1. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?

2. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), find the value of \(\cfrac{1}{α} + \cfrac{1}{β}\).

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

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

(a) 3 (b) √3 (c) -√3 (d) \(\pm\)√3

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

(a) 2 (b) 2√3 (c) 4 (d) 2-√3

5. Two chords, \(AB\) and \(CD\), of a circle with center \(O\), intersect at point \(P\). If \(\angle APC = 40°\), find the value of \(\angle AOC + \angle BOD\).

(a) 60° (b) 80° (c) 120° (d) None of these

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

7. 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

8. 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 \(n\).

9. 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) \(=\cfrac{3}{5}\)

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

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

12. In triangle ∆ABC, ∠B = 90°, AC = √13 cm, and AB + BC = 5 cm. Find the value of (cos A + cos C).

13. 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).

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

15. If \(∠A + ∠B = 90°\), then find the value of \(1 + \frac{\tan A}{\tan B}\).

16. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

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

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

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

20. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

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

22. If \(\sin(2x + y) = \cos(4x - y)\), find the value of \(\tan 3x\).

23. Given that \(\cos\alpha = \sin\beta\) and both \(\alpha\) and \(\beta\) are acute angles, find the value of \(\sin(\alpha + \beta)\).

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

25. If \(x + \frac{9}{x} = 6\), find the numerical value of \(x^2\).

26. From the equation \(5 \sin^2 \theta + 4 \cos^2 \theta = \frac{9}{2}\), find the value of \(\tan \theta\), where \(0^\circ < \theta < 90^\circ\).

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

28. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

29. Given: \(\tan A = \frac{x}{y}\), find the value of \(\frac{\cos A - \sin A}{\cos A + \sin A}\).

30. Given \(a = \cfrac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \cfrac{\sqrt{5} - 1}{\sqrt{5} + 1}\), find the value of \(\cfrac{a^2 + ab + b^2}{a^2 - ab + b^2}\).