4. 📚 Exercices corrigés

Ensembles de nombres et calculs – Tronc Commun Sciences BIOF
🔒 Document protégé – Copie, clic droit et sélection désactivés
✅ Exercice 1 – Symboles ∈ ou ∉

Compléter avec ∈ ou ∉ :

\(0 \ldots \mathbb{Z}^*\) \(0 \notin \mathbb{Z}^*\) (car \(\mathbb{Z}^*\) exclut 0)
\(\sqrt{97} \ldots \mathbb{R}^-\) \(\sqrt{97} \notin \mathbb{R}^-\) (car \(\sqrt{97} > 0\))
\(\frac{1}{12} \ldots \mathbb{D}\) \(\frac{1}{12} \in \mathbb{D}\) (car \(1/12 = 0,08333...\) décimal)
\(\frac{2}{3} \ldots \mathbb{Q}\) \(\frac{2}{3} \in \mathbb{Q}\) (rationnel)
\(4,1 \ldots \mathbb{Z}\) \(4,1 \notin \mathbb{Z}\) (n'est pas entier)
\(2 \ldots \mathbb{N}\) \(2 \in \mathbb{N}\)
\(-301 \ldots \mathbb{Q}^+\) \(-301 \notin \mathbb{Q}^+\) (négatif)
\(433 \ldots \mathbb{Z}^*\) \(433 \in \mathbb{Z}^*\)
\(0 \ldots \mathbb{N}\) \(0 \in \mathbb{N}\)
\(5,33 \ldots \mathbb{Q}\) \(5,33 = \frac{533}{100} \in \mathbb{Q}\)
\(5,33 \ldots \mathbb{D}\) \(5,33 \in \mathbb{D}\) (décimal)
\(\frac{17}{2} \ldots \mathbb{D}^+\) \(\frac{17}{2} = 8,5 \in \mathbb{D}^+\)
\(\sqrt{7} \ldots \mathbb{R}^-\) \(\sqrt{7} \notin \mathbb{R}^-\) (positif)
\(\frac{n(n+1)}{2} \ldots \mathbb{N}\) \(\frac{n(n+1)}{2} \in \mathbb{N}\) (somme des n premiers entiers)
\(\sqrt{16} + 2\sqrt{9} \ldots \mathbb{Q}\) \(4 + 2 \times 3 = 10 \in \mathbb{Q}\)
✅ Exercice 2 – Symboles ⊂ ou ⊄
\(\mathbb{R}^- \ldots \mathbb{R}^+\) \(\mathbb{R}^- \not\subset \mathbb{R}^+\)
\(\{0,2,3\} \ldots \mathbb{Z}\) \(\{0,2,3\} \subset \mathbb{Z}\)
\(\{-1\} \ldots \mathbb{Z}^+\) \(\{-1\} \not\subset \mathbb{Z}^+\)
\(\{1,3\} \ldots \mathbb{Z}\) \(\{1,3\} \subset \mathbb{Z}\)
\(\mathbb{N} \ldots \mathbb{Q}^-\) \(\mathbb{N} \not\subset \mathbb{Q}^-\) (les naturels sont positifs)
\(\mathbb{N} \ldots \mathbb{R}\) \(\mathbb{N} \subset \mathbb{R}\)
\(\mathbb{N} \ldots \mathbb{Z}\) \(\mathbb{N} \subset \mathbb{Z}\)
\(\mathbb{N} \ldots \mathbb{D}^+\) \(\mathbb{N} \subset \mathbb{D}^+\)
\(\mathbb{R}^- \ldots \mathbb{Z}\) \(\mathbb{R}^- \not\subset \mathbb{Z}\)
\(\mathbb{N} \ldots \mathbb{Z}^+\) \(\mathbb{N} \subset \mathbb{Z}^+\)
\(\mathbb{N}^* \ldots \mathbb{Z}^*\) \(\mathbb{N}^* \subset \mathbb{Z}^*\)
✅ Exercice 3 – Écriture en extension
\(\{-1\} \cap \mathbb{Z}^+\) \(\emptyset\) (aucun élément commun)
\(\{1,3\} \cap \mathbb{Z}\) \(\{1,3\}\)
\(\mathbb{N} \cap \mathbb{Q}^-\) \(\{0\}\)
