1. If \(x = \sqrt{5} + 1\), then show that \(x^2 - 2x = 4\).

2. If \(x = \frac{\sqrt{a + 1} + \sqrt{a - 1}}{\sqrt{a + 1} - \sqrt{a - 1}}\), then show that \(x^2 - 2ax + 1 = 0\).

3. If \(x = \frac{1 + \sin\theta}{\cos\theta}\), then show that \(x^2 - 2x\tan\theta - 1 = 0\).

4. If \(x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1\), then show that \(\cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1\).

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

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

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

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

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

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

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

12. If \(\frac{a^2}{b + c} = \frac{b^2}{c + a} = \frac{c^2}{a + b} = 1\), then show that \(\frac{a}{a + 1} + \frac{b}{b + 1} + \frac{c}{c + 1} = 2\).

13. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then show that \(c + \frac{1}{a} = 1\).

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

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

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

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

18. If \(x = \cfrac{\sqrt{a+2b} + \sqrt{a-2b}}{\sqrt{a+2b} - \sqrt{a-2b}}\), then show that \(bx^2 - ax + b = 0\).

19. If \(a=\sqrt{x}+\cfrac{1}{\sqrt{x}}\) and \(b=\sqrt{x}- \cfrac{1}{\sqrt{x}}\), then show that \(a^4+b^4-2a^2b^2=16\).

20. If \(x=\cfrac{\sqrt{a+2b}+\sqrt{a-2b}}{\sqrt{a+2b}-\sqrt{a-2b}}\), then show that \(bx^2 - ax + b = 0\).

21. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2 =\) constant.

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

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

24. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then show that \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1\).

25. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then show that \[ \cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1 \]

26. Find the value of \(k\) such that one of the roots of the quadratic equation \(x^2 + kx + 3 = 0\) is \(1\). Show the calculation.

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

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

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

30. If \[ \cfrac{x}{xa + yb + zc} = \cfrac{y}{ya + zb + xc} = \cfrac{z}{za + xb + yc} \] and \(x + y + z \ne 0\), then show that each of the above ratios is equal to \(\cfrac{1}{a + b + c}\).