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

2. 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

3. If \(A : B = 2 : 3\), \(B : C = 4 : 5\), and \(C : D = 6 : 7\), then find the value of \(A : D\).

4. \(x\) and \(y\) vary directly with the square of each other, and \(z\) varies inversely with the cube root. Given: \(y = 8\), \(z = 8\), \(x = 16\) When \(x = 24\), \(z = 27\), find the value of \(y =\) ?

(a) \(\pm{16}\) (b) \(\pm{14}\) (c) \(\pm{12}\) (d) \(\pm{10}\)

5. 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

6. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\).

(a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

7. ABCD is a cyclic quadrilateral and O is the center of the circle. Given: \(\angle COD = 130^\circ\), \(\angle BAC = 25^\circ\) Find the values of \(\angle BOC\) and \(\angle BCD\).

(a) 40\(^o\),90\(^o\) (b) 50\(^o\),90\(^o\) (c) 65\(^o\),50\(^o\) (d) None of the above

8. If \(a : b = 2 : 3\) and \(b : c = 4 : 5\), then what is the value of \(a^2 : b^2 : bc\)?

(a) 8:18:21 (b) 16:36:45 (c) 16:20:36 (d) 8:15:18

9. Find the value of \(x\):
\(x \tan^2 30° + 2x \sec^2 45° + 2x \csc^2 60° = 4\)

(a) \(\cfrac{7}{4}\) (b) \(\cfrac{4}{7}\) (c) \(\cfrac{3}{7}\) (d) \(\cfrac{1}{4}\)

10. 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.

11. 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.

12. If \(A : B = 6 : 7\) and \(B : C = 8 : 7\), then find the value of \(A : C\).

13. If \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).

(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

14. If \(a : b = 3 : 2\) and \(b : c = 3 : 2\), then find the value of the ratio \((a + b) : (b + c)\).

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

16. If the combined mean of the following data is 20.6, then find the value of \(a\):

17. In trapezium \(ABCD\), where \(AB ∥ DC\), points \(P\) and \(Q\) lie on sides \(AD\) and \(BC\) respectively such that \(PQ ∥ DC\). Given: \(PD = 18\) cm, \(BQ = 35\) cm, and \(QC = 15\) cm Find the length of side \(AD\).

(a) 60 cm (b) 30 cm (c) 12 cm (d) 15 cm

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

19. In the cyclic quadrilateral ABCD: \(\angle A = 4x^\circ\), \(\angle B = 7x^\circ\), \(\angle C = 5y^\circ\), and \(\angle D = y^\circ\) Find the value of the ratio \(x : y\).

20. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.

(a) undefined (b) positive (c) negative (d) zero

21. 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

22. Two concentric circles have radii of 13 cm and 15 cm, respectively. A chord AB of the larger circle intersects the smaller circle at points P and Q. If PQ = 10 cm, then AB will be:

(a) 28 cm (b) 20 cm (c) 18 cm (d) 16 cm

23. In triangles \(ABC\) and \(DEF\), if \(\angle A = \angle F = 40^\circ\), \(AB:ED = AC:EF\), and \(\angle F = 65^\circ\), find the value of \(AB\).

(a) 35° (b) 65° (c) 75° (d) 85°

24. 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

25. 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

26. The base length and width of a rectangular reservoir are 15 meters and 12 meters respectively. Water is filled into the reservoir from a nearby pond using a pump. If the pump can fill 36,000 liters of water per hour, then how long will it take for the pump to fill the reservoir up to a height of 7.2 decimeters? [Note: 1 liter = 1 cubic decimeter]

27. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

28. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

29. 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 |

30. A three-storey building has a flagpole of 3.6 meters mounted on its roof. From a point on the street, the angles of elevation to the top and bottom of the flagpole are 50° and 45°, respectively. Find the height of the building. [Assume: \(\tan 50^\circ = 1.2\)]