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

2. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

3. 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

4. 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

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

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

6. If \( \frac{a}{3} = \frac{b}{4} = \frac{c}{7} \), then what is the value of \( \frac{a + b + c}{c} \)?

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

7. 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

8. If \( (3x - 2y) : (3x + 2y) = 4 : 5 \), then what is the value of \(x : y\)?

(a) 1:6 (b) 1:1 (c) 2:1 (d) 6:1

9. If \(x\sin45^\circ \cos45^\circ \tan60^\circ = \tan^2 45^\circ - \cos^2 60^\circ\), then what is the value of \(x\)?

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

10. If \(\frac{a}{2} = \frac{b}{3} = \frac{c}{4} = \frac{3a - 2b + 4c}{p}\), then what is the value of \(p\)?

(a) 12 (b) 13 (c) 16 (d) 18

11. If \( \sin\theta + \csc\theta = 2 \), then what is the value of \( \sin^2\theta + \csc^2\theta \)?

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

12. Here’s the English translation of your question: **"If \( x = \frac{4ab}{a + b} \), then what is the value of \( \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \)?"** Would you like me to solve it step by step?

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

13. 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 \(k\)?

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

15. If \( u_i = \frac{x_i - 45}{10} \), \( ∑f_i u_i = -16 \), and \( ∑f_i = 200 \), then what is the value of \( \bar{x} \)?

16. If \( \sqrt{2} \sin(2x + 5^\circ) = \cot 45^\circ \), then what is the value of \( \sec 3x \)?

17. 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 \(k\)?

18. If \( \cos^2θ − \sin^2θ = \frac{1}{x} \), where \( x > 1 \), then what is the value of \( \cos^4θ − \sin^4θ \)?

19. If \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4} = \cfrac{2a - 3b + 4c}{p}\), then what is the value of \(p\)?

20. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

21. If \(x : y = 3 : 4\), then what is the value of \((3y - x) : (2x + y)\)?

22. If \(\cos 43^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 47^\circ\)?

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

24. If \(a : b = 2 : 3\) and \(b : c = 4 : 5\), then what is the value of \(a^2 : b^2 : bc\)?

(a) 8:18:21 (b) 16:36:45 (c) 16:20:36 (d) 8:15:18

25. If \(a : b = 2 : 1\) and \(x : y = 3 : 4\), then what is the value of \((2ax - by) : (2by - ax)\)?

(a) 2:1 (b) 3:1 (c) 4:1 (d) 5:1

26. If \((4x - 3y) : (2x - y) = 2 : 3\), then what is the value of \(x : y\)?

(a) 15:14 (b) 8:7 (c) 7:8 (d) 14:15

27. 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) \(\cfrac{1}{7430}\)

28. If \(3\sinθ + 4\cosθ = 5\), then what is the value of \(4\sinθ - 3\cosθ\)?

29. If \(\cos\theta = \frac{4}{5}\), then what is the value of \(\csc\theta \div (1 + \cot\theta)\)?

(a) \(\cfrac{2}{4}\) (b) \(\cfrac{5}{7}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{1}{3}\)

30. If \(a \propto b\) and when \(a = 2\), \(b = 14\), then what is the value of \(b\) when \(a = 5\)?