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

2. If \((5 + √3)(5 - √3) = 25 - x^2\), find the value of \(x\).

(a) 3 (b) √3 (c) -√3 (d) \(\pm\)√3

3. If \(x = \sqrt{7 + 4√3}\), find the value of \(x - \cfrac{1}{x}\).

(a) 2 (b) 2√3 (c) 4 (d) 2-√3

4. 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) \(\cfrac{1}{2709}\)

5. 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

6. If \(\sum_{i=1}^n (x_i - 3) = 0\) and \(\sum_{i=1}^n (x_i + 3) = 66\), then find the values of \(\bar{x}\) (the mean) and \(n\).

7. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

8. If \(x = 3 + 2\sqrt{2}\), then find the value of \(\left(\sqrt{x} + \cfrac{1}{\sqrt{x}}\right)\).

9. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

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

11. If \(x = \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

12. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

13. \((5 + \sqrt{3})(5 - \sqrt{3}) = 25 - x^2\); find the value of \(x\).

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

15. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

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

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

18. 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 \(x\).

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

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

21. If \(x = \sqrt{2} + 1\), find the values of \(x^4 + \frac{1}{x^4}\) and \(x^4 - \frac{1}{x^4}\).

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

23. If \(x - \cfrac{1}{x} = \sqrt{5}\), then find the value of \(x^4 + \cfrac{1}{x^4}\).

(a) 45 (b) 46 (c) 47 (d) 48

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

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

26. If \(\sum(x_i - 3) = 0\) and \(\sum(x_i + 3) = 66\), then find the values of \(\bar{x}\) (mean) and \(n\) (number of observations).

27. If \[ x \cos 60^\circ = \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} - \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] then find the value of \(x\).

28. If \(x = \sqrt{\cfrac{\sqrt{5}+1}{\sqrt{5}-1}}\), find the value of \(x^2 - x - 1\).

29. 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 \(x\).

30. If \(x = \cfrac{8ab}{a - b}\), then find the value of \(\cfrac{x + 4a}{x - 4a}\).