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

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

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

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

5. If \(x \propto y\) and \(y \propto z\), then show that \(x + y \propto z\).

6. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that \(x^2 : y^2 = (1 - a^2) : (1 - b^2)\).

7. If \(x \propto y\) and \(y \propto z\), then show that \[ \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]

8. If \(x ∝ y\), then show that \(x + y ∝ x - y\).

9. If \(x + y ∝ x - y\), then show that \(x^2 + y^2 ∝ xy\).

10. If \(x \propto y\), \(y \propto z\), and \(z \propto x\), then show that the product of the three non-zero proportionality constants is 1.

11. If \(x \propto y\) and \(y \propto z\), then show that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

12. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

13. If \(x ∝ \cfrac{1}{y}\) and \(y ∝ \cfrac{1}{z}\), then –

(a) \(x ∝ z\) (b) \(x∝ \cfrac{1}{z}\) (c) \(x∝z^2\) (d) \(x∝√z\)

14. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

15. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

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

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

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

18. If \(x+y ∝ x−y\), show that \(x^3+y^3 ∝ x^3−y^3\).

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

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

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

21. If \(x \propto y\) and \(y \propto z\), then prove that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

22. If \(x \alpha y\) and \(y \alpha z\), then prove that: \[ x^2 + y^2 + z^2 \alpha xy + yz + zx \]

23. If \(y\) is the geometric mean between \(x\) and \(z\), then what is the geometric mean between \(x^2 + y^2\) and \(y^2 + z^2\)?

24. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

25. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

26. If \(x \propto y\) and \(x \propto z\), then \((y + z) \propto\) ________.

27. If \(x \propto y\) and \(y \propto z\), then \((x + y + z) \propto (\sqrt{yz} + \sqrt{zx} + \sqrt{xy})\).

28. If \(x \propto y\), \(y \propto z\), and \(z \propto x\), then determine the product of the three nonzero proportionality constants.

29. If \(x \propto y\) and \(y \propto z\), prove that \(x^3 + y^3 + z^3 \propto xyz\).

30. If \(x \propto y\) and \(y \propto z\), then \(xy \propto z\).