1. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

2. If \[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3 \] then find the value of \[ \sin^4 \theta - \cos^4 \theta \]

3. If \[ x = \frac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}} \] then find the value of \[ \frac{x + \sqrt{3}}{x - \sqrt{3}} + \frac{x + \sqrt{5}}{x - \sqrt{5}} \]

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

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

6. If \( \tan\theta = \frac{x}{y} \), then find the value of \[ \frac{x \sin\theta - y \cos\theta}{x \sin\theta + y \cos\theta} \]

7. 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 \(x\).

8. If \[ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = 2\frac{51}{79} \] then find the value of \(\sin\theta\).

9. 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 \(x\).

10. If \(x = \cfrac{8ab}{a + b}\), then find the value of \[ \left(\cfrac{x + 4a}{x - 4a} + \cfrac{x + 4b}{x - 4b}\right) \]

11. If \[ \cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4} = \cfrac{2a - 3b + 4c}{p}, \] then find the value of \(p\).

12. If \[ \cfrac{3x - 5y}{3x + 5y} = \cfrac{1}{2}, \] then find the value of \[ \cfrac{3x^2 - 5y^2}{3x^2 + 5y^2}. \]

13. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

14. If \(\cot A = \frac{4}{7.5}\), then find the values of \(\cos A\) and \(\csc A\), and show that: \[ 1 + \cot^2 A = \csc^2 A \]

15. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x - \frac{1}{x} \]

16. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x + \frac{1}{x} \]

17. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x^2 + \frac{1}{x^2} \]

18. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x^3 + \frac{1}{x^3} \]

19. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then find the value of \[ \frac{a^2 + ab + b^2}{a^2 - ab + b^2} \]

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

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

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

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

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

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

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

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

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

(a) 2 (b) 2√3 (c) 4 (d) 2-√3

29. If \(p + q = \sqrt{13}\) and \(p - q = \sqrt{5}\), then find the value of \[ pq \]

(a) 2 (b) 18 (c) 9 (d) 8

30. If \(a + b = \sqrt{5}\) and \(a - b = \sqrt{3}\), then find the value of \[ a^2 + b^2 \]

(a) 8 (b) 4 (c) 2 (d) 1