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 consecutive positive odd numbers is 650. Find the numbers

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

5. The sum of the squares of two consecutive numbers is 313 — Form a quadratic equation with one variable from the given statement.

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

7. The product of two consecutive even numbers is 168. Find the two numbers.

8. If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya's formula (the quadratic formula).

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

10. If the sum of two numbers is three times their difference, find the ratio between the two numbers.

11. 25 is expressed as the sum of two numbers whose sum of reciprocals is \(\cfrac{1}{6}\). Determine the two numbers.

12. The product of two consecutive positive odd numbers is 143 — Form a quadratic equation with one variable from the given statement.

13. The product of two consecutive positive odd numbers is 143.

14. Form an equation with the product of two consecutive positive odd numbers equal to 143, and use Śrīdharāchārya’s formula (the quadratic formula) to find those two odd numbers.

15. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.

16. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

17. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

18. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

19. Prove that if the area of a square drawn on one side of any triangle is equal to the sum of the areas of squares drawn on the other two sides, then the angle opposite the first side is a right angle.

20. Prove that the area of the square drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares drawn on the other two sides.

21. The area of the square drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares drawn on the other two sides.