1. 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

2. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

10. If \((a^2 + b^2)(x^2 + y^2) = (ax + by)^2\), then what are the values of \(x\) and \(y\)?

(a) \(b:a\) (b) \(b^2:a^2\) (c) \(a^2:b^2\) (d) \(a:b\)

11. 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) \(\sqrt5\)

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

13. Sure! Here's the English translation of your math question: **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) \(\sqrt5\)

14. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

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

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

17. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

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

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

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

(a) 12:15 (b) 11:16 (c) 13:19 (d) 10:17

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

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

22. 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) \(\cfrac{1}{2}\)

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

24. 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

25. 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) \(2\sqrt{3}\) (b) \(\cfrac{1}{\sqrt{3}}\) (c) \(\sqrt{3}\) (d) \(\sqrt{3}\)

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

27. If \(x = \frac{8ab}{a + b}\), then what is the value of \[ \frac{x + 4a}{x - 4a} + \frac{x + 4b}{x - 4b}? \]

28. If \(x + \frac{1}{x} = 2\), then what is the value of \(x^{119} + \frac{1}{x^{119}}\)?

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

29. If the equation \(x^2 + k(4x + k - 1) + 2 = 0\) has equal roots, then what is the value of \(k\)?

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

30. If the sum of the roots of the equation \(x^2 - (k + 6)x + 2(2k - 1) = 0\) is half of their product, then what is the value of \(k\)?

(a) 6 (b) 7 (c) 1 (d) 5