1. Find the value of \[ \left( \sin^6 \alpha + \cos^6 \alpha + 3\sin^2 \alpha \cos^2 \alpha \right) \]

2. If \(0^\circ \leq \alpha < 90^\circ\), find the minimum value of \((\sec^2α + \cos^2α)\).

(a) 1 (b) 2 (c) \(\cfrac{5}{2}\) (d) 0

3. If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

4. Given that \(\cos\alpha = \sin\beta\) and both \(\alpha\) and \(\beta\) are acute angles, find the value of \(\sin(\alpha + \beta)\).

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

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

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

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

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

10. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(5x^2+2x+3=0\), determine the value of \(\cfrac{\alpha^2}{\beta}+\cfrac{\beta^2}{\alpha}\).

11. 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 \(\cfrac{32}{25}\).

12. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(7x^2 + 5x - 4 = 0\), determine the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

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

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

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

16. If \(0° \le \alpha < 90°\), then the minimum value of \(sec^2 \alpha + \cos^2 \alpha\) is 2.

17. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \[ 5x^2 + 2x - 3 = 0, \] find the value of \(\alpha^2 + \beta^2\).

18. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation whose roots are \(\alpha\) and \(\beta\), find the value of \(\alpha^3 + \beta^3\).

19. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\).

20. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{α^2}{β} + \frac{β^2}{α}\).

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

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

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

24. 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) \(1:\sqrt 3\)

25. If \( \sin\alpha + \sin^2\alpha = 1 \), then what is the value of \( \cos^2\alpha + \cos^4\alpha \)?

26. 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

27. If \( \tan\alpha + \cot\alpha = \sqrt{3} \), then what is the value of \( \tan^3\alpha + \cot^3\alpha \)?

28. If \( \cot\alpha = \tan(\beta + \gamma) \), then what is the value of \( \sin(\alpha + \beta + \gamma) \)?

(a) 1 (b) 2 (c) 4 (d) \(\cfrac{3}{4}\)

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

30. 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 \(\cfrac{1}{\beta}\).