1. If \(\frac{x}{y} \propto z\), \(\frac{y}{z} \propto x\), and \(\frac{z}{x} \propto y\), then \(xyz = 1\).
2. If \(x \propto y\) and \(y \propto z\), then \(xy \propto z\).
3. If \(x \propto z\) and \(y \propto z\), then \(xy \propto z\).
4. If \( x \propto y \) and \( y \propto z \), then prove that \( x^2+y^2+z^2 \propto xy+yz+zx \).
5. If \(x \propto z\) and \(y \propto z\), then \(xy \propto z\).
6. If \( x \propto z \) and \( y \propto z \), then \( xy \propto z \).
7. If \(x \propto z\) and \(y \propto z\), then will \(xy \propto z\)? ✅ (Note: In mathematical terms, this actually leads to \(xy \propto z^2\), not just \(z\).
8. If \( x \propto z \) and \( y \propto z \), then \( xy \propto z^2 \).
9. If \(x \propto z\) and \(y \propto z\), then \(xy \propto z^2\).
10. If \(x \propto y\) and \(y \propto z\), then show that \(x + y \propto z\).
11. If \(x \propto z\) and \(y \propto z\), then \(x^2 + y^2 \propto z^2\).
12. If \(x \propto y\) and \(y \propto z\), prove that \(x^3 + y^3 + z^3 \propto xyz\).
13. 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 \(0\).
14. If \( x \propto y \) and \( z \propto y \), then \( x^2 + z^2 \propto y^2 \).
15. If \(x \propto \cfrac{1}{y}\) and \(z \propto \cfrac{1}{y}\), then \(x \propto y\).
16. If \( x \propto y \) and \( y \propto z \), prove that \( x^2 + y^2 + z^2 \propto xy + yz + zx \).
17. If \(x\propto y, y\propto z\), then prove that \(x^2+y^2+z^2\propto xy+yz+zx\).
18. If \(x \propto y\) and \(y \propto z\), then show that \[ \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]
19. If \(x \propto y\) and \(y \propto z\), then prove that \(x^2 + y^2 + z^2 \propto xy + yz + zx\)
20. If \(x \propto y\) and \(y \propto z\), then prove that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]
21. If \(x \propto y\) and \(y \propto z\), then show that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]
22. If \(x\) is in inverse proportion to \(y\), and \(y\) is in inverse proportion to \(z\), then \(x\) will The statement is true.
23. If \( x \propto yz \) and \( y \propto zx \), then show that \( z \ne 0 \) is a constant.
24. If \(x \propto y\) and \(x \propto z\), then \((y + z) \propto\) ________.
25. If \(x \propto \sqrt{y}\) and \(z \propto \sqrt[3]{x}\), then what is the relationship between \(y\) and \(z\)?
(a) \(z^6 \propto y\) (b) \(y^6 \propto z\) (c) \(z^2 \propto y\) (d) \(y^2 \propto z\)
26. If \(y - z \propto \frac{1}{x}\), \(z - x \propto \frac{1}{y}\), and \(x - y \propto \frac{1}{z}\), then what is the sum of the three constants of proportionality?
(a) 0 (b) 2 (c) 4 (d) 6
27. If \((y - z) \propto \frac{1}{x}\), \((z - x) \propto \frac{1}{y}\), and \((x - y) \propto \frac{1}{z}\), then find the sum of the three proportionality constants.
28. If \(x \propto y\), \(y \propto z\), and \(z \propto x\), then determine the product of the three nonzero proportionality constants.
29. If \(x \propto \cfrac{1}{z}, z\propto \cfrac{1}{y}\), then the relation between \(x\) and \(y\) is _____.
30. If \(x \propto \cfrac{1}{z}\) and \(z \propto \cfrac{1}{y}\), then the relation between \(x\) and \(y\) is inverse proportionality.
