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

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

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. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that \(x^2 : y^2 = (1 - a^2) : (1 - b^2)\).

6. If \(x = a \cos\theta + b \sin\theta\) and \(y = a \sin\theta - b \cos\theta\), then prove that \(x^2 + y^2 = a^2 + b^2\).

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

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

9. 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) \(-r^2\)

10. If \(x = \cfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \cfrac{x^2 + xy + y^2}{x^2 - xy + y^2} = \cfrac{12}{11} \]

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

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

13. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

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

15. If \(x = \frac{\sqrt{a + 1} + \sqrt{a - 1}}{\sqrt{a + 1} - \sqrt{a - 1}}\), then show that \(x^2 - 2ax + 1 = 0\).

16. If \(x = \frac{4ab}{a + b}\), then show that \[ \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} = 2 \]

17. If \(x = a\sinθ\) and \(y = b\tanθ\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\)

18. If \(a=\sqrt{x}+\cfrac{1}{\sqrt{x}}\) and \(b=\sqrt{x}- \cfrac{1}{\sqrt{x}}\), then show that \(a^4+b^4-2a^2b^2=16\).

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

20. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2 =\) constant.

21. If \[ \cfrac{x}{xa + yb + zc} = \cfrac{y}{ya + zb + xc} = \cfrac{z}{za + xb + yc} \] and \(x + y + z \ne 0\), then show that each of the above ratios is equal to \(\cfrac{1}{a + b + c}\).

22. If \(x = \cfrac{4ab}{a + b}\), show that \[ \cfrac{x + 2a}{x - 2a} + \cfrac{x + 2b}{x - 2b} = 2 \] [Given: \(a \ne 0\), \(b \ne 0\), and \(a \ne b\)]

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

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

24. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

25. Given \(x = a \sec\theta\) and \(y = b \tan\theta\), prove that \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

26. 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) \(a:b\)

27. 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

28. If \((u^2 + v^2) \propto (x^2 + y^2)\) and\(uv \propto xy\), then show that\((u + v) \propto (x + y)\) when \(\frac{u}{v} + \frac{v}{u} = \frac{x}{y} + \frac{y}{x}\)."

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

30. If \(x : a = y : b = z : c\), then show that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)