1. Given: \[ x \cos \theta = 3 \quad \text{and} \quad 4 \tan \theta = y \] Find the relation between \(x\) and \(y\) eliminating \(\theta\).

2. Write the relation between \(x\) and \(y\) by eliminating \(\theta\) from the equations \(2x = 3\sin\theta\) and \(5y = 3\cos\theta\).

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

4. If \(\sec \theta = \frac{x}{y}\) and \(x \ne y\), then which one is greater between \(x\) and \(y\)?

5. If \(\cos \theta = \frac{x}{y}\) and \(x \ne y\), then which one is smaller between \(x\) and \(y\), and why?

6. \(x \propto y^2\) 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) \(x^2 = 2ay\)

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

8. Eliminate \(\theta\) from the given equations \(2x = 3\sin\theta\) and \(5y = 3\cos\theta\), and express the relationship between \(x\) and \(y\).

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

10. If \((a^2 + b^2)(x^2 + y^2) = (ax + by)^2\), then what are the values of \(x\) and \(y\)?

(a) \(b:a\) (b) \(b^2:a^2\) (c) \(a^2:b^2\) (d) \(a:b\)

11. If in triangle \( \triangle ABC \), the sum of the medians AD, BE, and CF is \(x\), and the sum of the sides is \(y\), then the relationship between \(x\) and \(y\) is —

(a) \(x\gt y\) (b) \(x\lt y\) (c) \(x= y\) (d) None of the above

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

13. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

14. \(x\) and \(y\) vary directly with the square of each other, and \(z\) varies inversely with the cube root. Given: \(y = 8\), \(z = 8\), \(x = 16\) When \(x = 24\), \(z = 27\), find the value of \(y =\) ?

(a) \(\pm{16}\) (b) \(\pm{14}\) (c) \(\pm{12}\) (d) \(\pm{10}\)

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

16. If \(x\) and \(y\) are inversely proportional, then \(xy=\) constant.

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

18. \(y\) varies inversely with the square of \(x\), and \(y = 3\) when \(x = 16\). Find the value of \(y\) when \(x = 2\).

19. If \(x \tan 30^\circ + y \cot 60^\circ = 0\) and \(2x - y \tan 45^\circ = 1\), then calculate and write the values of \(x\) and \(y\).

20. A line parallel to side \(BC\) of triangle \(∆ABC\) intersects sides \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. If \(AX = 2.4\) cm, \(AY = 3.2\) cm and \(YC = 4.8\) cm, find the length of side \(AB\).

(a) 3.6 cm (b) 6 cm (c) 6.4 cm (d) 7.2 cm

21. In the adjacent figure, within triangle \(ABC\), if \(DE \parallel PQ \parallel BC\) and \(AD = 3\) cm, \(DP = x\) cm, \(PB = 4\) cm, \(AE = 4\) cm, \(EQ = 5\) cm, and \(QC = y\) cm, then determine the values of \(x\) and \(y\).

22. \(y\) is in direct proportion with the square root of \(x\), and \(y = 9\) when \(x = 9\). Find the value of \(x\) when \(y = 6\).

23. Given \(x \propto y\), and \(y = 8\) when \(x = 2\); then when \(y = 16\), find the value of \(x\).

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

24. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.

(a) undefined (b) positive (c) negative (d) zero

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

26. If a right rectangular prism (cuboid) has \(x\) diagonals, \(y\) faces, and \(z\) dimensions, then \(x + y + z = 13\).

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

28. If \(x = 3 + \sqrt{8}\) and \(y = 3 - \sqrt{8}\), then find the value of \(x^{-3} + y^{-3}\).

(a) 199 (b) 195 (c) 198 (d) 201

29. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \cfrac{1}{\sqrt{3} + \sqrt{2}}\), then find the value of \((x + y)^2 + (x - y)^2\).

30. If a cuboid has \(x\) faces, \(y\) edges, \(z\) vertices, and \(p\) diagonals, then find the value of \((x - y + z + p)\).