1. If \( 3x = \csc \alpha \) and \( \cfrac{3}{x} = \cot \alpha \), then find the value of \( 3\left(x^2-\cfrac{1}{x^2}\right) \).

(a) \(\cfrac{1}{27}\) (b) \(\cfrac{1}{81}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{1}{9}\)

2. If \( 3x = \csc α \) and \( \cfrac{3}{x} = \cot α \), then find the value of \( 3\left(x^2 - \cfrac{1}{x^2}\right) \).

(a) \(\cfrac{1}{27}\) (b) \(\cfrac{1}{81}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{1}{9}\)

3. If \(3x = \csc \alpha\) and \(y = \cot \alpha\), then the value of \(3\left(x^2 - \cfrac{1}{x^2}\right)\) is -

4. If \(3x = \csc\alpha\) and \(\frac{3}{x} = \cot\alpha\), find the value of \(\left(x^2 - \frac{1}{x^2}\right)\).

5. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

6. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

7. If \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2+8x+2=0\), find the value of \(\cfrac{1}{\alpha^2}+\cfrac{1}{\beta^2}\).

8. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x - 10 = 0\), then find \(\alpha^2 + \beta^2\).

9. If the quadratic equation \(x^2 + px + q = 0\) has roots \(\alpha\) and \(\beta\), then find the value of \(\alpha^3 + \beta^3\).

10. If \(\csc θ + \cot θ = \sqrt{3}\), then find the values of both \(\csc θ\) and \(\cot θ\).

11. If \( \csc^2 θ = 2\cot θ \) and \( 0° < θ < 90° \), then find the value of \( θ \).

12. If \( \alpha \) and \( \beta \) are the roots of the equation \(x^2 - 22x + 105 = 0\), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).

13. If \( \sqrt{2} \cos(\alpha - \beta) = 1 \) and \( \alpha + \beta = \cfrac{\pi}{2} \), then find the values of \( \alpha \) and \( \beta \).

14. If \( \alpha \) and \( \beta \) are the roots of the equation \( 5x^2 - 3x + 6 = 0 \), Find the value of \( \left( \frac{1}{\alpha} + \frac{1}{\beta} \right) \).

15. If \( \csc\theta + \cot\theta = 3 \), then find the values of both \( \csc\theta \) and \( \cot\theta \).

16. If \(\alpha + \beta = 90^\circ\) and \(\alpha : \beta = 2 : 1\), then find the value of \(\sin \alpha : \sin \beta\).

(a) \(3:1\) (b) \(1:3\) (c) \(\sqrt3:1\) (d) \(1:\sqrt 3\)

17. 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

18. If \( \csc^2 θ = 2 \cot θ \), then find the value of \( θ \), where \( 0^\circ < θ < 90^\circ \).

19. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

20. If \( \csc \theta + \cot \theta = \sqrt{3} \), then find the value of \( \sin \theta \), where \( 0^\circ < \theta < 90^\circ \).

21. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - 22x + 105 = 0 \), find the value of \( \alpha - \beta \).

22. If \( \cosθ = \cfrac{3}{5} \), then find the value of \( \cotθ + \cscθ \).

(a) 1 (b) 2 (c) 3 (d) 4

23. If \( \csc A = \sqrt2 \), then find the value of
\(\cfrac{2\sin^2A + 3\cot^2A}{4\tan^2A - \cos^2A}\).

(a) \(\cfrac{8}{7}\) (b) \(\cfrac{7}{8}\) (c) \(\cfrac{1}{8}\) (d) \(\cfrac{1}{7}\)

24. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2-3x+6=0\), determine the value of \(\left(\cfrac{1}{\alpha}+\cfrac{1}{\beta}\right)\).

25. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?

26. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2 + 2x - 3 = 0\), then the value of \(\alpha^2 + \beta^2\) will be \(\cfrac{32}{25}\).

27. If the roots of the equation \(3x^2+8x+2=0\) are \(\alpha\) and \(\beta\), then determine the equation whose roots are \(\cfrac{1}{\alpha}\) and \(\cfrac{1}{\beta}\).

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

29. If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), find the value of \(\left(1+\cfrac{\alpha}{\beta}\right)\left(1+\cfrac{\beta}{\alpha}\right)\).

30. If \( \sin \theta - \cos \theta = 0 \) (where \( 0^\circ \lt \theta \lt 90^\circ \)) and \( \sec \theta + \csc \theta = x \), then find the value of \( x \).

(a) 1 (b) 2 (c) √2 (d) 2√2