1. Problem 7: If the perimeter of a circle is equal to that of a square, find the ratio of their areas. 2. Formula: - Perimeter of circle (circumference) = $2\pi r$ - Perimeter of square = $4a$ - Area of circle = $\pi r^2$ - Area of square = $a^2$ 3. Since perimeters are equal, $2\pi r = 4a \Rightarrow a = \frac{\pi r}{2}$. 4. Ratio of areas = $\frac{\text{Area of circle}}{\text{Area of square}} = \frac{\pi r^2}{a^2} = \frac{\pi r^2}{(\frac{\pi r}{2})^2} = \frac{\pi r^2}{\frac{\pi^2 r^2}{4}} = \frac{4}{\pi}$. 5. Using $\pi = \frac{22}{7}$, ratio = $\frac{4}{\frac{22}{7}} = \frac{4 \times 7}{22} = \frac{28}{22} = \frac{14}{11}$. 6. Final answer for problem 7 is option B: 14:11. --- 7. Problem 8: Find the radius of a circle whose circumference equals the sum of circumferences of two circles with diameters 36 cm and 20 cm. 8. Circumference formula: $C = 2\pi r$ 9. Radii: $r_1 = \frac{36}{2} = 18$ cm, $r_2 = \frac{20}{2} = 10$ cm. 10. Sum of circumferences = $2\pi r_1 + 2\pi r_2 = 2\pi (r_1 + r_2) = 2\pi (18 + 10) = 2\pi \times 28$. 11. Let radius of new circle be $R$, then circumference $= 2\pi R = 2\pi \times 28$. 12. So, $R = 28$ cm. 13. Final answer for problem 8 is option C: 28 cm. --- 14. Problem 9: Diameter of wheel is 1.26 m, find distance covered in 500 revolutions. 15. Circumference of wheel = $\pi d = \pi \times 1.26$ m. 16. Distance covered in 500 revolutions = $500 \times \pi \times 1.26$ m. 17. Calculate: $500 \times 3.1416 \times 1.26 = 1978.4$ m = 1.9784 km. 18. Final answer for problem 9 is option D: 1.98 km. --- 19. Problem 10: In a circle with center O, area of sector OAPB is $\frac{5}{36}$ times area of circle. Find angle $x$. 20. Area of sector = $\frac{x}{360} \times \pi r^2$. 21. Given $\frac{x}{360} \pi r^2 = \frac{5}{36} \pi r^2$. 22. Cancel $\pi r^2$: $\frac{x}{360} = \frac{5}{36}$. 23. Solve for $x$: $x = 360 \times \frac{5}{36} = 50$ degrees. 24. Final answer for problem 10 is option A: 50°. --- 25. Problem 11: Assertion and Reason about area of circular path. 26. Outer radius $R = \frac{10}{2} = 5$ m, inner radius $r = \frac{6}{2} = 3$ m. 27. Area of path = $\pi (R^2 - r^2) = \pi (25 - 9) = 16\pi$ m². 28. Assertion is true. 29. Reason states area = $\pi (R_1^2 + R_2^2)$ which is incorrect; it should be difference. 30. Reason is false. 31. Final answer for problem 11 is option C: Assertion true, Reason false. --- 32. Problem 12: Wire length 22 cm bent into circle, find area. 33. Circumference = length = 22 cm. 34. Circumference formula: $2\pi r = 22 \Rightarrow r = \frac{22}{2\pi} = \frac{11}{\pi}$ cm. 35. Area = $\pi r^2 = \pi \left(\frac{11}{\pi}\right)^2 = \pi \times \frac{121}{\pi^2} = \frac{121}{\pi} \approx 38.5$ cm². 36. Given area is 40 cm², close but not exact. 37. Assertion is false. 38. Reason is true. 39. Final answer for problem 12 is option D: Assertion false, Reason true.