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

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

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

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

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

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

7. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ 3x^2 - 5xy + 3y^2 \]

8. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

9. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).

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

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

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

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

14. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), find the value of \(3x^2 + 5xy + 3y^2\).

15. If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

16. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is —

(a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

17. If \(x = 7 + 4\sqrt{3}\), then find the value of \(\cfrac{x^3}{x^6 + 7x^3 + 1}\).

(a) \(\cfrac{1}{2737}\) (b) \(\cfrac{1}{2730}\) (c) \(\cfrac{1}{2710}\) (d) \(\cfrac{1}{2709}\)

18. If \(x = 2 + \sqrt{3}\), then the value of \(x + \frac{1}{x}\) will be \(2\sqrt{3}\).

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

20. If \(x = \sqrt{3} + \sqrt{2}\), find the value of \(x^3 + \cfrac{1}{x^3}\).

21. If \((x + 1)\cot^2\frac{\pi}{2} = 2\cos^2\frac{\pi}{3} + \frac{3}{4}\sec^2\frac{\pi}{4} + 4\sin^2\frac{\pi}{6}\), then find the value of \(x\).

22. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

23. If \(x = \frac{\sqrt{3}}{2}\), then find the value of \[ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} - \sqrt{1 - x}} \]

24. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then what is the value of \(\frac{x^2 + y^2 + xy}{x^2 + y^2 - xy}\)?

(a) \(\cfrac{6}{7}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{11}\) (d) None of the above

25. If \(\cfrac{(5 + \sqrt{3})}{(5 - \sqrt{3})} = x - \sqrt{15}y\), then find the value of \(x + y\).

26. If \[ x \cot\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{3}\right) + \frac{3}{4} \sec^2\left(\frac{\pi}{4}\right) + 4 \sin\left(\frac{\pi}{6}\right) \] then find the value of \(x\).

27. If \(x = 2\), \(y = 3\), and \(z = 6\), then calculate the value of \[ \frac{3\sqrt{x}}{\sqrt{y} + \sqrt{z}} - \frac{4\sqrt{y}}{\sqrt{z} + \sqrt{x}} + \frac{\sqrt{z}}{\sqrt{x} + \sqrt{y}} \]

28. If \(x=\cfrac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\) and \(xy=1\), then find the value of \(x^2 - xy + y^2\).

29. If \(x=2+\sqrt3\) and \(y = 2-\sqrt3\), find the simplest value of \(3x^2-5xy+3y^2\).

30. If \(a = 3 + 2\sqrt2\), then find the value of \(\cfrac{a^6 + a^4 + a^2 + 1}{a^3}\).