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

2. 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

3. If \(\sum_{i=1}^n (x_i - 3) = 0\) and \(\sum_{i=1}^n (x_i + 3) = 66\), then find the values of \(\bar{x}\) (the mean) and \(n\).

4. If the average of the first four numbers out of five is 26, and the average of the last four numbers is 25, then find the difference between the first and the last number.

5. 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 \(K \ne 0\)).

6. If the combined average of the following frequency distribution table is 54, then find the value of \(k\).

7. If the combined average of the numbers in the following frequency distribution table is 24, find the value of \(p\).

8. 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) \(A=M\)

9. 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 \((n - 1)\bar{x} + K\).

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 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

12. 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

13. 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

14. 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

15. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

16. 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).

17. ABCD is a cyclic quadrilateral and O is the center of the circle. If ∠COD = 120° and ∠BAC = 30°, then find the values of ∠BOC and ∠BCD.

18. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

19. AOB is a diameter of a circle. C is a point on the circle. If ∠OBC = 60°, then find the value of ∠OCA.

20. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

21. If the surface area of a sphere is \(A\) and its volume is \(V\), then find the value of \(\cfrac{A^3}{V^2}\).

22. If the median of the following frequency distribution table is 27, then find the value of \(a\): | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | |----------------|------|--------|--------|--------|--------| | Frequency | 3 | a | 20 | 12 | 7 |

23. The volume of a right circular cone is \(V\) cubic units. If the area of the base is \(A\) square units and the height is \(H\) units, then find the value of \(\frac{AH}{V}\).

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. If the combined (weighted) mean of the following frequency distribution table is 54, then find the value of K: | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|-------|-------|-------|--------| | Frequency | 7 | 11 | K | 9 | 13 |

26. If \(n\) is an even number, then the median will be the average of the \(\left(\cfrac{n}{2}\right)\)-th and \(\left(\cfrac{n}{2}−1\right)\)-th observations.

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

28. Five times a positive integer is 3 less than twice the square of that integer. Form the required quadratic equation to find the integer, and then solve the equation to determine its value.

29. If the current price of a machine is ₹\(b\) and its value depreciates by \(r\%\) every year, then after \(n\) years, the value of the machine will be ₹\(b\left(1 - \cfrac{r}{100}\right)^n\).

30. If the monthly simple interest on ₹\(x\) at an annual rate of 5% is ₹1, then find the value of \(x\).

(a) ₹ 200 (b) ₹ 220 (c) ₹ 240 (d) ₹ 260