\(\mathbb{N} \cap \mathbb{R}\) \(\mathbb{N}\)
\(\mathbb{N} \cap \mathbb{Z}\) \(\mathbb{N}\)
\(\mathbb{N} \cap \mathbb{R}^*\) \(\mathbb{N}^*\)
\(\mathbb{D} \cap \mathbb{Q}\) \(\mathbb{D}\)
\(\mathbb{Z}^- \cap \mathbb{Z}^*\) \(\mathbb{Z}^- \setminus \{0\}\)
\(\mathbb{R} \cap \mathbb{R}^+\) \(\mathbb{R}^+\)
\(\{0,2,3\} \cap \mathbb{Z}\) \(\{0,2,3\}\)
\(\mathbb{N} \cap \mathbb{D}^+\) \(\mathbb{N}\)
\(\mathbb{R} \cap \mathbb{Z}\) \(\mathbb{Z}\)
\(\mathbb{N} \cap \mathbb{Z}^+\) \(\mathbb{N}\)
\(\mathbb{N}^* \cap \mathbb{Z}^*\) \(\mathbb{N}^*\)
\(\{-1\} \cup \{1,3,4\}\) \(\{-1,1,3,4\}\)
\(\{1,3\} \cup \mathbb{Z}\) \(\mathbb{Z}\)
\(\mathbb{N} \cup \mathbb{Q}^*\) \(\mathbb{Q}\)
\(\mathbb{N} \cup \mathbb{R}\) \(\mathbb{R}\)
\(\mathbb{N} \cup \mathbb{R}^+\) \(\mathbb{R}^+\)
\(\{0,2,3\} \cup \mathbb{Z}\) \(\mathbb{Z}\)
✅ Exercice 4 – Calculs (fractions et racines)
\( A = \left( \frac{3}{4} - \frac{5}{3} \right) \times \frac{2 - \frac{4}{7}}{3} \times \frac{1}{\frac{4}{3} - \frac{1}{2}} \)
\( \frac{3}{4} - \frac{5}{3} = \frac{9 - 20}{12} = -\frac{11}{12} \)
\( 2 - \frac{4}{7} = \frac{14 - 4}{7} = \frac{10}{7} \)
\( \frac{4}{3} - \frac{1}{2} = \frac{8 - 3}{6} = \frac{5}{6} \)
\( A = -\frac{11}{12} \times \frac{10/7}{3} \times \frac{1}{5/6} = -\frac{11}{12} \times \frac{10}{21} \times \frac{6}{5} \)
\( A = -\frac{11 \times 10 \times 6}{12 \times 21 \times 5} = -\frac{660}{1260} = -\frac{11}{21} \)
\( B = \frac{10101}{10101} \)
\( B = 1 \)
\( C = \) (expression complexe)
Après simplification, \( C = \frac{3(\sqrt{2} - \sqrt{3}) + \frac{1}{\sqrt{6}(1/\sqrt{2} - 1/\sqrt{3})}}{\sqrt{2} + \sqrt{3} - \frac{1}{\sqrt{6}(1/\sqrt{2} + 1/\sqrt{3})}} \)
\( C = 1 \)
\( D = 2 + \frac{1}{4 + \frac{1}{\sqrt{5+2}}} \)
\( \sqrt{5+2} = \sqrt{7} \approx 2,64575 \)
\( D = 2 + \frac{1}{4 + \frac{1}{\sqrt{7}}} = 2 + \frac{1}{\frac{4\sqrt{7} + 1}{\sqrt{7}}} = 2 + \frac{\sqrt{7}}{4\sqrt{7} + 1} \)
✅ Exercice 5 – Calculs avec racines
\( A = 3\left(1 + \frac{1}{3} - \frac{3}{2}\right) + \left(2 - \frac{1}{3}\right)\left(2 - \frac{3}{2}\right) \)
\( 1 + \frac{1}{3} - \frac{3}{2} = \frac{6 + 2 - 9}{6} = -\frac{1}{6} \)
\( 3 \times (-\frac{1}{6}) = -\frac{1}{2} \)
\( 2 - \frac{1}{3} = \frac{5}{3} \), \( 2 - \frac{3}{2} = \frac{1}{2} \)
\( \frac{5}{3} \times \frac{1}{2} = \frac{5}{6} \)
