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

3. Write the value of \((x - y + z + p)\), where \(x\), \(y\), \(z\), and \(p\) represent the number of faces, edges, vertices, and diagonals of a cuboid respectively.

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

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

6. 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

7. 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

8. 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

9. 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)?

10. 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?

11. 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

12. 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

13. 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) \(\cfrac{\sqrt5}{\sqrt2}\)

14. 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 \(\frac{xy}{z}\)?

15. 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 \(\frac{YZ}{X}\)?

16. If a cuboid has \( x \) faces, \( y \) edges, \( z \) vertices, and \( p \) diagonals, determine the value of \( x+y+z+p \).

17. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

18. 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

19. If the average of the numbers 6, 7, \(x\), 8, \(y\), 14 is 9, then –

(a) x+y=21 (b) x+y=29 (c) x-y=21 (d) x+y=19

20. 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) \(\cfrac{180}{π}\)

21. 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) \(a:b\)

22. If \(\sum \limits_{i=1}^n (x_i - 7) = -8\) and \(\sum \limits_{i=1}^n (x_i + 3) = 72\), then what are the values of \(\bar{x}\) (the mean of \(x_i\)) and \(n\) (the number of terms)?

(a) \(\bar{x}=5, n=8\) (b) \(\bar{x}=6, n=8\) (c) \(\bar{x}=4, n=7\) (d) \(\bar{x}=8, n=6\)

23. If the mean of the numbers \(x_1, x_2, x_3, x_4, ..., x_n\) is \(\bar{x}\), then the value of \((x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + ... + (x_n - \bar{x})\) will be —

(a) 0 (b) 1 (c) 3 (d) 5

24. The numbers 11, 12, 14, \(x - 2\), \(x + 4\), \(x + 9\), 32, 38, 47 are arranged in ascending order, and their median is 24. Find the value of \(x\).

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

26. 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) \(\cfrac{1}{8}\)

27. If a rectangular prism has the number of faces \(= x\), the number of edges \(= y\), and the number of vertices \(= z\), then \(z - \cfrac{y}{x} =\) _____.

28. If a rectangular prism has \(d\) diagonals, \(v\) vertices, and \(e\) edges, then the value of \(v+d-e\) is ____.

29. The numbers 11, 12, 14, \(x-2\), \(x+4\), \(x+9\), 32, 38, 47 are arranged in ascending order, and their median is 24. Find the value of \(x\).

30. If the average of \(n\) natural numbers is \(\cfrac{n+8}{4}\), then what is the value of \(n\)?

(a) 4 (b) 6 (c) 8 (d) 0