1. 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

2. 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

3. If \(\sin(3x - 20^\circ) = \cos(3y + 20^\circ)\), then what is the value of \(x + y\)?

(a) 60° (b) 30° (c) 45° (d) 90°

4. From the equation \( \sin(90^\circ + \theta) = \cos(120^\circ - 3\theta) \), the value of \( \theta \) is —

(a) 30° (b) 60° (c) 45° (d) None of the above

5. The minimum value of \( \cos^2\theta + \sec^2\theta \) is:

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

6. If \(a \sin^2 \theta + b \cos^2 \theta = c\), then what is the value of \(\tan^2 \theta\)?

(a) \(\cfrac{b+c}{a+c}\) (b) \(\cfrac{b-c}{c-a}\) (c) \(\cfrac{a-b}{b-c}\) (d) \(\cfrac{c-a}{a-b}\)

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

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

9. The value of \( \sin^2 5\theta + \cos^2 5\theta \) is 5

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

11. If sinA+sinB=2, then what is the value of cosA-cosB

(a) 0 (b) 1 (c) 2 (d) 3

12. 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) -\(\cfrac{1}{3}\)

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

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

14. 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

15. 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

16. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.

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

17. If the mode of the numbers 64, 60, 48, x, 43, 48, 43, 34 is 43. Then the value of \((x+3)\) is.

(a) 44 (b) 45 (c) 46 (d) 48

18. The value of (sin43°cos47° +cos43°sin47°) is:

(a) 0 (b) 1 (c) sin4° (d) cos4°

19. If A + B = 90° and tanA = \(\cfrac{3}{4}\), then the value of cotB is -

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{4}{3}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{5}{3}\)

20. If sinθ + cosθ = √2 (where 0° < θ < 90°), then the value of θ is

(a) 30° (b) 45° (c) 60° (d) 90°

21. ABC is a triangle. Find the value of \(\sin\left(\cfrac{B+C}{2}\right)\)

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

22. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be –

(a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12

23. If \(\cfrac{p}{q} = \cfrac{5}{7}\) and \(p - q = -2\), then the value of \(p + q\) is –

(a) 12 (b) 13 (c) 14 (d) 15

24. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is –

(a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3

25. 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

26. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—

(a) 2 (b) 18 (c) 9 (d) 8

27. If \(tanα + cotα = 2\), then the value of \(tan^{13}α + cot^{13}α\) is—?

(a) 13 (b) 2 (c) 1 (d) 0

28. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—?

(a) \(1\) (b) \(2\) (c) \(\sqrt2\) (d) \(2\sqrt2\)

29. 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

30. 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