1. If \((3a + 7b) : (5a - 3b) = 5 : 3\), then determine the value of \(a : b\).

2. 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

3. 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

4. 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

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

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

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

8. 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

9. 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

10. 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

11. Here’s the English translation: *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 \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

12. Translate and write: If by applying Sridharacharya's formula to the equation \(5x^2 + 2x - 7 = 0\), we get \(x = \cfrac{k \pm 12}{10}\), then what will be the value of \(k\)? Calculate and write it.

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

14. Here’s the English translation of your math problem: If the ascending data set is \[ 27,\ 31,\ 46,\ 52,\ x,\ y + 2,\ 71,\ 79,\ 85,\ 90 \] and the median of the set is \(64\), then what is the value of \(x + y\)?

(a) 125 (b) 126 (c) 127 (d) 128

15. If \(a + b : \sqrt{ab} = 4 : 1\), then what is the ratio of \(a : b\)?

16. 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) \(\sqrt5\)

17. 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

18. 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

19. If \(\cfrac{x-3y}{2y}=\cfrac{6x-5y}{5x}\), then what is the value of \(x:y\)?

(a) 5:1 or 2:5 (b) 1:5 or 2:5 (c) 1:5 or 5:2 (d) None of the above

20. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?

21. If \( \tan θ + \cot θ = 2 \), then what is the value of \( \tan^7 θ + \cot^7 θ \)?

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

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

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. \(a\) is a positive number, and in the proportion \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), what is the value of \(a\)?

(a) \(\cfrac{9}{11}\) (b) \(\cfrac{3}{4}\) (c) \(\cfrac{9}{20}\) (d) \(\cfrac{9}{16}\)

28. If \((a + b) : \sqrt{ab} = 2 : 1\), then what is the ratio \(a : b\)?

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

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