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

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

3. 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

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

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

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 \[ \frac{3a^2 + 5ab + 3b^2}{3a^2 - 5ab + 3b^2} \]

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

9. If \(\sin(A + B) = 1\) and \(\cos(A - B) = 1\), then find the value of \(\cot 2A\), given that \(0^\circ \leq (A + B) \leq 90^\circ\) and \(A \geq B\).

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

11. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).

(a) 54° (b) 72° (c) 36° (d) 27°

12. 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

13. In triangles \(ABC\) and \(DEF\), if \(\angle A = \angle F = 40^\circ\), \(AB:ED = AC:EF\), and \(\angle F = 65^\circ\), find the value of \(AB\).

(a) 35° (b) 65° (c) 75° (d) 85°

14. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of \((3ax - by) : (4by - 7ax)\).

15. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

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

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

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

19. 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

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

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

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

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

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

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

26. In the given figure, \(O\) is the center of the circle, and \(AB\) is the diameter. If \(\angle ADC = 120^\circ\), then determine the value of \(\angle BAC\).

(a) 50° (b) 60° (c) 40° (d) 30°

27. Three circles with centers \(A\), \(B\), and \(C\) externally touch each other. If \(AB = 5\) cm, \(BC = 7\) cm, and \(CA = 6\) cm, then find the radius of the circle centered at \(C\).

28. 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 \(x\).

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 \(A : B = 6 : 7\) and \(B : C = 8 : 7\), then find the value of \(A : C\).