1. 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

2. 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

3. 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

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

5. 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.

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

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

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

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

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

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

16. If the number of faces of a right-angled quadrilateral is \( x \) and the number of diagonals is \( y \), determine the value of \( (x^2 - y^2) \).

(a) 32 (b) 48 (c) 2 (d) 20

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 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

19. 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) \(\cfrac{180}{π}\)

20. 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) \(a:b\)

21. 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) \(\bar{x}=8, n=6\)

22. 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 \(x\).

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

24. If \(x + y = z\) and \(2x - z = y\), then what is the value of \(x : y : z\)?

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

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

26. If a right circular cone has a volume of \(x\) cubic units and a base area of \(y\) square units, then its height is _____.

27. 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 \(x\).

28. Three friends—A, B, and C—started a business by investing \(x\), \(2x\), and \(y\) taka respectively. If the profit at the end of the term is \(z\) taka, then what will be A’s share of the profit?

(a) ₹ \(\cfrac{xz}{3x+y}\) (b) ₹ \(\cfrac{2xz}{3x+y}\) (c) ₹ \(\cfrac{z}{2x+y}\) (d) ₹\(\cfrac{xyz}{3x+y}\)

29. 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.

30. If the volume of a cone is \(x\), the base area is \(y\), and the height is \(z\), then the value of \(\cfrac{x}{yz}\) will be 3.