1. 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

2. 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

3. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

4. 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) \(\cfrac{180}{π}\)

5. If \((a^2 + b^2)(x^2 + y^2) = (ax + by)^2\), then what are the values of \(x\) and \(y\)?

(a) \(b:a\) (b) \(b^2:a^2\) (c) \(a^2:b^2\) (d) \(a:b\)

6. 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

7. If the base area of a cone is \(x\) square units, its height is \(y\) units, and its volume is \(z\) cubic units, then what is the value of \(\frac{xy}{z}\)?

8. If the volume of a right circular cone is \(X\) cubic units, the area of its base is \(Y\) square units, and its height is \(Z\) units, then what is the value of \(\frac{YZ}{X}\)?

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

10. If a right rectangular prism (cuboid) has \(x\) vertices, \(y\) edges, and \(z\) faces, then what is the value of \((x - y + z)\)?

11. If a cuboid has \(x\) faces, \(y\) edges, \(z\) vertices, and \(p\) diagonals, then find the value of \((x - y + z + p)\).

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 \(4\), \(5 + x\), \(2x + 1\), 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

15. 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

16. If the distance between the points \((x, -7)\) and \((3, -3)\) is 5 units, then the values of \(x\) are:

(a) 0 or 6 (b) 2 or 3 (c) 5 or 1 (d) -6 or 0

17. If the area of a circular region is \(x\) square units, the circumference is \(y\) units, and the diameter is \(z\) units, then the value of \(\cfrac{x}{yz}\) is —

(a) \(\cfrac{1}{2}\) (b) \(\cfrac{1}{4}\) (c) 1 (d) \(\cfrac{1}{8}\)

18. If \(x\) is the mean proportional between \((x + 2)\) and \((x - 3)\), then the value of \(x\) is ___.

19. A brass plate has a square base with side length \(x\) cm, thickness 1 mm, and a total weight of 4725 grams. If the weight of 1 cubic centimeter of brass is 8.4 grams, then calculate and write the value of \(x\).

20. If a cuboid has \(x\) vertices, \(y\) faces, and \(z\) edges, then the value of \((x + y - z)\) will be 2.

21. If \(x-2\) and \(x+3\) have \(x\) as their mean proportional, then the value of \(x\) is _____.

22. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

23. If \(x\) is the geometric mean between \((a^2bc)\) and \((4bc)\), then the value of \(x\) is ______.

24. If \(x\) is the mean proportional between \((x + 2)\) and \((x - 3)\), then the value of \(x\) is ________.

25. If the rationalizing factor of \(\sqrt{5}\) is \(\sqrt{x}\), then calculate and write the smallest possible value of \(x\), where \(x\) is a whole number.

26. If the median of the following data is 28.5 and the total frequency is 60, then find the values of \(x\) and \(y\).

27. If \(x \tan 30^\circ + y \cot 60^\circ = 0\) and \(2x - y \tan 45^\circ = 1\), then calculate and write the values of \(x\) and \(y\).

28. In the adjacent figure, if \(LM \parallel AB\), and \(AL = (x - 3)\) units, \(AC = 2x\) units, \(BM = (x - 2)\) units, and \(BC = (2x + 3)\) units, then find the value of \(x\).

29. In the adjacent figure, within triangle \(ABC\), if \(DE \parallel PQ \parallel BC\) and \(AD = 3\) cm, \(DP = x\) cm, \(PB = 4\) cm, \(AE = 4\) cm, \(EQ = 5\) cm, and \(QC = y\) cm, then determine the values of \(x\) and \(y\).

30. Given \(x \propto y\), and \(y = 8\) when \(x = 2\); then when \(y = 16\), find the value of \(x\).

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