1. If \(x : y = 2 : 3\), then what is the value of \((2x + 3y) : (2x + 5y)\)?

(a) 12:15 (b) 11:16 (c) 13:19 (d) 10:17

2. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?

3. If \(a : b = 2 : 1\) and \(x : y = 3 : 4\), then what is the value of \((2ax - by) : (2by - ax)\)?

(a) 2:1 (b) 3:1 (c) 4:1 (d) 5:1

4. If \((2x - 3y) : (5x - 2y) = 1 : 19\), then what is the value of \((2x + 3y) : (3x - 2y)\)?

(a) 9:19 (b) 16:9 (c) 9:16 (d) 19:9

5. If \(x:y = 3:4\), find the value of \((3y - x) : (2x + y)\).

6. If \( (3x - 2y) : (3x + 2y) = 4 : 5 \), then what is the value of \(x : y\)?

(a) 1:6 (b) 1:1 (c) 2:1 (d) 6:1

7. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \]

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

8. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

9. If \((4x - 3y) : (2x - y) = 2 : 3\), then what is the value of \(x : y\)?

(a) 15:14 (b) 8:7 (c) 7:8 (d) 14:15

10. If \(x + y = z\) and \(2x - z = y\), then what is the value of \(x : y : z\)?

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

11. If \((3x - 2y) : (3x + 2y) = 4 : 5\), then what is the value of \((x + y) : (x - y)\)?

12. Here’s the English translation of your math problem: If the ascending data set is \[ 27,\ 31,\ 46,\ 52,\ x,\ y + 2,\ 71,\ 79,\ 85,\ 90 \] and the median of the set is \(64\), then what is the value of \(x + y\)?

(a) 125 (b) 126 (c) 127 (d) 128

13. If \(\cfrac{x-3y}{2y}=\cfrac{6x-5y}{5x}\), then what is the value of \(x:y\)?

(a) 5:1 or 2:5 (b) 1:5 or 2:5 (c) 1:5 or 5:2 (d) None of the above

14. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:

(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

15. 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.

16. If x : y = 3 : 4, then determine the value of (3y − x) : (2x + y).

17. If \((5x - 3y) : (2x + 4y) = 11 : 12\), then determine the value of \(x : y\).

18. If \( (7a - 5b) : (3a + 4b) = 7 : 11 \), then what is the value of \( (5a - 3b) : (6a + 5b) \)?

(a) 777:244 (b) 777:247 (c) 247:778 (d) 247:787

19. Here’s the English translation of your question: **"If \( x = \frac{4ab}{a + b} \), then what is the value of \( \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \)?"** Would you like me to solve it step by step?

(a) 1 (b) -2 (c) 2 (d) -1

20. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of \((3ax - by) : (4by - 7ax)\).

21. The modal class of the above frequency distribution is 15–20. So, the mode is calculated as: \[ \text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \] Where: - \(l = 15\) (lower boundary of modal class) - \(f_1 = 28\) (frequency of modal class) - \(f_0 = 18\) (frequency of class before modal class) - \(f_2 = 17\) (frequency of class after modal class) - \(h = 5\) (class width) Substituting the values: \[ = 15 + \left(\frac{28 - 18}{2 \times 28 - 18 - 17}\right) \times 5 = 15 + \frac{10}{21} \times 5 = 15 + \frac{50}{21} = 15 + 2.38 = 17.38 \quad \text{(approx)} \] ✅ Therefore, the mode is approximately **17.38**.

22. If the compound ratio of \(2 : 3\), \(3 : 5\), and \(x : 8\) is \(1 : 4\), then what is the value of \(x\)?

(a) 3 (b) 4 (c) 5 (d) 6

23. If \((5x - 3y) : (2x + 4y) = 11:12\), determine the value of \(x:y\).

24. 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.

25. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of the ratio \((3ax - by) : (4by - 7ax)\).

26. If \((3x - 2y) : (x + 3y) = 5 : 6\), then find the value of \((2x - 5y) : (3x + 4y)\).

27. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

28. 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 |

29. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)?

(a) 1 (b) 2 (c) 3 (d) 4

30. If \(4x = 5y = 6z\), what is the value of the ratio \(x : y : z\)?

(a) 12:10:15 (b) 10:12:15 (c) 15:12:10 (d) 15:10:12