1. If \(\cos^4\theta - \sin^4\theta = \frac{2}{3}\), then what is the value of \(1 - 2\sin^2\theta\)?

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

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

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 \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), 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

7. 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 \(n\).

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

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

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

11. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

12. 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).

13. If \( \csc^2 θ = 2 \cot θ \), then find the value of \( θ \), where \( 0^\circ < θ < 90^\circ \).

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

15. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of \((3ax - by) : (4by - 7ax)\).

16. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

17. ABCD is a cyclic quadrilateral and O is the center of the circle. If ∠COD = 120° and ∠BAC = 30°, then find the values of ∠BOC and ∠BCD.

18. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

19. AOB is a diameter of a circle. C is a point on the circle. If ∠OBC = 60°, then find the value of ∠OCA.

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

21. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).

22. If \( \cos^2 θ - \sin^2 θ = \frac{1}{2} \), then find the value of \( \cos^4 θ - \sin^4 θ \).

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

24. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \tan^2 θ \).

25. If \( \sin10θ = \cos8θ \) and \(10θ\) is a positive acute angle, find the value of \( \tan9θ \).

26. If \(u_i = \frac{x_i − 35}{10}\), \(Σf_iu_i = 30\), and \(Σf_i = 60\), then find the value of \(\bar{x}\).

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

28. \(A \propto B\); when \(A = 2\), then \(B = 14\). Find the value of \(B\) when \(A = 5\).

29. If \(r \cos \theta = \frac{1}{2}\) and \(r \sin \theta = \frac{\sqrt{3}}{2}\), then find the value of \(r\), where \(0^\circ < \theta < 90^\circ\).

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