1. If \(x ∝ \cfrac{1}{y}\) and \(y ∝ \cfrac{1}{z}\), then determine whether \(x\) and \(z\) are in direct or inverse proportion.

2. If \(x ∝ y\) and \(y ∝ z\), then prove that \((x^2 + y^2 + z^2) ∝ (xy + yz + zx)\).

3. If \(x ∝ y^2\) and \(y = 2a , x = a\), then show that \(y^2 = 4ax\).

4. If \(x ∝ y\), \(y ∝ z\), and \(z ∝ x\), then find the product of the three constants of proportionality.

5. If \(y - z \propto \cfrac{1}{x}\), \(z - x \propto \cfrac{1}{y}\), and \(x - y \propto \cfrac{1}{z}\), then the sum of the three proportional constants is \(0\).

6. If \(x \propto \cfrac{1}{y}\) and \(y \propto \cfrac{1}{z}\), then \(x \propto\) _____.

7. If \(x\propto \cfrac{1}{y}\) and \(y\propto \cfrac{1}{z}\), then \(x^2 \propto \)_____.

8. If \(x ∝ y\), \(y ∝ z\), and \(z ∝ x\), then determine the relationship among the three proportional constants.

9. If \(x ∝ y\) and \(x ∝ z\), then show that \(x ∝ (y - z)\).

10. A rhombus and a square are situated on the same base. If the area of the square is \(x^2\) square units and the area of the rhombus-shaped figure is \(y\) square units, then –

11. If \(x\) and \(y\) are the factors of \(30\), then which of the following is correct –

(a) \(x \propto y\) (b) \(x^2 \propto y\) (c) \(x \propto \cfrac{1}{y}\) (d) \(x \propto y^2\)

12. If \(x \propto \cfrac{1}{z}, z\propto \cfrac{1}{y}\), then the relation between \(x\) and \(y\) is _____.

13. If \(x \propto \cfrac{1}{z}\) and \(z \propto \cfrac{1}{y}\), then the relation between \(x\) and \(y\) is inverse proportionality.

14. \(x ∝ y\), and \(y = 8\) when \(x = 2\). Determine the value of \(x\) when \(y = 16\).

(a) 2 (b) 8 (c) 6 (d) 4

15. If the median of the following data is 28.5 and the total frequency is 60, then find the values of \(x\) and \(y\).

16. \(x ∝ y^2\) and \(y = 4\) when \(x = 8\); when \(x = 32\), find the positive value of \(y\).

(a) 4 (b) 8 (c) 16 (d) 32

17. \(x = \cfrac{k}{y}\) and \(y = \cfrac{m}{z}\), where \(k, m\) are non-zero proportional constants. ∵ \(x = \cfrac{k}{y} = \cfrac{k}{\cfrac{m}{z}} = \cfrac{kz}{m}\) ∴ \(x ∝ z\) [∵ \(k, m\) are non-zero proportional constants]

18. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |