1. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is —

(a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

2. If \(x = 3 + \sqrt{5}\) and \(xy = 4\), determine the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\).

3. If \(x=3+\sqrt5\) and \(xy=4\), find the value of \(\cfrac{x^2-3xy+y^2}{x^2+3xy+y^2}\).

4. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then find the simplest value of \(\frac{x^2 - xy + y^2}{x^2 + xy + y^2}\).

5. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).

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

7. Given: \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] Find the value of: \[ \frac{x^2 + 3xy + y^2}{x^2 - 3xy + y^2} \]

8. If \(x=\cfrac{\sqrt5+1}{\sqrt5-1}\) and \(xy=1\), then find the value of \(\cfrac{3x^2+5xy+3y^2}{3x^2-5xy+3y^2}\).

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

10. 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

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

12. 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) \(\sqrt5\)

13. If \(x : a = y : b = z : c\), then show that \(\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}\).

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

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

16. 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

17. 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

18. If \(x=\cfrac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\) and \(xy=1\), then find the value of \(x^2 - xy + y^2\).

19. If \(x = \cfrac{\sqrt3 - \sqrt2}{\sqrt3 + \sqrt2}\) and \(xy = 1\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

20. If \(x:y = 3:4\), find the value of \((3y - x) : (2x + y)\).

21. 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.

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

23. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \cfrac{xyz}{abc}\)

24. If \(x : a = y : b = z : c\), then prove that \[\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\]

25. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ xy + \frac{1}{xy} \]

26. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ 3x^2 - 5xy + 3y^2 \]

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

28. If \(x : a = y : b = z : c\), then show that \(\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\).

29. Given: \[ x = 3 + \sqrt{3}, \quad xy = 6 \] Find the value of \((x + y)\).

30. If \(x : a = y : b = z : c\), then show that \[ \frac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \frac{xyz}{abc} \]