1. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

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

3. If \(x = \cfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \cfrac{x^2 + xy + y^2}{x^2 - xy + y^2} = \cfrac{12}{11} \]

4. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(xy = 1\), prove that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

5. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

6. If \[ \sin\theta + \cos\theta = \frac{7}{5} \quad \text{and} \quad \sin\theta \cos\theta = \frac{12}{25}, \] then show that \( \sin\theta = ? \)

7. If \[ \frac{x^2 - yz}{a} = \frac{y^2 - zx}{b} = \frac{z^2 - xy}{c} \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz \]

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

9. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

10. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

11. If \(\frac{x}{y} = \frac{a + 2}{a - 2}\), then prove that \[ \frac{x^2 - y^2}{x^2 + y^2} = \frac{4a}{a^2 + 4} \]

12. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

13. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

14. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

15. 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

16. If \[ \frac{x^2}{by + cz} = \frac{y^2}{cz + ax} = \frac{z^2}{ax + by} = 1 \] then prove that \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

17. If \[ \cfrac{x^2}{by + cz} = \cfrac{y^2}{cz + ax} = \cfrac{z^2}{ax + by} = 1 \] then show that \[ \cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1 \]

18. If \[ \cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}, \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz. \]

19. If \[ \cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}, \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz. \]

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

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

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

23. If \((b + c - a)x = (c + a - b)y = (a + b - c)z = 2\), then show that \[ \left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right) = abc \]

24. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

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

26. For what value of \(k\) will the system of equations \[ x + 5y = 8 \quad \text{and} \quad 2x - ky = 13 \] have no solution?

(a) 1 (b) 5 (c) 10 (d) -10

27. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

28. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

29. If \[ \cfrac{a+b}{b+c}=\cfrac{c+d}{d+a} \] then prove that \[ c=a \quad \text{or} \quad a+b+c+d=0. \]

30. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]