Maths BAC : Testez vos connaissances en algèbre et analyse

Question 1 sur 10 10:00

[{"id":69912,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A : 2x + 3","option_b":"B : 2x - 5","option_c":"C : x² + 3","option_d":"D : 2x² + 3x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle (x^n)' = n*x^(n-1). Ici, f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69913,"question":"L'équation x² - 4x + 3 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = (-4)² - 4*1*3 = 16 - 12 = 4 \u003E 0. L'équation admet donc deux solutions réelles distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69914,"question":"Quelle est la limite de la fonction f(x) = (2x² + 1)\/(x² - 3) quand x tend vers l'infini ?","option_a":"A : 0","option_b":"B : 2","option_c":"C : -3","option_d":"D : +∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) = (2 + 1\/x²)\/(1 - 3\/x²). Quand x → ∞, f(x) → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69915,"question":"La fonction f(x) = e^(3x) est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée f'(x) = 3e^(3x) \u003E 0 pour tout x ∈ ℝ. La fonction est donc strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69916,"question":"Quelle est la solution de l'inéquation ln(x) \u003E 2 ?","option_a":"A : x \u003E e²","option_b":"B : x \u003E 2","option_c":"C : x \u003C e²","option_d":"D : x \u003C 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction ln est croissante. ln(x) \u003E 2 ⇔ x \u003E e² (car ln(e²) = 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69917,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une fonction paire vérifie f(-x) = f(x). Ici, f(-x) = -x³ - 3x² - 2x ≠ f(x). La fonction n'est pas paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69918,"question":"Quelle est la primitive de f(x) = 4x³ + 2x qui s'annule en x = 1 ?","option_a":"A : x⁴ + x² - 2","option_b":"B : x⁴ + x²","option_c":"C : 4x⁴ + 2x²","option_d":"D : x⁴ + x² - 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive générale est F(x) = x⁴ + x² + C. F(1) = 0 ⇒ 1 + 1 + C = 0 ⇒ C = -2. Donc F(x) = x⁴ + x² - 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69919,"question":"L'équation sin(x) = 0 admet une infinité de solutions dans ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les solutions sont x = kπ pour tout k ∈ ℤ. Il y a donc une infinité de solutions.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69920,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) (3x² + 2x) dx ?","option_a":"A : 2","option_b":"B : 1","option_c":"C : 3","option_d":"D : 4","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale se calcule ainsi : [x³ + x²](0 à 1) = (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":69921,"question":"La fonction f(x) = 1\/x est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0 car le dénominateur s'annule.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]