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. 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 |

3. 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 |

4. 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 |

5. From the following cumulative frequency distribution table, construct a frequency distribution table and determine the **mode** of the data: | Class Boundary | Less than 10 | Less than 20 | Less than 30 | Less than 40 | Less than 50 | Less than 60 | Less than 70 | Less than 80 | |--------------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------| | Cumulative Frequency | 4 | 16 | 40 | 76 | 96 | 112 | 120 | 125 |

6. Calculate the average (mean) from the following frequency distribution table: | Class Interval | 45–54 | 55–64 | 65–74 | 75–84 | 85–94 | 95–104 | |----------------|-------|-------|-------|-------|-------|--------| | Frequency | 8 | 13 | 19 | 32 | 12 | 6 |

7. Find the median of the following frequency distribution: | Class Interval | 1–5 | 6–10 | 11–15 | 16–20 | 21–25 | 26–30 | 31–35 | |----------------|-----|------|--------|--------|--------|--------|--------| | Frequency | 2 | 3 | 6 | 7 | 5 | 4 | 3 |

8. From the following frequency distribution table of candidates' ages in an entrance examination, find the **mode**: | Age (in years) | 16–18 | 18–20 | 20–22 | 22–24 | 24–26 | |----------------|--------|--------|--------|--------|--------| | Number of Candidates | 45 | 75 | 38 | 22 | 20 |

9. From the following frequency distribution of candidates' ages in an entrance examination, find the median: | Class Interval | 1–5 | 6–10 | 11–15 | 16–20 | 21–25 | 26–30 | 31–35 | |----------------|-----|------|-------|-------|-------|-------|-------| | Frequency | 2 | 3 | 6 | 7 | 5 | 4 | 3 |

10. Find the mode of the following frequency distribution: | Class Interval | 50–59 | 60–69 | 70–79 | 80–89 | 90–99 | 100–109 | |----------------|--------|--------|--------|--------|--------|----------| | Frequency | 5 | 20 | 40 | 50 | 30 | 6 |

11. Find the median of the following frequency distribution: | Class Interval | 0–10 | 10–30 | 30–60 | 60–70 | 70–90 | |----------------|------|-------|-------|-------|-------| | Frequency | 15 | 25 | 30 | 4 | 10 |

12. If the median of the following data is 32 and the total frequency is 100, find the values of \(x\) and \(y\): | Class Interval | 0–10 | 10–20 | 20–30 | |----------------|------|--------|--------| | Frequency | 10 | \(x\) | 25 | | Class Interval | 30–40 | 40–50 | 50–60 | |----------------|--------|--------|--------| | Frequency | 30 | \(y\) | 10 |

13. Find the mode of the following frequency distribution: | Class Interval | 0–5 | 5–10 | 10–15 | |----------------|-----|------|--------| | Frequency | 5 | 12 | 18 | | Class Interval | 15–20 | 20–25 | 25–30 | 30–35 | |----------------|--------|--------|--------|--------| | Frequency | 28 | 17 | 12 | 8 |

14. Find the median from the following grouped frequency distribution: | Class Interval | Frequency | |----------------|-----------| | 0–10 | 4 | | 10–20 | 7 | | 20–30 | 10 | | 30–40 | 15 | | 40–50 | 10 | | 50–60 | 8 | | 60–70 | 5 |

15. Determine the mode of the following frequency distribution: | Class Interval | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 | |---------------|------|------|------|------|------|------|------| | Frequency | 5 | 12 | 18 | 28 | 17 | 12 | 8 |

16. Prepare a greater-than cumulative frequency table from the following frequency distribution and draw an ogive on graph paper. | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | |----------------|------|--------|--------|--------|--------|--------| | Frequency | 7 | 10 | 23 | 50 | 6 | 4 |

