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

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

3. 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.

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

5. 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

6. If the volume of a cone is \(x\), the base area is \(y\), and the height is \(z\), then the value of \(\cfrac{x}{yz}\) will be 3.

7. 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

8. 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 \(0\).

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

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

10. Three friends—A, B, and C—started a business by investing \(x\), \(2x\), and \(y\) taka respectively. If the profit at the end of the term is \(z\) taka, then what will be A’s share of the profit?

(a) ₹ \(\cfrac{xz}{3x+y}\) (b) ₹ \(\cfrac{2xz}{3x+y}\) (c) ₹ \(\cfrac{z}{2x+y}\) (d) ₹\(\cfrac{xyz}{3x+y}\)

11. Write the value of \((x - y + z + p)\), where \(x\), \(y\), \(z\), and \(p\) represent the number of faces, edges, vertices, and diagonals of a cuboid respectively.

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

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

13. Here is the English translation: If \( y - z ∝ \frac{1}{x} \), \( z - x ∝ \frac{1}{y} \), and \( x - y ∝ \frac{1}{z} \), then the sum of the three proportionality constants.

(a) 0 (b) 1 (c) -1 (d) 2