Quiz : Maîtrisez les fondamentaux de la Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":54911,"question":"Quelle est la limite de la suite définie par u_{n+1} = 0.5u_n + 1 avec u_0 = 2 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite converge vers la solution de l'équation L = 0.5L + 1, soit L = 2. C'est un exemple classique de suite arithmético-géométrique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54912,"question":"La fonction f(x) = x^3 - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = 3x^2 - 3. En x=1, f'(1)=0 et f''(1)=6\u003E0, donc c'est un minimum local. L'affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54913,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x+1)","option_b":"B. 2\/(2x+1)","option_c":"C. 2x\/(2x+1)","option_d":"D. 1\/(x+0.5)","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée de ln(u) est u'\/u. Ici u=2x+1, donc u'=2. Ainsi f'(x)=2\/(2x+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54914,"question":"Dans une expérience aléatoire, P(A) = 0.4 et P(B) = 0.5. Si A et B sont indépendants, quelle est P(A ∩ B) ?","option_a":"A. 0.2","option_b":"B. 0.9","option_c":"C. 0.1","option_d":"D. 0.5","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54915,"question":"L'équation différentielle y' + 2y = 0 admet-elle pour solution y = Ce^{-2x} ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La solution générale de y' + 2y = 0 est y = Ce^{-2x}, où C est une constante réelle. L'affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54916,"question":"Quelle est la valeur de la somme S = 1 + 2 + 4 + ... + 2^n ?","option_a":"A. 2^{n+1} - 1","option_b":"B. 2^n - 1","option_c":"C. n*2^n","option_d":"D. (n+1)*2^n","option_e":"","option_f":"","bonne_reponse":"A","explication":"Il s'agit d'une suite géométrique de raison 2. La somme des n+1 premiers termes est S = 2^{n+1} - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54917,"question":"Si X suit une loi binomiale B(10, 0.3), quelle est son espérance ?","option_a":"A. 3","option_b":"B. 7","option_c":"C. 0.3","option_d":"D. 10","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'espérance d'une loi binomiale B(n,p) est E(X) = n*p = 10*0.3 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54918,"question":"La fonction f(x) = x^2 - 4x + 5 est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"f''(x) = 2 \u003E 0 pour tout x ∈ ℝ, donc f est convexe sur ℝ. L'affirmation est vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54919,"question":"Quelle est la solution de l'équation différentielle y'' + y = 0 avec y(0) = 1 et y'(0) = 0 ?","option_a":"A. y = cos(x)","option_b":"B. y = sin(x)","option_c":"C. y = e^x","option_d":"D. y = x^2","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation caractéristique est r^2 + 1 = 0, donc r = ±i. La solution générale est y = A cos(x) + B sin(x). Avec les conditions initiales, A=1 et B=0, donc y = cos(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54920,"question":"Dans un arbre pondéré, si P(A) = 0.6 et P(B|A) = 0.25, quelle est P(A ∩ B) ?","option_a":"A. 0.15","option_b":"B. 0.85","option_c":"C. 0.35","option_d":"D. 0.4","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(A ∩ B) = P(A) × P(B|A) = 0.6 × 0.25 = 0.15. C'est une application directe de la formule des probabilités conditionnelles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]