1. If \(2\sin 2\theta - \sqrt{3} = 0\), then what is the value of \(\csc \theta\)?

(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{2}{\sqrt3}\) (d) 2

2. If \(x = \cfrac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}}\), then what is the value of \(\cfrac{x + \sqrt{5}}{x - \sqrt{5}} + \cfrac{x + \sqrt{3}}{x - \sqrt{3}}\)?

(a) -1 (b) 0 (c) 1 (d) None of the above

3. If \( x = 2 + \sqrt{3} \), then what is the value of \( x + \frac{1}{x} \)?

(a) -4 (b) 2 (c) 4 (d) \(2+\sqrt3\)

4. If \(\csc\theta + \cot\theta = 2 + \sqrt{3}\), then what is the value of \(\csc\theta - \cot\theta\)?

(a) \(\sqrt3+\sqrt2\) (b) \(\sqrt2-3\) (c) \(\sqrt3-2\) (d) \(2-\sqrt3\)

5. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

6. If \( \tan\alpha + \cot\alpha = \sqrt{3} \), then what is the value of \( \tan^3\alpha + \cot^3\alpha \)?

7. If \(x = \frac{\sqrt{3}}{2}\), then what is the value of \(\frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}\)?

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

8. If \(\sqrt{5} - \sqrt{3} = a\), then what is the value of \(\sqrt{5} + \sqrt{3}\)?

(a) \(\cfrac{2}{a}\) (b) \(\cfrac{a}{2}\) (c) 0 (d) None of the above

9. If \(\tan(\theta + 15^\circ) = \sqrt{3}\), then what is the value of \(\sin \theta\)?

10. If \(x = \cfrac{1}{2 + \sqrt{3}}\) and \(y = \cfrac{1}{2 - \sqrt{3}}\), then what is the value of \(\cfrac{1}{1 + x} + \cfrac{1}{1 + y}\)?

11. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

12. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

13. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then what is the value of \(\frac{x^2 + y^2 + xy}{x^2 + y^2 - xy}\)?

(a) \(\cfrac{6}{7}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{11}\) (d) None of the above

14. If \(\sqrt{5} - \sqrt{3} = a\), then what is the value of \(\sqrt{5} + \sqrt{3}\)?

(a) \(\cfrac{2}{a}\) (b) \(\cfrac{a}{2}\) (c) \(0\) (d) None of the above

15. If \(x = 7 + 4\sqrt{3}\), then what is the value of \(\cfrac{x^3}{x^6 + 3x^3 + 1}\)?

(a) \(2705\) (b) \(7430\) (c) \(\cfrac{1}{2705}\) (d) \(\cfrac{1}{7430}\)

16. If \( (\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 21 - x^2 \), then what is the value of \( x \)?

17. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

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

18. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?

(a) 1 (b) 1 (c) 0 (d) None of the above

19. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)?

(a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1

20. If \(a + b + c = 0\), then what is the value of \(\cfrac{a^3 + b^3 + c^3}{abc} - 3\)?

(a) 1 (b) 0 (c) -1 (d) None of the above

21. If \(x = \cfrac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}}\), then what is the value of \(bx^2 - ax + b\)?

(a) 0 (b) 1 (c) 2 (d) 3

22. If \(x + \cfrac{1}{x} = -2\), then what is the value of \(x^7 + \cfrac{1}{x^6}\)?

(a) -1 (b) 1 (c) 2 (d) None of the above

23. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)?

(a) 1 (b) 2 (c) 3 (d) 4

24. If \(\cfrac{1}{3x}+\cfrac{1}{4x}+\cfrac{1}{2x}+\cfrac{1}{6x}=5\), then what is the value of \(x\)?

(a) \(\cfrac{1}{7}\) (b) \(\cfrac{1}{5}\) (c) \(\cfrac{1}{4}\) (d) None of the above

25. If \( \csc^2 A = 4 - \sec^2 A \), then what is the value of \( A \)?

(a) \(\cfrac{\pi}{6}\) (b) \(\cfrac{\pi}{3}\) (c) \(\cfrac{\pi}{2}\) (d) \(\cfrac{\pi}{4}\)

26. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

27. If \( \sin^2A + \sin^4A = 1 \), then what is the value of \( \tan^2A - \tan^4A \)?

(a) 1 (b) 0 (c) -1 (d) 2

28. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then what is the value of \( \cos\theta + \sin\theta \)?

(a) \(2 cos\theta\) (b) \(\sqrt2 sin \theta\) (c) \(2 sin\theta\) (d) \(\sqrt2 cos \theta\)

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

(a) \(-\cfrac{1}{\sqrt3}\) (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(-\cfrac{1}{3}\)

30. If \( r \sin\theta = \frac{7}{2} \) and \( r \cos\theta = \frac{7\sqrt{3}}{2} \), then what are the values of \( r \) and \( \theta \)?

(a) \(r=7, \theta=30^0\) (b) \(r=\cfrac{1}{7}, \theta=30^0\) (c) \(r=7, \theta=60^0\) (d) \(r=\cfrac{1}{7}, \theta=60^0\)