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. 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

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

4. 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

5. 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

6. 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.

7. 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

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

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

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

11. 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) \(-r^2\)

12. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

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

14. 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

15. If \(\sum(x_i - 3) = 0\) and \(\sum(x_i + 3) = 66\), then find the values of \(\bar{x}\) (mean) and \(n\) (number of observations).

16. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

17. If \(x = \cfrac{1}{2 + \sqrt{3}}\) and \(y = \cfrac{1}{2 - \sqrt{3}}\), then what is the value of \(\cfrac{1}{1 + x} + \cfrac{1}{1 + y}\)?

18. In triangle XYZ, ∠Y is a right angle. Given: XY = \(2\sqrt{6}\) and XZ − YZ = 2 Find the value of sec X + tan X.

19. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

20. If \(x^2 - px + 12 = (x - 3)(x - a)\) is an identity, then the values of \(a\) and \(p\) respectively are?

(a) \(a = 4, p = 7\) (b) \(a = 7, p = 4\) (c) \(a = 4, p =-7\) (d) \(a =-4, p = 7\)

21. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then what is the value of \(\frac{x^2 + y^2 + xy}{x^2 + y^2 - xy}\)?

(a) \(\cfrac{6}{7}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{11}\) (d) None of the above

22. If \(x = \cfrac{\sqrt3 - \sqrt2}{\sqrt3 + \sqrt2}\) and \(xy = 1\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

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

24.

25. If \(x + y = \sqrt{7}\) and \(xy = 1\), then what is the value of \(x - y\)?

26. If two circles touch each other externally at point A, and the number of direct common tangents is \(= x\), and the number of transverse common tangents is \(= y\), then \(x - y =-----\).

27. If \(x = 3 + 2\sqrt{2}\) and \(xy = 1\), then what is the value of \(\cfrac{x^2 + y^2}{xy}\)?

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

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

29. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is —

(a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

30. If ???? = 3 + 2 and ???? = 3 − 2 , then what is the value of 8 ???? ???? ( ???? 2 + ???? 2 ) ?

(a) 24 (b) 80 (c) 16 (d) 8