1. Given that the combined (weighted) mean of the following frequency distribution is 50 and the total frequency is 120, find the values of \( f_1 \) and \( f_2 \). | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|--------|--------|--------|---------| | Frequency | 17 | \( f_1 \) | 32 | \( f_2 \) | 19 |

2. The expenses of a hostel are partly constant and partly directly proportional to the number of boarders. When the number of boarders is 120, the total expense is ₹2000, and when the number of boarders is 100, the expense is ₹1700. Calculate and write the number of boarders when the expense is ₹1880.

3. Let's calculate the difference between the upper boundary of the median class and the lower boundary of the modal class from the given frequency distribution table:

4. Here is the English translation: Values related to A and B are: | | A | B | |-----|-----|-----| | | 25 | 10 | | | 30 | 12 | | | 45 | 18 | | | 250 | 100 | Determine if there is any proportional relationship between A and B, and find the constant of proportionality if it exists.

5. Given that the combined mean of the following distribution is 50 and the total frequency is 120, find the values of \(f_1\) and \(f_2\): | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|--------|--------|--------|---------| | Frequency | 17 | \(f_1\) | 32 | \(f_2\) | 19 |

6. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

7. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

8. In a partnership business, the capital ratio of three individuals is 3:8:5, and the profit of the first person is ₹60 less than the profit of the third person. Find the total profit of the business.

9. A hostel's expenses are partly fixed and partly dependent on the number of residents. When the number of residents is 120, the total expense is ₹2000. When the number of residents is 100, the expense is ₹1700. What will be the number of residents if the total expense is ₹1880?

10. A and B started a business by investing ₹12,500 and ₹8,500 respectively. They agreed that 40% of the profit would be shared equally, and the remaining 60% would be divided in proportion to their capital investments. If A's profit is ₹1950, what is B's profit?

11. What is the third proportional of 4 and 18?

(a) 80 (b) 82 (c) 81 (d) 83

12. If a book is sold for ₹880 and the person incurs a 12% loss, at what price should it be sold to gain a 10% profit?

(a) 1050 (b) 1100 (c) 1000 (d) 1160

13. A man deposited a total of ₹29,000 in a bank in the names of his two daughters, aged 12 and 14, in such a way that when each turns 18, they would receive the same amount including principal and interest at a 10% rate of interest. Find how much money was deposited in the name of the elder daughter.

(a) 15000 (b) 16000 (c) 18000 (d) 20000

14. Determine the mode from the following frequency distribution:

15. In a joint business, the capital ratio of three individuals is 5:8:3, and the second person's profit is ₹100 more than the third person's. What is the total profit in the business?

16. If half of a principal is deposited at a 6% interest rate, one-fourth at an 8% interest rate, and the remaining part at a 12% interest rate in a bank, and the total interest received at the end of the year is ₹800, the principal amount is-

(a) ₹ 1000 (b) ₹ 10,000 (c) ₹ 8000 (d) ₹ 80,000

17. Gautam and Samaresh started a joint business with ₹12,500 and ₹8,500 respectively. They agreed that 40% of the profit would be equally distributed between them, and the remaining profit would be shared in proportion to their investments. If Gautam received ₹1,950 as his share of the profit, determine Samaresh’s profit amount.

18. In a partnership business, the capital ratio of three individuals is 3:8:5, and the profit of the first person is 70 rupees less than the profit of the third person. What was the total profit in the business?

19. The outer dimensions of a box with a lid are 12 cm in length, 10 cm in width, and 8 cm in height. The total inner surface area of the box is 376 square cm. If the walls of the box have equal thickness, what is their thickness?

20. In a partnership business, the capital ratio of three individuals is \(\cfrac{1}{4}:\cfrac{1}{3}:\cfrac{1}{8}\). If the second person's profit is ₹500 more than the third person's profit, then the total profit in the business was –?

21. From the given cumulative frequency distribution table, create a frequency distribution table and determine the mode of the data.

22. In a partnership business, the capital ratio of three individuals is 3:8:5, and the profit of the first person is 60 rupees less than the profit of the third person. What is the total profit of the business?

23. Find the modal value (mode) from the following frequency distribution:

24. From the following cumulative frequency distribution table, prepare a frequency distribution table and determine the mode of the data:

25. If ₹800 is deposited in a bank at an annual simple interest rate of 10%, and the total amount received (principal + interest) is ₹1200, then calculate and write for how long the money was deposited in the bank.

26. A wooden box with a 5 cm thick lid is made from wooden planks. Its weight is 115.5 kg, and when filled with rice, it weighs 880.5 kg. The internal length and breadth of the box are 12 decimeters and 8.5 decimeters respectively, and the weight of rice per cubic decimeter is 1.5 kg. **(1) Calculate and write the internal height of the box.** **(2) At a rate of ₹1.50 per square decimeter, calculate the total cost to paint the outer surface of the box (excluding the base).**

27. The third proportional of 8 and 12.

(a) 12 (b) 16 (c) 18 (d) 20

28. For the following data of handicraft scores awarded to 45 girl students in our village, calculate the **median** of the frequency distribution:

29. If the median of the following data is 28.5 and the total frequency is 60, then find the values of \(x\) and \(y\).

30.