1. The compound ratio of \(ab : c^2\), \(bc : a^2\), and \(ca : b^2\) is 1:1.

2. The compound ratio of \(ab:c^2\), \(bc:a^2\), and \(ca:b^2\) is 1:1.

3. The inverse ratio of the mixed ratio of \( ab:c^2, bc:a^2 \) and \( ca:b^2 \) is \( 1:1 \).

4. The compound ratio of \( ab:c^2 \), \( bc:a^2 \), and \( ca:b^2 \) is 1:1.

5. The compound ratio of \(ab:c^2\), \(bc:a^2\), and \(ca:b^2\) is \(1:1\).

6. The compound ratio of \(ab : c^2\), \(bc : a^2\), and \(ca : b^2\) is \(1 : 1\).

7. If 7, x, y, 189 are in continued proportion, then the values of x and y respectively will be:

(a) 63,21 (b) 21,23 (c) 21,63 (d) 23,21

8. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

9. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

10. ABCD is a cyclic quadrilateral where AB \(\parallel\) DC, and O is the center of the circle. Given: \(\angle\)BOC = 80°, and \(\angle\)ACO = 10° Find the measure of \(\angle\)BAD.

11. 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

12. In triangle ABC, points D and E lie on sides AB and AC respectively such that DE || BC and AD : DB = 3 : 1. If EA = 3.3 cm, find the length of AC.

13. The length, width, and height of a rectangular prism are in the ratio 5:3:1. If the volume of the prism is 120 cubic centimeters, determine the total surface area.

14. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.

15. If the height ratio of two right circular cones is 1:3 and the ratio of their radii is 3:1, then show that the ratio of their volumes is 3:1.

16. A rectangular box has a length, width, and height ratio of 3:2:1. If its volume is 384 cubic centimeters, find the total surface area of the box.

17. Three jars with equal diameter and height were partially filled with dilute sulfuric acid: - the first jar was filled to \(\frac{2}{3}\) of its height, - the second to \(\frac{5}{6}\), and - the third to \(\frac{7}{9}\). If the acid from these three jars is poured into a single jar with a diameter of 2.1 decimetres, and the height of the acid in this jar becomes 4.1 decimetres, then calculate and write the height of each of the original three jars, given that their diameter was 1.4 decimetres.

18. In the adjacent figure, O is the center of the circle. Given: \(\angle\)OAB = 40°, \(\angle\)ABC = 120°, \(\angle\)BCO = y°, and \(\angle\)COA = x° Find the values of x and y.

19. 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.

20. Here is the English translation: If \( y - z ∝ \frac{1}{x} \), \( z - x ∝ \frac{1}{y} \), and \( x - y ∝ \frac{1}{z} \), then the sum of the three proportionality constants.

(a) 0 (b) 1 (c) -1 (d) 2

21. The ratio of the total surface areas of a solid sphere and a solid hemisphere of the same radius is 2:1.

22. If a planet's gravitational force on its satellite is directly proportional to the planet's mass (M) and inversely proportional to the square of their distance (D), and the square of the satellite's orbital period (T) is directly proportional to the cube of the distance and inversely proportional to the gravitational force, then for two sets of values \(m_1, d_1, t_1\) and \(m_2, d_2, t_2\) corresponding to M, D, and T respectively, prove that: \[ m_1 t_1^2 d_2^3 = m_2 t_2^2 d_1^3 \]