Quiz : Maîtrisez les maths de 4ème année secondaire !

Question 1 sur 10 10:00

[{"id":20318,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1) \/ (x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 3","option_b":"B. 0","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"La limite d'un polynôme de même degré au numérateur et au dénominateur est égale au rapport des coefficients dominants. Ici, 3x² \/ x² = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20319,"question":"Une fonction f est continue en un point a si et seulement si lim(x→a) f(x) = f(a).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est la définition exacte de la continuité d'une fonction en un point a.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20320,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A. 2x \/ (x² + 1)","option_b":"B. 1 \/ (x² + 1)","option_c":"C. 2x","option_d":"D. 1 \/ x","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la dérivée des fonctions composées : f'(x) = (1 \/ (x² + 1)) * 2x = 2x \/ (x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20321,"question":"Soit X une variable aléatoire suivant une loi normale N(0, 1). Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"A. 0.3413","option_b":"B. 0.6826","option_c":"C. 0.9544","option_d":"D. 0.9974","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour une loi normale centrée réduite, P(-1 ≤ X ≤ 1) ≈ 0.6826 (règle des 68-95-99.7).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20322,"question":"La fonction f(x) = x³ - 3x + 2 admet un extremum local en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(x) = 3x² - 3. f'(1) = 0 et f''(1) = 6 \u003E 0, donc x=1 est un minimum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20323,"question":"Quelle est la solution de l'inéquation x² - 5x + 6 \u003E 0 ?","option_a":"A. x ∈ ]2, 3[","option_b":"B. x ∈ ]-∞, 2[ ∪ ]3, +∞[","option_c":"C. x ∈ ]-∞, 2]","option_d":"D. x ∈ [3, +∞[","option_e":"","option_f":"","bonne_reponse":"B","explication":"Les racines sont x=2 et x=3. Le polynôme est positif à l'extérieur des racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20324,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Quelle est la primitive F de f telle que F(0) = 3 ?","option_a":"A. F(x) = x² + x + 3","option_b":"B. F(x) = x² + x","option_c":"C. F(x) = x² + x + 1","option_d":"D. F(x) = 2x² + x + 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"F(x) = ∫(2x + 1) dx = x² + x + C. Avec F(0) = 3, on trouve C=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20325,"question":"Une fonction strictement croissante sur ℝ est toujours dérivable.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une fonction peut être strictement croissante sans être dérivable (exemple : f(x) = x³ pour x ≥ 0 et f(x) = 0 pour x \u003C 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20326,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"C","explication":"∫(2x + 1) dx = x² + x. Évalué entre 0 et 1 : (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":20327,"question":"Soit X une variable aléatoire discrète prenant les valeurs 1, 2 et 3 avec les probabilités respectives 0.2, 0.5 et 0.3. Quelle est l'espérance E(X) ?","option_a":"A. 1.8","option_b":"B. 2.1","option_c":"C. 2.3","option_d":"D. 2.5","option_e":"","option_f":"","bonne_reponse":"C","explication":"E(X) = 1*0.2 + 2*0.5 + 3*0.3 = 0.2 + 1 + 0.9 = 2.1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]