1. 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 _____

2. If the combined average of the numbers 7, \(x - 3\), 10, \(x + 3\), and \(x - 5\) is 15, then the median is _____.

(a) 16 (b) 10 (c) 18 (d) 24

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

4. If the average of the numbers \(7, x - 3, x + 3, 10, x - 5\) is \(15\), then their median will be \(10\).

5. 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

6. 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 \(K \ne 0\)).

7. If the annual compound interest rate is \(r\%\), and the interest earned is _____, then the compound amount of \(p\) rupees in \(n\) years will be \(p\left(1+\cfrac{r}{400}\right)^{4n}\) rupees.

8. 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

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

10. 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

11. 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

12. If the current population of a village is \(p\) and the annual population growth rate is \(2r\%\), then the population after \(n\) years will be _____.

13. If the population of a village is \(p\) and the annual population growth rate is \(2r\%\), then after \(n\) years, the population will be _____.

14. 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

15. If the population of a village is \(p\) and the annual population growth rate is \(2r\%\), then the population after \(n\) years will be _____.

16. Got it — sticking strictly to translation. Here's the English version without any extra commentary: If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 5x^2 - 3x + 6 = 0 \), then \( \alpha + \beta = -\frac{-3}{5} = \frac{3}{5} \) and \( \alpha\beta = \frac{6}{5} \) \(\therefore \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} \) \( = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2} \) (Answer)

17. ABCD is a cyclic quadrilateral. If \(\angle ADB = x^\circ\) and \(\angle ABD = y^\circ\), then the measure of \(\angle BCD\) will be \((x + y)^\circ\).

18. If the average of the quantities \(x_1, x_2, x_3,.......,x_{10}\) is 20, then the average of \(x_1+4, x_2+4, x_3+4,...., x_{10}+4\) will be—?

(a) 20 (b) 24 (c) 40 (d) 10

19. 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 \(x\).

20. If the current population of a village is \(P\) and the annual population growth rate is \(2r\%\), then after \(n\) years the population will be ———.

21. If the population of a village is \(p\) and the annual growth rate is \(4r\%\), then after \(n\) years the population will be \[ p\left(1+\frac{r}{100}\right)^n \]

22. If the current population of a city is \(P\) and the annual population growth rate is \(r\%\), then after \(n\) years, the population of the city will be \(P\left(1-\frac{r}{100}\right)^n\).

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

24. If the population of a village is \(P\) and the annual population growth rate is \(4r\%\), then after \(n\) years, the population will be \(P\left(1-\cfrac{r}{25}\right)^n\).

25. If the population of a village is \(P\) and the annual growth rate of the population is \(2r\%\), then the population after \(n\) years will be ___.

26. 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 \(x\).

27. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

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. Translate and write: If by applying Sridharacharya's formula to the equation \(5x^2 + 2x - 7 = 0\), we get \(x = \cfrac{k \pm 12}{10}\), then what will be the value of \(k\)? Calculate and write it.

30. If \(x\) is the mean proportional between \((x + 2)\) and \((x - 3)\), then the value of \(x\) is ________.