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Question 1 sur 10 10:00

[{"id":74472,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2x","option_d":"3x² + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynomiale s'obtient en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74473,"question":"La suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0 est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, u_{n+1} - u_n = u_n + 1, qui n'est pas constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74474,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^{-2x}","option_b":"y = Ce^{2x}","option_c":"y = C + e^{-2x}","option_d":"y = Cx + e^{-2x}","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation est de la forme y' + ay = 0, dont la solution générale est y = Ce^{-ax}. Ici, a = 2, donc y = Ce^{-2x}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74475,"question":"Si P(A) = 0,3 et P(B) = 0,5 avec A et B indépendants, quelle est P(A ∩ B) ?","option_a":"0,15","option_b":"0,8","option_c":"0,2","option_d":"0,0","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour deux événements indépendants, P(A ∩ B) = P(A) * P(B) = 0,3 * 0,5 = 0,15.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74476,"question":"Le nombre complexe z = 3 - 4i a pour module |z| = 5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le module d'un nombre complexe z = a + bi est |z| = √(a² + b²). Ici, |z| = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74477,"question":"Quelle est la limite de la suite u_n = (n² + 1)\/(2n² - 3) quand n tend vers l'infini ?","option_a":"1\/2","option_b":"0","option_c":"∞","option_d":"2","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour les suites rationnelles, la limite est le rapport des coefficients des termes de plus haut degré. Ici, lim u_n = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74478,"question":"Si f est une fonction dérivable sur [a; b], alors f est continue sur [a; b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Toute fonction dérivable sur un intervalle est continue sur cet intervalle (théorème fondamental de l'analyse).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74479,"question":"Quelle est la forme algébrique de z = 5(cos(π\/3) + i sin(π\/3)) ?","option_a":"5\/2 + (5√3\/2)i","option_b":"5√3\/2 + (5\/2)i","option_c":"5 + 5i","option_d":"5\/2 - (5√3\/2)i","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant cos(π\/3) = 1\/2 et sin(π\/3) = √3\/2, on obtient z = 5*(1\/2) + 5*(√3\/2)i = 5\/2 + (5√3\/2)i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74480,"question":"La fonction f(x) = x^3 - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Oui, un minimum","option_b":"Oui, un maximum","option_c":"Non","option_d":"Oui, un point d'inflexion","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(x) = 3x² - 3. En x = 1, f'(1) = 0 et f''(1) = 6 \u003E 0, donc c'est un maximum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":74481,"question":"Si une variable aléatoire X suit une loi binomiale B(n=10, p=0,4), quelle est son espérance E(X) ?","option_a":"4","option_b":"6","option_c":"0,4","option_d":"10","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour une loi binomiale B(n, p), l'espérance est E(X) = n*p = 10*0,4 = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]