1. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

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

2. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

3. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

4. If the numbers \((2 + x)\), \((8 + x)\), and \((26 + x)\) are in continued proportion, then what is the value of \(x\)?

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

5. 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

6. 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

7. 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

8. If \(x\sin45^\circ \cos45^\circ \tan60^\circ = \tan^2 45^\circ - \cos^2 60^\circ\), then what is the value of \(x\)?

(a) 1 (b) \(\cfrac{2}{\sqrt3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)

9. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

10. If a cuboid has number of faces = x, number of edges = y, number of vertices = z, and number of diagonals = p, then what is the value of (x − y + z + p)?

11. If \(\sin 3x = 1\), what is the value of \(\tan 2x\)?

12. If the compound ratio of \(2 : 3\), \(3 : 5\), and \(x : 8\) is \(1 : 4\), then what is the value of \(x\)?

(a) 3 (b) 4 (c) 5 (d) 6

13. If 8, 14, \(7x + 2\), and 28 are in continued proportion, then what is the value of \(x\)?

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

14. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

15. \(x \propto y\) and \(y = 8\) when \(x = 2\); if \(y = 16\), what is the value of \(x\)?

(a) 2 (b) 4 (c) 6 (d) 8

16. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

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

18. 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

19. 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

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

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

21. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \( \frac{x}{y} \)?

(a) \(\cfrac{π}{180}\) (b) \(\cfrac{π}{90}\) (c) \(\cfrac{90}{π}\) (d) \(\cfrac{180}{π}\)

22. If the complement of a \(3x^\circ\) angle is \(6x^\circ\), then what is the value of \(x\)?

(a) 10 (b) 9 (c) 20 (d) 12

23. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

24. 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°

25. \(\theta\) is a positive acute angle, and if \( \tan\theta = \cot\theta \), then what is the value of \(\theta\)?

(a) 40° (b) 45° (c) 60° (d) 20°

26. If \(\sqrt{x} + \sqrt{y} = \sqrt{18 + 6\sqrt{5}}\), what is the value of \(x\)?

(a) 8 (b) 15 (c) 6 (d) 12

27. If \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), and the combined mean is \(13.5 + p\), then what is the value of \(p\)?

(a) 0 (b) 1 (c) 0.1 (d) 0.01

28. If \(\sin^2 x + \sin^2 y = 1\), then what is the value of \(\sin \frac{(x + y)}{2} + \cos \frac{(x + y)}{2}\)?

29. If \( u_i = \frac{x_i - 45}{10} \), \( ∑f_i u_i = -16 \), and \( ∑f_i = 200 \), then what is the value of \( \bar{x} \)?

30. If \(u_i = \frac{x_i - 30}{10}\), \(∑f_i = 50\), and \(∑u_i f_i = 25\), then what is the value of \(\bar{x}\)?