1. The difference between two positive integers is 3, and the sum of their squares is 117. Find the two numbers.

(a) 6,9 (b) 3,6 (c) 4,9 (d) 5,6

2. The difference between two positive integers is 3, and the sum of their squares is 117. Determine the two numbers.

3. The sum of the squares of two numbers is 29, and their product is 10. Find the two numbers.

4. The distance between two pillars is 150 meters. One pillar's height is three times the other. The angles of elevation to their tops from the midpoint of the line joining their bases are complementary. Find the height of the smaller pillar.

5. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.

6. The sum of the squares of two consecutive positive even numbers is 340. Find the two numbers.

7. The difference between two given angles is 40°, and their sum is \(\frac{\pi}{2}\) radians. Find the radian measures of the two angles.

8. The distance between two pillars is 150 meters. One is three times as tall as the other. From the midpoint of the line connecting the bases of the two pillars, the angles of elevation to their tops are complementary. What is the height of the shorter pillar?

9. Translate: Two positive integers differ by 3 and the sum of their squares is 117; find the numbers and write them.

10. If the volumes of two cones are in the ratio 1:4 and the ratio of the diameters of their bases is 4:5, find the ratio of their heights

(a) 1:5 (b) 5:4 (c) 5:16 (d) 25:64

11. If the ratio of the areas of two similar triangles is 64:49, then find the ratio of their corresponding sides.

12. Two identical circles, each with a radius of 10 cm, intersect each other, and the length of their common chord is 12 cm. Find the distance between the centers of the two circles.

13. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent between the two circles.

14. If the ratio of the curved surface areas of two spheres is 1 : 4, find the ratio of their volumes.

15. Three friends purchase a bus by investing ₹1,20,000, ₹1,50,000, and ₹1,10,000 respectively. The first friend works as the driver, while the other two work as conductors. They decide that \(\frac{2}{5}\) of the income will be distributed based on work in the ratio 3 : 2 : 2, and the remaining amount will be divided according to their capital investment. If the total income in a certain month is ₹29,260, determine how much each person will receive.

16. If the product of two consecutive odd numbers is 783, find the two numbers.

17. The volume of a cone is jointly proportional to the square of the radius of the base and its height. If the ratio of the radii of two cones is 2:3 and their heights are in the ratio 5:4, then find the ratio of their volumes.

18. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent to the two circles.

19. If two arms are externally tangent to each other, the distance between their centers will be the sum of their radii.

20. 25 is expressed as the sum of two numbers whose sum of reciprocals is \(\cfrac{1}{6}\). Determine the two numbers.

21. If the volume ratio of two solid spheres is 1:8, find the ratio of their curved surface areas.

(a) 1:2 (b) 1:4 (c) 1:8 (d) 1:16

22. The radii of two circles are 8 cm and 3 cm, respectively, and the distance between their centers is 13 cm. Find the length of a common external tangent of the circles.

(a) 10 cm (b) 14 cm (c) 15 cm (d) 12 cm

23. If the sum of two angles is 135° and their difference is \(\frac{\pi^c}{12}\), find their sexagesimal and circular measures.

24. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), find the values of the two angles in degrees and radians.

25. If the ratio of the heights of two right circular cylinders is 1:2, and the ratio of the perimeters (circumferences) of their bases is 3:4, then find the ratio of their volumes.

26. Two equal circles, each with a radius of 10 cm, intersect each other. The length of their common chord is 12 cm. Find the distance between their centers.

27. The ratio of the radii of two right circular solid cylinders is 2:3, and the ratio of their heights is 5:3. Find the ratio of their curved surface areas.

28. If the volumes of two right circular solid cylinders are equal and the ratio of their heights is 1:2, then find the ratio of their radii.

(a) 1:\(\sqrt2\) (b) \(\sqrt2\):1 (c) 1:2 (d) 2:1

29. The radii of two circles are 8 cm and 3 cm respectively, and the distance between their centers is 1.3 cm. Find the length of a direct common tangent between the two circles.

30. Under the condition that no interest will be charged if the money is repaid within two years, three friends borrowed ₹6000, ₹8000, and ₹5000 respectively from a cooperative bank to jointly purchase four cycle rickshaws. After two years, after deducting all expenses, their total income amounted to ₹30,400. They divided this income in proportion to their capital contributions and then repaid their individual bank loans. Now, calculate how much money each person will have left in hand and what the ratio of the remaining amounts will be.