1. If \(a \propto b\), \(b \propto \cfrac{1}{c}\), and \(c \propto d\), then what will be the proportional relationship between \(a\) and \(d\)?

2. \(x \propto y^2\) and \(y = 2a\). When \(y = a\), determine the relationship between \(x\) and \(y\).

(a) \(y^2 = 4ax \) (b) \(x^2 = 4ay \) (c) \(x=y\) (d) \(x^2 = 2ay\)

3. \(x \propto y^2\) and \(y = 2a\). When \(y = a\), determine the relationship between \(x\) and \(y\).

4. \(x \propto y^2\), and \(y = 2a\) when \(x = a\). Determine the relationship between \(x\) and \(y\).

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

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

7. \(a \propto b^2\) and \(1+b \propto 6\), and if \(a = 1\) when \(c = 9\) and \(b = 5\), determine the value of \(c\).

8. If \(a ∝ \frac{1}{c}\) and \(c ∝ \frac{1}{b}\), then determine the relationship between \(a\) and \(b\).

9. 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) \(y^2 \propto z\)

10. If \(a \propto b\) and \(b \propto c\), then prove that \(a + b + c \propto 5abc\).

11. If \( a \propto b \) and \( b \propto \cfrac{1}{c} \), then \( a \propto c \) will hold.

12. If \(a\propto b\) and \(b\propto c\), then show that \((pa+qb+rc) \propto (l\sqrt{bc}+m\sqrt{ca}+n\sqrt{ab})\).

13. If \(b\propto a^3\) and \(a\) increases in the ratio \(2:3\), determine the ratio in which \(b\) will increase.

14. If \( a \propto b \) and \( b \propto c \), then prove that \( a^3 + b^3 + c^3 \propto 3abc \).

15. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When \(x = 1\), \(y = -1\); and when \(x = 3\), \(y = 5\). Determine the relationship between x and y.

16. If \(a \propto b\), and \(b \propto c\), then show that \(a^3 b^3 + b^3 c^3 + c^3 a^3 \propto abc(a^3 + b^3 + c^3)\).

17. If \(a \propto b\), \(b \propto \frac{1}{c}\), and \(c \propto d\), then \(a \propto \frac{1}{d}\).

18. If \( a \propto b \) and \( b \propto c \), then prove that \( a^3 + b^3 + c^3 \propto 3abc \).

19. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\).

20. If \(x\) is in inverse proportion to \(y\), and \(y\) is in inverse proportion to \(z\), then \(x\) will The statement is true.

21. Given: \(a \cos θ = 3\) and \(b \tan θ = 4\), Find the relation between \(a\) and \(b\) that eliminates \(θ\).

22. Given: \(x = a \sec \theta\), \(y = b \tan \theta\); eliminate \(\theta\) and establish a relation between \(x\) and \(y\).

23. If \( x \cos\theta = 3 \) and \( y \cot\theta = 4 \), then find the relationship between \( x \) and \( y \) eliminating \( \theta \).

24. If \(x = a\sec\theta\) and \(y = b\tan\theta\), then find the relation between \(x\) and \(y\) eliminating \(\theta\).

25. Given that \(a \propto b\) and \(b \propto c\), show that \(a^3 + b^3 + c^3 \propto 5abc\).

26. If \(a \propto b\) and \(b \propto c\), then show that \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]

27. If \(a \propto b^3\) and \(c \propto \sqrt{a}\), then which of the following is correct?

(a) \(b \propto c^2\) (b) \(b \propto c^3\) (c) \(b^3 \propto c^3\) (d) \(b^3 \propto c^2\)

28. If \(a \propto \frac{1}{b}\) and \(b \propto \frac{1}{c}\), then which of the following is correct?

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

29. If \(a \propto (b + c)\), \(b \propto (c + a)\), and \(c \propto (a + b)\), and \(k, l, m\) are non-zero distinct constants respectively, then what is the value of \[ \frac{k}{k+1} + \frac{l}{l+1} + \frac{m}{m+1} \, ? \]

(a) 1 (b) 3 (c) 5 (d) 7

30. If \(x \propto yz\), \(y \propto ab^2\), and \(z \propto \cfrac{b}{a}\), then \(x \propto\) _____ (express in terms of \(a\) and \(b\)).