1. 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

2. 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

3. 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

4. The minimum and maximum values of \(a + b\sin\theta\) are 5 and 6 respectively. Find the values of \(a\) and \(b\).

(a) \(a=1, b=5\) (b) \(a=-5, b=1\) (c) \(a=5, b=1\) (d) \(a=-1, b=5\)

5. Find the minimum value of \(9 \tan^2 \theta + 4 \cot^2 \theta\).

6. The expression \(x^4 - 2x^2 + k\) will be a perfect square when the value of \(k\) is:

(a) \(\cfrac{1}{4}\) (b) 0 (c) \(\cfrac{1}{2}\) (d) 1

7. If \(2 \cos^2\theta + 3 \sin \theta = 3\) and \(0^\circ < \theta < 90^\circ\), find the value of \(\theta\).

8. Find the minimum value of the expression \((4\csc^2 θ + 9\sin^2 θ)\) for \(0° < θ < 90°\).

9. If \(0^\circ < \theta < 90^\circ\), then determine the minimum value of \((9 \tan^2 \theta + 4 \cot^2 \theta)\).

10. If \(x \propto \cfrac{1}{y}\), then the value of \(x + y\) will be minimum when _____.

11. If the roots of the quadratic equation \(5x^2+13x+k=0\) are reciprocals of each other, then the value of \(k\) is:

(a) 3 (b) 4 (c) 5 (d) -5

12. If \(0° ≤ \alpha ≤ 90°\), then the minimum value of \((sec^2 \alpha + cos^2 \alpha)\) is **2**.

13. Find the value of \(\cfrac{4}{\sec^2\theta}+\cfrac{1}{1+\cot^2\theta}+3\sin^2\theta\).

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

15. The English translation is: "If \(x^2 = \sin^2 30° + 4\cot^2 45° - \sec^2 60°\), determine the value of \(x\)."

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

17. If \( 0° < θ < 90° \), then find the minimum value of \( 9 \tan^2 θ + 4 \cot^2 θ \).

18. Determine the value(s) of \(\theta\) within the range \(0^\circ \leq \theta \leq 90^\circ\) for which the equation \(2\cos^2\theta - 3\cos\theta + 1 = 0\) holds true.

19. Minimum value of \( \tan \theta + \cot \theta \)

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

20. The value of \(\frac{1}{\sec^2\theta} + \frac{1}{1 + \cot^2\theta} + 3\sin^2\theta\) is _____

21. Find the numerical value of \(\frac{9}{\csc^2\theta} + 4\cos^2\theta + \frac{5}{1 + \tan^2\theta}\).

22. If \(0^\circ < \theta \leq 90^\circ\), then what is the minimum value of \((4\csc^2\theta + 9\sin^2\theta)\)?

23. What is the value of \(\frac{4}{\sec^2\theta} + \frac{1}{1 + \cot^2\theta} + 3\sin^2\theta\)?

24. If \(0^\circ \le \theta \le 90^\circ\), then find the minimum value of \[ 9\tan^2\theta + 4\cot^2\theta \]

25. If a cuboid has \(x\) edges and \(y\) faces, what is the minimum value of \(a\) such that \((x + y + a)\) becomes a perfect square?

26. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

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

27. In triangle ABC , what is the value of sin⁡\(\cfrac{(B+C)}{2} \) ?

(a) sin⁡\(\cfrac{A}{2}\) (b) sinA (c) cosA (d) cos⁡ \(\cfrac{A}{2}\)

28. 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

29. If the number of vertices, faces, and edges of a cuboid are \( p \), \( q \), and \( r \) respectively, what is the value of \( \frac{3(p + r)}{2q} \) ?

(a) 10 (b) 12 (c) 5 (d) 6

30. In a circle with center \( O \), \( AB \) and \( CD \) are two equal-length chords. \( E \) is the midpoint of \( CD \), and \( \angle AOB = 70^\circ \). The value of angle \( \angle COE \) is:

(a) 70° (b) 110° (c) 35° (d) 55°