1. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then find the value of \[ \frac{a^2 + ab + b^2}{a^2 - ab + b^2} \]

2. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then find the value of \[ \frac{3a^2 + 5ab + 3b^2}{3a^2 - 5ab + 3b^2} \]

3. Given \(a = \cfrac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \cfrac{\sqrt{5} - 1}{\sqrt{5} + 1}\), find the value of \(\cfrac{a^2 + ab + b^2}{a^2 - ab + b^2}\).

4. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then find the value of \[ \frac{(a - b)^3}{(a + b)^3} \]

5. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then find the value of \[ \frac{a^3 + b^3}{a^3 - b^3} \]

6. If \( a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \) and \( b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1} \), then find the value of \[ \frac{a^2 + ab + b^2}{a^2 - ab + b^2} \]

7. If \(a = \cfrac{\sqrt5+1}{\sqrt5-1}\) and \(ab = 1\), then what is the value of \(\cfrac{3a^2+5ab+3b^2}{3a^2-5ab+3b^2}\)?

8. If \(a = \cfrac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \cfrac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then find \(\cfrac{a^2 + 5ab + b^2}{a^2 - 5ab + b^2}\).

9. If \(a =\cfrac{\sqrt{5}+1}{\sqrt{5}-1}\) and \(ab = 1\), then find the value of \(\left(\cfrac{a}{b}+\cfrac{b}{a}\right)\).

10. If \( a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \) and \( b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1} \), then find the value of \[ \frac{3a^2 + 5ab + 3b^2}{3a^2 - 5ab + 3b^2} \]

11. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

12. If \(a = \cfrac{1}{2+\sqrt3}\) and \(b = \cfrac{1}{2-\sqrt3}\), then what is the value of \(\cfrac{1}{a+1}+\cfrac{1}{b+1}\)?

13. If \(a + b + c = 0\), then what is the value of \(\cfrac{a^3 + b^3 + c^3}{abc} - 3\)?

(a) 1 (b) 0 (c) -1 (d) None of the above

14. 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

15. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

16. If \(a + \cfrac{1}{b} = 1\) and \(b + \cfrac{1}{c} = 1\), then find the values of \((c + \cfrac{1}{a})\) and \((abc + 1)\).

(a) 1 and 0 (b) 0 and 1 (c) 0 and 0 (d) 1 and 1

17. If \(a + b = \sqrt{5}\) and \(ab = \sqrt{3}\), then the value of \((a^2 + b^2)\) is __.

18. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

19. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

20. In the adjacent figure, if \(LM \parallel AB\), and \(AL = (x - 3)\) units, \(AC = 2x\) units, \(BM = (x - 2)\) units, and \(BC = (2x + 3)\) units, then find the value of \(x\).

21. If \(a + \cfrac{1}{b} = 1\) and \(b + \cfrac{1}{c} = 1\), then what is the value of \(c + \cfrac{1}{a}\)?

22. In triangle ABC, if AB\(^2\) + BC\(^2\) = AC\(^2\) and AC = \(\sqrt{2}\) × BC, then what type of triangle is it?

(a) Right-angled (b) Right-angled isosceles (c) Equilateral (d) Isosceles

23. 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

24. If \(a + b = 3\) and \(a - b = \sqrt{5}\), then find the value of \(ab\).

25. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

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

27. If \(a + b + c = 0\), then the value of \(\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab}\) is?

(a) \(0\) (b) \(1\) (c) \(-1\) (d) \(3\)

28. If \(x = \frac{8ab}{a + b}\), then what is the value of \[ \frac{x + 4a}{x - 4a} + \frac{x + 4b}{x - 4b}? \]

29. If \(x = \cfrac{4ab}{a+b}\), then what is the value of \(\cfrac{x+2a}{x-2a}+\cfrac{x+2b}{x-2b}\)?

30. If \(a = \cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(ab = 1\), then prove that \(\cfrac{a^2+ab+b^2}{a^2-ab+b^2}=\cfrac{12}{11}\).