Quiz — _énoncé_Série n°9(Isométries - Dérivabilités).pdf

Question 1 sur 10 20:00

[{"id":7175,"question":"Parmi les transformations suivantes, laquelle est une isométrie ?","option_a":"Projection orthogonale","option_b":"Homothétie de rapport 2","option_c":"Rotation de 90°","option_d":"Symétrie centrale","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une rotation est une isométrie car elle conserve les distances entre les points. Les autres transformations ne conservent pas nécessairement les distances.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Projection orthogonale\", \"b\": \"Homothétie de rapport 2\", \"c\": \"R","_debug_options_count":4},{"id":7176,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en x = 0 car elle présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","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":7177,"question":"Quelle est la dérivée de f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3","option_c":"2x - 5","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc f'(x) = 2x + 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\": \"2x + 3\", \"b\": \"x² + 3\", \"c\": \"2x - 5\", \"d\": \"x + 3\"}}","_debug_options_count":4},{"id":7178,"question":"Une translation est une isométrie qui conserve l'orientation des figures.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une translation est une isométrie directe, ce qui signifie qu'elle conserve l'orientation des figures (pas de retournement).","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":7179,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) = 0 pour tout x, alors f est :","option_a":"Constante","option_b":"Croissante","option_c":"Décroissante","option_d":"Périodique","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est nulle partout, la fonction est constante (théorème des accroissements finis).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Constante\", \"b\": \"Croissante\", \"c\": \"Décroissante\", \"d\": \"Pério","_debug_options_count":4},{"id":7180,"question":"Quelle est la nature de la transformation définie par (x, y) ↦ (x+2, y-3) ?","option_a":"Rotation","option_b":"Translation","option_c":"Symétrie axiale","option_d":"Homothétie","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette transformation est une translation de vecteur (2, -3), car elle ajoute une constante aux coordonnées.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Rotation\", \"b\": \"Translation\", \"c\": \"Symétrie axiale\", \"d\": \"Hom","_debug_options_count":4},{"id":7181,"question":"La dérivée de sin(x) est :","option_a":"cos(x)","option_b":"-cos(x)","option_c":"sin(x)","option_d":"-sin(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction sinus est la fonction cosinus.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"cos(x)\", \"b\": \"-cos(x)\", \"c\": \"sin(x)\", \"d\": \"-sin(x)\"}}","_debug_options_count":4},{"id":7182,"question":"Une symétrie centrale est une isométrie indirecte.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une symétrie centrale est une isométrie indirecte car elle inverse l'orientation des figures (retournement).","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":7183,"question":"Soit f(x) = x³ - 3x² + 2. Que vaut f'(1) ?","option_a":"0","option_b":"-3","option_c":"3","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de f(x) est f'(x) = 3x² - 6x. Donc f'(1) = 3(1)² - 6(1) = 3 - 6 = -3.","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\": \"-3\", \"c\": \"3\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":7184,"question":"Toute fonction continue est dérivable.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction peut être continue sans être dérivable (exemple : f(x) = |x| en x = 0). La dérivabilité implique la continuité, mais l'inverse n'est pas vrai.","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}]