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

2. \(x \propto y\) and \(y = 8\) when \(x = 2\); if \(y = 16\), what is the value of \(x\)?

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

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

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

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

6. In triangle \(∆PQR\), points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\), respectively. Given: \(PX = 2\) units, \(XQ = 3.5\) units, \(YR = 7\) units, and \(PY = 4.25\) units. Determine whether line segment \(XY\) is parallel to side \(QR\), with justification.

7. 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 \(y\).

8. 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.

9. 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

10. \(x \propto \sqrt{y}\) and \(y = a^2\), if \(x = 2a\), then find the value of \(x^2 : y\).

11. If \(\cfrac{1}{3x}+\cfrac{1}{4x}+\cfrac{1}{2x}+\cfrac{1}{6x}=5\), then what is the value of \(x\)?

(a) \(\cfrac{1}{7}\) (b) \(\cfrac{1}{5}\) (c) \(\cfrac{1}{4}\) (d) None of the above

12. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

13. 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) \(a:b\)

14. 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

15. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

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

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

18. If \(x ∝ y^2\) and \(y = 2a , x = a\), then show that \(y^2 = 4ax\).

19. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

20. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

21. If \(x = \cfrac{1}{2 + \sqrt{3}}\) and \(y = \cfrac{1}{2 - \sqrt{3}}\), then what is the value of \(\cfrac{1}{1 + x} + \cfrac{1}{1 + y}\)?

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

23. Given \(x = 3 + \sqrt{3}\) and \(y = 6\), find the value of \((x + y)^2\).

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

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

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

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

28. If \(x = a\cos(90^\circ - \theta)\), \(y = b\cot(90^\circ - \theta)\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\).

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

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