1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

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

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

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

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

9. If \( (7a - 5b) : (3a + 4b) = 7 : 11 \), then what is the value of \( (5a - 3b) : (6a + 5b) \)?

(a) 777:244 (b) 777:247 (c) 247:778 (d) 247:787

10. If \(\sqrt{10} - 3 = k\), then what is the value of \(\sqrt{10} + 3\)?

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

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

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

12. If \(5\cos\theta + 12\sin\theta = 13\), then what is the value of \(\tan\theta\)?

(a) \(\cfrac{13}{15}\) (b) \(\cfrac{12}{5}\) (c) \(\cfrac{5}{13}\) (d) \(\cfrac{5}{12}\)

13. If \( \sin 51^\circ = \cfrac{a}{\sqrt{a^2 + b^2}} \), then what is the value of \( \tan 51^\circ + \tan 39^\circ \)?

(a) \(\cfrac{a^2+b^2}{ab}\) (b) \(\cfrac{a^2-b^2}{ab}\) (c) \(\cfrac{a-b}{ab}\) (d) \(\cfrac{a+b}{ab}\)

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

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

15. If \(a = \frac{1}{5 + 2\sqrt{6}}\), then what is the value of \(a^2 - \frac{1}{a^2}\)?

(a) \(4\sqrt6\) (b) -\(4\sqrt6\) (c) \(40\sqrt6\) (d) -\(40\sqrt6\)

16. 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) \(\sqrt5\)

17. If \(\sum f_iu_i = 10\), class width = 20, \(\sum f_i = 40 + k\), the combined mean is 54, and the assumed mean is 50, then what is the value of \(k\)?

18. 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

19. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?

20. If \(\sin^2 x + \sin^2 y = 1\), then what is the value of \(\sin \frac{(x + y)}{2} + \cos \frac{(x + y)}{2}\)?

21. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then what is the value of \(\frac{a^2 + ab + b^2}{a^2 - ab + b^2}\)?

22. If the mean of a statistical distribution is 4.1, \(∑f_i x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?

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

24. If \(\sec^2 \theta + \tan^2 \theta = \frac{13}{12}\), then what is the value of \(\sec^4 \theta - \tan^4 \theta\)?

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

26. If \(m = \sqrt{\cfrac{n}{n + \cfrac{1}{2}}}\) and \(m = \cfrac{1}{2}\), then what is the value of \(n\)?

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

28. If \((2x - 3y) : (5x - 2y) = 1 : 19\), then what is the value of \((2x + 3y) : (3x - 2y)\)?

(a) 9:19 (b) 16:9 (c) 9:16 (d) 19:9

29. If \(x + \cfrac{1}{x} = 3\), then what is the value of \(\cfrac{(x^2 + 3x + 1)}{(x^2 + 7x + 1)}\)?

(a) \(\cfrac{3}{5}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{2}{4}\) (d) \(\cfrac{1}{2}\)

30. 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