17. Find the median from the following frequency distribution: | Number Range (\(x_i\)) | 0–4 | 5–9 | 10–14 | 15–19 | 20–24 | 25–29 | 30–34 | 35–39 | |------------------------|-----|-----|-------|-------|-------|-------|-------|-------| | Frequency (\(f_i\)) | 4 | 5 | 7 | 8 | 7 | 5 | 6 | 3 |

18. If the median of the following data is 32 and \(x + y = 100\), determine the values of \(x\) and \(y\). | Class Interval | 0–10 | 10–20 | 20–30 | |----------------|------|-------|-------| | Frequency | 10 | x | 25 | | Class Interval | 30–40 | 40–50 | 50–60 | |----------------|--------|--------|--------| | Frequency | 30 | y | 10 | ---

19. PQRS is a cyclic trapezium. PQ is a diameter of the circle, and PO || SR. If \(\angle\)QRS = 110°, then the value of \(\angle\)QSR is -

(a) 20° (b) 25° (c) 30° (d) 40°

20. **English Translation:** Draw the **less than ogive** (cumulative frequency curve) for the following frequency distribution: | Marks Obtained | 50–60 | 60–70 | 70–80 | 80–90 | 90–100 | |----------------|-------|-------|-------|-------|--------| | Frequency | 4 | 8 | 12 | 6 | 10 |

21. From the following table, calculate the average marks of 52 students in Class 10 of a school using both the **direct method** and the **assumed mean method**: | Number of Students | 4 | 7 | 10 | 15 | 8 | 5 | 3 | |--------------------|---|---|----|----|---|---|---| | Marks |30 |33 | 35 | 40 |43 |45 |48 |

22. Calculate the average marks obtained by the students, given the following cumulative frequency distribution: | Marks | Number of Students | |--------------|--------------------| | Less than 10 | 6 | | Less than 20 | 10 | | Less than 30 | 18 | | Less than 40 | 30 | | Less than 50 | 46 |

23. Find the mode from the following grouped frequency distribution: | Class Interval | Frequency | |----------------|-----------| | 3–6 | 2 | | 6–9 | 6 | | 9–12 | 12 | | 12–15 | 24 | | 15–18 | 21 | | 18–21 | 12 | | 21–24 | 3 |

24. Let's find the mode from the following frequency distribution: | Class Interval | 0–5 | 5–10 | 10–15 | 15–20 | 20–25 | 25–30 | 30–35 | |----------------|-----|------|-------|-------|-------|-------|-------| | Frequency | 5 | 12 | 18 | 28 | 17 | 12 | 8 |

25. **Find the average (mean) of the following data:** | Class Interval | 5–14 | 15–24 | 25–34 | 35–44 | 45–54 | 55–64 | |----------------|------|-------|-------|-------|-------|-------| | Frequency | 3 | 6 | 18 | 20 | 10 | 3 |

26. Using the direct method, calculate the mean of the following data: | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | |----------------|------|-------|-------|-------|-------| | Frequency | 4 | 6 | 10 | 6 | 4 |

27. Translate: ABCD is a cyclic trapezium with AD || BC. If \(\angle ABC = 75^\circ\), then find the measure of \(\angle BCD\).

(a) \(105^o\) (b) \(75^o\) (c) \(45^o\) (d) \(90^o\)

28. Draw a greater cumulative frequency ogive using the following statistical distribution and determine the median from it: | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | |---------------------|------|--------|--------|--------|--------|--------| | Frequency | 8 | 12 | 21 | 30 | 22 | 7 |

29. Let's determine the mode from the frequency distribution below: | Class Interval | 45–54 | 55–64 | 65–74 | 75–84 | 85–94 | 95–104 | |--------------------|--------|--------|--------|--------|--------|---------| | Frequency | 4 | 6 | 40 | 76 | 96 | 112 |

30. Total number of students = 84, and the distribution of students’ marks is as follows: | Marks Obtained | 0–10 | 10–30 | 30–60 | 60–70 | 70–90 | |----------------|------|--------|--------|--------|--------| Median class interval’s class width.

(a) 10 (b) 20 (c) 30 (d) 40