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. 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

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. 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

6. 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)?

7. 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

8. 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

9. 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

10. 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.

11. Translate: If the average of \(p\) numbers is \(b\), and the average of \((p + q)\) numbers is \(a\), then the average of the \(q\) numbers is _____

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

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

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

15. \(a\) is a positive number, and in the proportion \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), what is the value of \(a\)?

(a) \(\cfrac{9}{11}\) (b) \(\cfrac{3}{4}\) (c) \(\cfrac{9}{20}\) (d) \(\cfrac{9}{16}\)

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

17. 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

18. If \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).

(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

19. 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?