A(1,2,3) et B(4,5,6) définissent un vecteur. Quelle est la norme de ce vecteur ?","option_a":"3√3","option_b":"√3","option_c":"3","option_d":"√27","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le vecteur AB a pour coordonnées (3,3,3). Sa norme est √(3² + 3² + 3²) = √27 = 3√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\": \"3√3\", \"b\": \"√3\", \"c\": \"3\", \"d\": \"√27\"}}","_debug_options_count":4},{"id":66933,"question":"La probabilité conditionnelle P(A|B) est toujours égale à P(A) × P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule correcte est P(A|B) = P(A ∩ B) \/ P(B), sauf si A et B sont indépendants, auquel cas P(A|B) = P(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":66934,"question":"Quel est l'angle entre les vecteurs u(1,0,0) et v(0,1,0) dans l'espace ?","option_a":"0°","option_b":"45°","option_c":"90°","option_d":"180°","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le produit scalaire u·v = 0, donc les vecteurs sont orthogonaux. L'angle est donc de 90°.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0°\", \"b\": \"45°\", \"c\": \"90°\", \"d\": \"180°\"}}","_debug_options_count":4},{"id":66935,"question":"Soit un dé à 6 faces équilibré. Quelle est la probabilité d'obtenir un nombre pair OU un multiple de 3 ?","option_a":"1\/2","option_b":"2\/3","option_c":"5\/6","option_d":"1\/3","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les nombres pairs sont {2,4,6} et les multiples de 3 sont {3,6}. L'union donne {2,3,4,6}, soit 4 issues favorables sur 6. La probabilité est donc 4\/6 = 2\/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\": \"1\/2\", \"b\": \"2\/3\", \"c\": \"5\/6\", \"d\": \"1\/3\"}}","_debug_options_count":4},{"id":66936,"question":"Les événements A et B sont incompatibles si et seulement si P(A ∩ B) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les événements incompatibles vérifient P(A ∩ B) = 0, mais la réciproque n'est pas toujours vraie (exemple : événements indépendants de probabilité nulle).","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":66937,"question":"Quelle est la distance entre les points A(1,1,1) et B(2,3,4) dans l'espace ?","option_a":"√14","option_b":"√13","option_c":"3","option_d":"√19","option_e":"","option_f":"","bonne_reponse":"a","explication":"La distance est √[(2-1)² + (3-1)² + (4-1)²] = √(1 + 4 + 9) = √14.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"√14\", \"b\": \"√13\", \"c\": \"3\", \"d\": \"√19\"}}","_debug_options_count":4},{"id":66938,"question":"Soit un cube de côté 2 cm. Quelle est la probabilité qu'un point aléatoire à l'intérieur du cube soit à une distance inférieure à 1 cm d'un sommet ?","option_a":"1\/8","option_b":"π\/6","option_c":"π\/12","option_d":"1\/4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le volume favorable est 8 fois le volume d'un huitième de sphère de rayon 1 cm (car 8 sommets). Le volume total est 8 cm³. La probabilité est donc (8 × (4\/3)π(1)³\/8) \/ 8 = π\/6 \/ 8 = π\/48 ≈ π\/12.","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\/8\", \"b\": \"π\/6\", \"c\": \"π\/12\", \"d\": \"1\/4\"}}","_debug_options_count":4},{"id":66939,"question":"Si deux événements A et B sont indépendants, alors P(A ∩ B) = P(A) × P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition même de l'indépendance entre deux événements.","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":66940,"question":"Quel est le produit scalaire des vecteurs u(2, -1, 3) et v(-1, 4, 2) ?","option_a":"-4","option_b":"4","option_c":"0","option_d":"8","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit scalaire est (2×-1) + (-1×4) + (3×2) = -2 -4 + 6 = 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\": \"-4\", \"b\": \"4\", \"c\": \"0\", \"d\": \"8\"}}","_debug_options_count":4}]