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

(a) 4 (b) 6 (c) 8 (d) 10

2. If the middle number of the first \((2n + 1)\) consecutive natural numbers is \(\frac{n + 103}{3}\), then find the value of \(n\).

3. If \(m = \sqrt{\cfrac{n}{n + \cfrac{1}{2}}}\) and \(m = \cfrac{1}{2}\), then what is the value of \(n\)?

4. If the average of the \(n\) numbers \(x_1, x_2, x_3, ..., x_n\) is \(\bar{x}\), then the average of \(Kx_1, Kx_2, Kx_3, ..., Kx_n\) is —— (where \(K \ne 0\)).

5. 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

6. If the average of the first ten natural numbers is A and the median is M, then the relationship between A and M is —

(a) \(A\gt M\) (b) \(A\lt M\) (c) \(A=\cfrac{1}{M}\) (d) \(A=M\)

7. In a group of \(n\) numbers with an average of \(\bar{x}\), if the sum of the first \((n - 1)\) numbers is \(K\), then the \(n\)th number will be \((n - 1)\bar{x} + K\).

8. If the combined average of the data set 7, 5, 2, 10, x, 6, 4, y, 8, 1 is 5, then what is the value of \(x + y\)?

9. If the following data has a combined (weighted) average of 20.6, then what is the value of \(a\)?

10. If the average of the numbers 6, 7, x, 8, y, and 14 is 9, then

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

11. If sinA+sinB=2, then what is the value of cosA-cosB

(a) 0 (b) 1 (c) 2 (d) 3

12. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

(a) 3 (b) -3 (c) \(\cfrac{1}{3}\) (d) -\(\cfrac{1}{3}\)

13. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

14. If the average of \( x_1, x_2, x_3, \dots, x_n \) is \( \bar{x} \), then what is the average of \( ax_1, ax_2, ax_3, \dots, ax_n \) ?

(a) \(\bar{x}\) (b) \(a\bar{x}\) (c) \(n\bar{x}\) (d) None of these

15. 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

16. 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

17. If the average of the numbers 6, 7, x, 8, y, and 16 is 9, then—?

(a) x + y = 21 (b) x + y = 17 (c) x - y = 21 (d) x - y = 19

18. 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

19. 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

20. If sin 51° = \(\cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of tan 51° + tan 39°?

(a) \(\cfrac{a^2-b^2}{ab}\) (b) \(\cfrac{a^2+b^2}{2ab}\) (c) \(\cfrac{a^2+b^2}{ab}\) (d) \(\cfrac{a^2-b^2}{2ab}\)

21. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?

(a) 1 (b) 1 (c) 0 (d) None of the above

22. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)?

(a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1

23. If \(a + b + c = 0\), then what is the value of \(\cfrac{a^3 + b^3 + c^3}{abc} - 3\)?

(a) 1 (b) 0 (c) -1 (d) None of the above

24. If \(x = \cfrac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}}\), then what is the value of \(bx^2 - ax + b\)?

(a) 0 (b) 1 (c) 2 (d) 3

25. 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

26. If \(x + \cfrac{1}{x} = -2\), then what is the value of \(x^7 + \cfrac{1}{x^6}\)?

(a) -1 (b) 1 (c) 2 (d) None of the above

27. If \(2\sin 2\theta - \sqrt{3} = 0\), then what is the value of \(\csc \theta\)?

(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{2}{\sqrt3}\) (d) 2

28. If PQ is a tangent to a circle centered at O and touches the circle at point C, then what is the value of \(\angle\)BCP?

(a) 90\(^o\) (b) 80\(^o\) (c) 70\(^o\) (d) None of the above

29. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

30. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)?

(a) 1 (b) 2 (c) 3 (d) 4