1. If \(∠A + ∠B = 90^\circ\), then find the value of \(1 + \tan A \div \tan B\).

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

3. 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

4. If \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), and the combined mean is \(13.5 + p\), then what is the value of \(p\)?

(a) 0 (b) 1 (c) 0.1 (d) 0.01

5. 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 \(n\).

6. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\).

(a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

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 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).

10. If \(a + b = 3\) and \(a - b = \sqrt{5}\), then find the value of \(ab\).

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

12. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).

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

14. If \(\angle A + \angle B = 90^\circ\), then prove that \(1 + \frac{\tan A}{\tan B} = \csc^2 B\).

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

16. If \(\frac{x}{y} = \frac{a + 2}{a - 2}\), then find the value of \(\frac{x^2 - y^2}{x^2 + y^2}\).

17. If \( \csc \theta + \cot \theta = \sqrt{3} \), then find the value of \( \sin \theta \), where \( 0^\circ < \theta < 90^\circ \).

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

19. If \[ \frac{2x}{3} = \frac{4y}{5} = \frac{7z}{9} \] then find the value of \[ \frac{4x + 12y - 21z}{3y} \]

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

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

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

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

25. If \(b + c = a^2\), \(c + a = b^2\), and \(a + b = c^2\), then find the value of \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c}\).

(a) 2 (b) \(\infty\) (c) 0 (d) 1

26. 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

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

28. If \( \frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 5 \), then find the value of \( \tan\theta \).

29. If \( \frac{a}{2} = \frac{b}{3} = \frac{c}{4} = \frac{2a - 3b + 4c}{p} \), then find the value of \( p \).

30. If \(\sin A + \sin B = 2\), where \(0^\circ \leq A \leq 90^\circ\) and \(0^\circ \leq B \leq 90^\circ\), then find the value of \(\cos A + \cos B\).