1. 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 |

2. 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 |

3. 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 |

4. 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 | ---

5. 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 |

6. 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 |

7. 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 |

8. In a circle centered at O, AC is the diameter, and DC || EB. Given that \(\angle\)AOB = 80° and \(\angle\)ACE = 10°, determine the measure of \(\angle\)BED.

9. In the trapezium ABCD, BC || AD, and AD = 4 cm. The diagonals AC and BD intersect at a point such that \(\frac{AO}{OC}=\frac{DO}{OD}\). Find the length of BC.

10. 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°

11. In the cyclic quadrilateral ABCD, AB is the diameter and AB || DC. If \(\angle\)ACD = 25°, determine the value of \(\angle\)CAD.

12. If \( BC || AD \), \( \angle CDE = 80^\circ \), and \( \angle CBD = 30^\circ \), determine the values of \( \angle ABD \), \( \angle ACD \), and \( \angle BAD \).

13. In a circle centered at O, if the lengths of chords PQ and RS are in the ratio 1:1, then the ratio of \(\angle\)POQ to \(\angle\)ROS is _____ |

14. In trapezium ABCD, BC || AD and AD = 4 cm. Diagonals AC and BD intersect at point O such that \(\frac{AO}{OC} = \frac{DO}{OB} = \frac{1}{2}\). Write the length of BC.

15. 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) \(90^o\)

16. In the adjacent figure, if DE || BC, BE || XC, and \(\frac{AD}{DB} = \frac{2}{1}\), then find the value of \(\frac{AX}{XB}\).

17. In the given figure, LM || AB and AL = (x - 3) units, AC = 2x units, BM = (x - 2) units, and BC = (2x + 3) units. Determine the value of \(x\).

18. If \( DE || BC \) and \( BD = (x - 3) \) cm, \( AB = 2x \) cm, \( CE = (x - 2) \) cm, and \( AC = (2x + 3) \) cm, determine the value of \( x \).

19. In the adjacent figure, DE || BC in triangle \( \triangle ABC \). If \( AD = 5 \) cm, \( DB = 6 \) cm, and \( AE = 7.5 \) cm, then calculate and write the length of \( AC \).

20. x and y are two variables, and their corresponding values are: | x | 18 | 8 | 12 | 6 | |----|-----|-----|-----|-----| | y | 3 | \(\frac{27}{4}\) | \(\frac{9}{2}\) | 9 | Determine whether there is any proportional relationship between x and y.

21. 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 |

22. 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 |

23. 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 |

24. > In a medical entrance examination, the frequency distribution of the marks obtained by 200 candidates is given below: > > | Marks Obtained | 400–450 | 450–500 | 500–550 | 550–600 | 600–650 | 650–700 | 700–750 | 750–800 | > |----------------|---------|---------|---------|---------|---------|---------|---------|---------| > | No. of Students| 20 | 30 | 28 | 26 | 24 | 22 | 18 | 32 | > > Draw an ogive curve and determine the median using it.

25. 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 |

26. 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 |

27. 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 |

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. 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 |

30. 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 |