Quiz — 664c5eab53d31_Enoncé - prototypes d'examen 3-.pdf

Question 1 sur 10 20:00

[{"id":15739,"question":"Quelle est la forme canonique de l'expression quadratique f(x) = 2x² - 8x + 5 ?","option_a":"A) 2(x - 2)² - 3","option_b":"B) 2(x - 4)² + 5","option_c":"C) 2(x - 1)² - 3","option_d":"D) 2(x - 2)² + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La forme canonique s'obtient par complétion du carré : f(x) = 2(x² - 4x) + 5 = 2[(x - 2)² - 4] + 5 = 2(x - 2)² - 8 + 5 = 2(x - 2)² - 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\": \"A) 2(x - 2)² - 3\", \"b\": \"B) 2(x - 4)² + 5\", \"c\": \"C) 2(x - 1)²","_debug_options_count":4},{"id":15740,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l'intervalle ]-∞, 0[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 6x + 2. Pour x \u003C 0, f'(x) \u003E 0 (exemple : f'(-1) = 3 + 6 + 2 = 11 \u003E 0), donc la fonction est bien croissante sur ]-∞, 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":15741,"question":"Quelle est la solution de l'équation ln(x) + ln(2x - 1) = ln(3) ?","option_a":"A) x = 1","option_b":"B) x = 2","option_c":"C) x = 3\/2","option_d":"D) x = 1\/2","option_e":"","option_f":"","bonne_reponse":"c","explication":"ln(x(2x - 1)) = ln(3) ⇒ x(2x - 1) = 3 ⇒ 2x² - x - 3 = 0 ⇒ x = [1 ± √(1 + 24)]\/4 = [1 ± 5]\/4. Seule x = 3\/2 est valide (x \u003E 0 et 2x - 1 \u003E 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) x = 1\", \"b\": \"B) x = 2\", \"c\": \"C) x = 3\/2\", \"d\": \"D) x = 1\/2\"}","_debug_options_count":4},{"id":15742,"question":"La droite d'équation y = 2x + 1 est parallèle à la droite d'équation 4x - 2y + 3 = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La pente de y = 2x + 1 est 2. La pente de 4x - 2y + 3 = 0 est 2 (y = 2x + 3\/2). Les droites sont donc parallèles (mêmes pentes).","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":15743,"question":"Quelle est la valeur de la limite lim(x→+∞) (3x² - 5x + 2)\/(2x² + x - 1) ?","option_a":"A) 0","option_b":"B) 3\/2","option_c":"C) +∞","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient (3 - 5\/x + 2\/x²)\/(2 + 1\/x - 1\/x²). Quand x→+∞, les termes en 1\/x tendent vers 0, donc la limite est 3\/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\": \"A) 0\", \"b\": \"B) 3\/2\", \"c\": \"C) +∞\", \"d\": \"D) -∞\"}}","_debug_options_count":4},{"id":15744,"question":"Le triangle ABC est rectangle en A si et seulement si AB² + AC² = BC².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de Pythagore : dans un triangle rectangle, le carré de l'hypoténuse est égal à la somme des carrés des deux autres côtés.","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":15745,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) * sin(x) ?","option_a":"A) e^(2x)(2sin(x) + cos(x))","option_b":"B) e^(2x)(2sin(x) - cos(x))","option_c":"C) e^(2x)(sin(x) + 2cos(x))","option_d":"D) e^(2x)(sin(x) - 2cos(x))","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle du produit : f'(x) = 2e^(2x)sin(x) + e^(2x)cos(x) = e^(2x)(2sin(x) + cos(x)).","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) e^(2x)(2sin(x) + cos(x))\", \"b\": \"B) e^(2x)(2sin(x) - cos(x))\",","_debug_options_count":4},{"id":15746,"question":"La fonction f(x) = |x - 2| est dérivable en x = 2.","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 = 2 car la dérivée à gauche (-1) et à droite (1) ne sont pas égales.","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":15747,"question":"Quelle est la solution de l'inéquation (x - 1)(x + 3) \u003E 0 ?","option_a":"A) x ∈ ]-∞, -3[ ∪ ]1, +∞[","option_b":"B) x ∈ ]-3, 1[","option_c":"C) x ∈ ]-∞, -1[ ∪ ]3, +∞[","option_d":"D) x ∈ ]-1, 3[","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit (x - 1)(x + 3) est positif lorsque les deux facteurs sont de même signe. Les solutions sont donc x \u003C -3 ou x \u003E 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\": \"A) x ∈ ]-∞, -3[ ∪ ]1, +∞[\", \"b\": \"B) x ∈ ]-3, 1[\", \"c\":","_debug_options_count":4},{"id":15748,"question":"La suite u_n = n²\/(n + 1) est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = n\/(1 + 1\/n). Quand n→+∞, u_n → +∞, donc la suite diverge.","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}]