\( A = -\frac{1}{2} + \frac{5}{6} = \frac{-3 + 5}{6} = \frac{2}{6} = \frac{1}{3} \)
\( C = \sqrt{96} + 2\sqrt{6} - 2\sqrt{24} - 3\sqrt{54} \)
\( \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} \)
\( 2\sqrt{24} = 2\sqrt{4 \times 6} = 4\sqrt{6} \)
\( 3\sqrt{54} = 3\sqrt{9 \times 6} = 9\sqrt{6} \)
\( C = 4\sqrt{6} + 2\sqrt{6} - 4\sqrt{6} - 9\sqrt{6} = (4+2-4-9)\sqrt{6} = -7\sqrt{6} \)
\( D = \frac{1}{3}\sqrt{363} + \sqrt{108} - \sqrt{300} + \frac{2}{\sqrt{12}} - 2\sqrt{\frac{75}{36}} \)
\( \sqrt{363} = \sqrt{121 \times 3} = 11\sqrt{3} \) → \( \frac{1}{3} \times 11\sqrt{3} = \frac{11}{3}\sqrt{3} \)
\( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \)
\( \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} \)
\( \frac{2}{\sqrt{12}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)
\( 2\sqrt{\frac{75}{36}} = 2 \times \frac{\sqrt{75}}{6} = \frac{\sqrt{75}}{3} = \frac{5\sqrt{3}}{3} \)
\( D = \frac{11}{3}\sqrt{3} + 6\sqrt{3} - 10\sqrt{3} + \frac{\sqrt{3}}{3} - \frac{5\sqrt{3}}{3} \)
\( D = \left(\frac{11}{3} + 6 - 10 + \frac{1}{3} - \frac{5}{3}\right)\sqrt{3} = \left(\frac{11 + 1 - 5}{3} - 4\right)\sqrt{3} = \left(\frac{7}{3} - 4\right)\sqrt{3} = -\frac{5}{3}\sqrt{3} \)
✅ Exercice 6 – Calculs supplémentaires
\( E = \frac{\sqrt{2} - \frac{1}{\sqrt{2}}}{\sqrt{2} + \frac{1}{\sqrt{8}}} \)
\( \sqrt{2} - \frac{1}{\sqrt{2}} = \frac{2 - 1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \)
\( \sqrt{2} + \frac{1}{\sqrt{8}} = \sqrt{2} + \frac{1}{2\sqrt{2}} = \frac{4 + 1}{2\sqrt{2}} = \frac{5}{2\sqrt{2}} \)
\( E = \frac{1/\sqrt{2}}{5/(2\sqrt{2})} = \frac{1}{\sqrt{2}} \times \frac{2\sqrt{2}}{5} = \frac{2}{5} \)
\( F = 6 - \frac{\frac{5}{3} + \frac{3}{2}}{\frac{3}{2} - \frac{5}{4}} \)
\( \frac{5}{3} + \frac{3}{2} = \frac{10 + 9}{6} = \frac{19}{6} \)
\( \frac{3}{2} - \frac{5}{4} = \frac{6 - 5}{4} = \frac{1}{4} \)
\( \frac{19/6}{1/4} = \frac{19}{6} \times 4 = \frac{76}{6} = \frac{38}{3} \)
\( F = 6 - \frac{38}{3} = \frac{18 - 38}{3} = -\frac{20}{3} \)
\( G = \frac{\frac{3}{1} + \frac{1}{3}}{\frac{1}{4} + \frac{2}{2}} \)
\( \frac{3}{1} + \frac{1}{3} = 3 + \frac{1}{3} = \frac{10}{3} \)
\( \frac{1}{4} + \frac{2}{2} = \frac{1}{4} + 1 = \frac{5}{4} \)
\( G = \frac{10/3}{5/4} = \frac{10}{3} \times \frac{4}{5} = \frac{40}{15} = \frac{8}{3} \)
📌 Exercices corrigés – Ensembles de nombres et calculs – Tronc Commun Sciences BIOF