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

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

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

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

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

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

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

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

9. 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

10. 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 \(x\).

11. 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 \(x\).

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

13. Two chords, \(AB\) and \(CD\), of a circle with center \(O\), intersect at point \(P\). If \(\angle APC = 40°\), find the value of \(\angle AOC + \angle BOD\).

(a) 60° (b) 80° (c) 120° (d) None of these

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

15. If \(x = \sqrt{\cfrac{\sqrt{5}+1}{\sqrt{5}-1}}\), find the value of \(x^2 - x - 1\).

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

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

18. In triangle ∆ABC, ∠B = 90°, AC = √13 cm, and AB + BC = 5 cm. Find the value of (cos A + cos C).

19. 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?

20. 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 \(BC - XY\)?

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

22. If \(b + c = a^2\), \(c + a = b^2\), and \(a + b = c^2\), then find the value of \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c}\).

(a) 2 (b) \(\infty\) (c) 0 (d) 1

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

24. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

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

26. If \(x^2 + ax + b = (x+2)(x-2)\), find the values of \(a\) and \(b\).

(a) 1,2 (b) 2,1 (c) 0,4 (d) 0,-4

27. If the quadratic equation \(ax^2 + 7x + b = 0\) has roots \(\cfrac{2}{3}\) and \(-3\), then find the values of \(a\) and \(b\).

28. In the given figure, \( LM \parallel AB \) and \( AL = (x - 3) \) units, \( AC = 2x \) units, \( BM = (x - 2) \) units, and \( BC = (2x + 3) \) units. Find the value of \( x \).

29. If the quadratic equation \(ax^2 + 7x + b = 0\) has two roots \(\frac{2}{3}\) and \(-3\), then let us find the values of \(a\) and \(b\).

30. If the roots of the equation \(ax^2 + bx + 35 = 0\) are \(-5\) and \(-7\), find the values of \(a\) and \(b\).