1. If \(A ∝ \frac{1}{C}\), and \(C ∝ \frac{1}{B}\), then show that \(A ∝ B\).

2. If \(b ∝ a^3\) and the increase in \(a\) is in the ratio \(2 : 3\), then determine the ratio of increase in \(b\).

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

4. If \(b ∝ a^3\) and \(a\) increases in the ratio \(2:3\), then determine the ratio in which \(b\) increases.

5. If \(b ∝ a^2\) and \(a\) increases in the ratio \(2:3\), then determine the ratio in which \(b\) increases.

6. If a cube has a side length of \(a\) units and a diagonal length of \(d\) units, then the relationship between \(a\) and \(d\) is—?

(a) \(\sqrt2a=d\) (b) \(\sqrt3a=d\) (c) \(a=\sqrt3d\) (d) \(a=\sqrt2d\)

7. If \(b ∝ a^3\) and \(a\) increases in the ratio \(2 : 3\), then find the ratio in which \(b\) increases.

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

9. If \( x ∝ y^2 \) and \( y = 2a \) when \( x = a \), then find the relationship between \( x \) and \( y \).