1. If \(x = \cfrac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}}\), then what is the value of \(\cfrac{x + \sqrt{5}}{x - \sqrt{5}} + \cfrac{x + \sqrt{3}}{x - \sqrt{3}}\)?
(a) -1 (b) 0 (c) 1 (d) None of the above
2. If \(\sqrt{x} + \sqrt{y} = \sqrt{18 + 6\sqrt{5}}\), what is the value of \(x\)?
(a) 8 (b) 15 (c) 6 (d) 12
3. If \(\sqrt{5} - \sqrt{3} = a\), then what is the value of \(\sqrt{5} + \sqrt{3}\)?
(a) \(\cfrac{2}{a}\) (b) \(\cfrac{a}{2}\) (c) 0 (d) None of the above
4. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then what is the value of \(\frac{a^2 + ab + b^2}{a^2 - ab + b^2}\)?
5. If \(\sqrt{5} - \sqrt{3} = a\), then what is the value of \(\sqrt{5} + \sqrt{3}\)?
(a) \(\cfrac{2}{a}\) (b) \(\cfrac{a}{2}\) (c) \(0\) (d) None of the above
6. If \(p + q = \sqrt{13}\) and \(p - q\sqrt{5}\), then what is the value of \(pq\)?
(a) 2 (b) 18 (c) 9 (d) 8
7. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"
(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4
8. At an annual simple interest rate of 12%, if the ratio of principal to interest after \(x\) years is 25:24, what is the value of \(x\)?
(a) 8 (b) 10 (c) 12 (d) 5
9. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—
10. If a right-angled quadrilateral has \(x\) number of vertices, \(y\) number of edges, and \(z\) number of faces, then what is the value of \(x - y + z\)?
(a) 8 (b) 6 (c) 2 (d) 12
11. If a right-angled quadrilateral has \(a\) number of vertices, \(b\) number of edges, and \(c\) number of faces, then what is the value of \(2a - b + 3c\)?
(a) 16 (b) 18 (c) 20 (d) 22
12. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?
(a) 1 (b) 1 (c) 0 (d) None of the above
13. What is the value of \(\cfrac{\sin \theta}{\csc \theta} + \cfrac{\cos \theta}{\sec \theta} - 2\)?
(a) 0 (b) -1 (c) 4 (d) None of the above
14. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)?
(a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1
15. What is the value of \(\cfrac{3\sqrt{8} - 2\sqrt{12} + \sqrt{20}}{3\sqrt{18} - 2\sqrt{27} + \sqrt{45}}\)?
(a) \(\cfrac{3}{2}\) (b) \(\cfrac{1}{2}\) (c) \(\cfrac{2}{3}\) (d) \(\cfrac{13}{12}\)
16. 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
17. If \(x = \cfrac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}}\), then what is the value of \(bx^2 - ax + b\)?
(a) 0 (b) 1 (c) 2 (d) 3
18. If a cuboid has \(x\) number of vertices, \(y\) number of edges, and \(z\) number of faces, then what is the value of \((x - y + z)^2\)?
(a) 2 (b) 4 (c) 6 (d) None of the above
19. If \(x + \cfrac{1}{x} = -2\), then what is the value of \(x^7 + \cfrac{1}{x^6}\)?
(a) -1 (b) 1 (c) 2 (d) None of the above
20. If \(2\sin 2\theta - \sqrt{3} = 0\), then what is the value of \(\csc \theta\)?
(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{2}{\sqrt3}\) (d) 2
21. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?
(a) -11 (b) 11 (c) 12 (d) 14
22. 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
23. If \(\cfrac{1}{3x}+\cfrac{1}{4x}+\cfrac{1}{2x}+\cfrac{1}{6x}=5\), then what is the value of \(x\)?
(a) \(\cfrac{1}{7}\) (b) \(\cfrac{1}{5}\) (c) \(\cfrac{1}{4}\) (d) None of the above
24. What is the value of \(\cfrac{9}{\csc^2\theta} + 4\cos^2\theta + \cfrac{5}{1 + \tan^2\theta}\)?
(a) 1 (b) 0 (c) 3 (d) 9
25. If \( \csc^2 A = 4 - \sec^2 A \), then what is the value of \( A \)?
(a) \(\cfrac{\pi}{6}\) (b) \(\cfrac{\pi}{3}\) (c) \(\cfrac{\pi}{2}\) (d) \(\cfrac{\pi}{4}\)
26. If \( \tan\left(\cfrac{\pi}{2} - \cfrac{\alpha}{2}\right) = \sqrt{3} \), then what is the value of \( \cos\alpha \)?
(a) \(\cfrac{1}{2}\) (b) \(\cfrac{\sqrt3}{2}\) (c) \(\cfrac{1}{\sqrt2}\) (d) 1
27. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?
(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)
28. If \(a, b, c, d\) are in continued proportion, then what is the value of \(\frac{abc(a + b + c)}{ab + bc + ca}\)?
(a) \(\cfrac{1}{a^2}\) (b) \(\cfrac{1}{c^2}\) (c) \(b^2\) (d) \(a^2\)
29. If \( \sin^2A + \sin^4A = 1 \), then what is the value of \( \tan^2A - \tan^4A \)?
(a) 1 (b) 0 (c) -1 (d) 2
30. What is the value of \( \tan 4^\circ \cdot \tan 43^\circ \cdot \tan 47^\circ \cdot \tan 86^\circ \)?
(a) -1 (b) 1 (c) 0 (d) -2
