Solution Explanation

The uploaded image is a mathematics test in French, containing various types of problems. I will address the questions step-by-step, explaining each one clearly.

Question (1): Complète

1. Si $5a = 3b$, alors $a : b = ?$
   • From $5a = 3b$, divide both sides by $b$:
     $\frac{5a}{b} = 3 \implies a : b = \frac{3}{5}.$

2. Si $\text{card}(X) = 5$ et $\text{card}(X \times Y) = 15$, alors $\text{card}(Y) = ?$
   • The cardinality of the Cartesian product $X \times Y$ is given by:
     $\text{card}(X \times Y) = \text{card}(X) \cdot \text{card}(Y).$
     Substituting:
     $15 = 5 \cdot \text{card}(Y) \implies \text{card}(Y) = 3.$

3. Si $(x + 5; 8) = (1; 6y + x)$, alors $y = ?$
   • For two ordered pairs to be equal, their corresponding components must be equal: $x + 5 = 1 \implies x = -4,$
     and
     $8 = 6y + x.$
     Substituting $x = -4$:
     $8 = 6y - 4 \implies 6y = 12 \implies y = 2.$

4. Si $f(x) = 5x - 7$, alors $f(3) = ?$
   • Substitute $x = 3$ into the function $f(x)$:
     $f(3) = 5(3) - 7 = 15 - 7 = 8.$

5. Si $f(x) = 6x$, alors $f(2) + f(2) = ?$
   • Calculate $f(2)$:
     $f(2) = 6(2) = 12.$
     Then:
     $f(2) + f(2) = 12 + 12 = 24.$

6. Si $3a + 4b + b(4) = 13$, alors $b = ?$
   • Rewrite as $3a + 4b + 4b = 13$:
     $3a + 8b = 13.$
     Since $a$ is not specified, we cannot uniquely determine $b$ without further information.

7. Si $\frac{4}{x} = \frac{7}{y} = \frac{2}{y - x}$, alors $a = ?$
   • Use proportionality relationships:
     • From $\frac{4}{x} = \frac{7}{y}$, cross-multiply:
       $4y = 7x.$
     • From $\frac{7}{y} = \frac{2}{y - x}$, cross-multiply:
       $7(y - x) = 2y.$
       Simplify:
       $7y - 7x = 2y \implies 5y = 7x.$
       Combining $4y = 7x$ and $5y = 7x$, we solve for $a$.

8. Si $x - 6 = 0$, alors $x = ?$
   • Solve directly:
     $x - 6 = 0 \implies x = 6.$

Question (2): Choisis la bonne réponse entre les parenthèses

1. Si $3, 6, x, 4$ forment une proportion, alors $x = ?$
   • In a proportion:
     $\frac{3}{6} = \frac{x}{4}.$
     Cross-multiply:
     $3 \cdot 4 = 6 \cdot x \implies 12 = 6x \implies x = 2.$

2. Si $\frac{x}{y} = \frac{3}{5}$, alors $\frac{2x + y}{x + 2y} = ?$
   • Substitute $\frac{x}{y} = \frac{3}{5}$:
     $x = 3k, \, y = 5k.$
     Substituting into $\frac{2x + y}{x + 2y}$:
     $\frac{2(3k) + 5k}{3k + 2(5k)} = \frac{6k + 5k}{3k + 10k} = \frac{11k}{13k} = \frac{11}{13}.$

3. La moyenne proportionnelle de $\sqrt{x^2 - 4}$ et $x + 2$ est...?
   • The geometric mean of $a$ and $b$ is $\sqrt{a \cdot b}$:
     $\sqrt{\sqrt{x^2 - 4} \cdot (x + 2)}.$

4. Si $x + y = 5$, $x y = 6$, alors $x^2 y + y^2 x = ?$
   • Simplify using symmetric sums:
     $x^2 y + y^2 x = xy(x + y).$
     Substituting:
     $xy = 6, \, x + y = 5 \implies 6 \cdot 5 = 30.$

5. L'étendue des valeurs $5; 14; 4; 21; 16; 12$ est...?
   • Range ($\text{étendue}$) = max - min:
     $\text{étendue} = 21 - 4 = 17.$

Let me know if you'd like further clarifications!