1. If \((3x - 2y) : (x + 3y) = 5 : 6\), then find the value of \((2x - 5y) : (3x + 4y)\).

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

3. 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

4. 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

5. If \((7x - 5y) : (3x + 4y) = 7 : 11\), then show that \((3x - 2y) : (3x + 4y) = 137 : 473\).

6. If \((10x + 3y) : (5x + 2y) = 9 : 5\), then show that \((2x + y) : (x + 2y) = 11 : 13\).

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

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 \((5x - 3y) : (2x + 4y) = 11:12\), determine the ratio \(x:y\).

(a) 19 : 40 (b) 40 : 19 (c) 19 : 21 (d) 40 : 21

10. If \((5x - 3y) : (2x + 4y) = 11 : 12\), then determine the value of \(x : y\).

11. If \(xy = 5 : 1\), find the value of the ratio \(x - y^2 : x^2 + y^2\).

(a) 25:1 (b) 12:13 (c) 12:25 (d) None of the above

12. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\).

(a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

13. If \(a + b : \sqrt{ab} = 2 : 1\), then find the ratio \(a : b\).

14. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

15. Find the value of \(x\):
\(x \tan^2 30° + 2x \sec^2 45° + 2x \csc^2 60° = 4\)

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

16. If \(p:q=5:7\) and \(p-q=-4\), then find \(3p+4q\).

17. If \((5x - 3y) : (2x + 4y) = 11:12\), determine the value of \(x:y\).

18. 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.

19. If \(a : b = 3 : 2\) and \(b : c = 3 : 2\), then find the value of the ratio \((a + b) : (b + c)\).

20. If \(\cot A = \frac{4}{7.5}\), then find the values of \(\cos A\) and \(\csc A\), and show that: \[ 1 + \cot^2 A = \csc^2 A \]

21. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]