1. If \(a \propto \frac{1}{b^2}\), then the correct statement is: (a) \(b \propto \sqrt a\) (b) \(b \propto \cfrac {1}{\sqrt a}\) (c) \(b \propto \cfrac {1}{-\sqrt a}\) (d) \(b \propto \cfrac {1}{\pm \sqrt a}\)

2. If \(\frac{x}{y} \propto (x + y)\) and \(\frac{y}{x} \propto (x - y)\), then find the value of \(x^2 - y^2\). (a) "Proportional to \(x\)" (b) "Proportional to \(y\)" (c) "Proportional to \(xy\)" (d) constant

3. If \((x^2 + y^2) \propto (x^2 - y^2)\), then which of the following is correct? (a) \(x\propto y\) (b) \(xy=\) constant (c) \(x^2\propto y\) (d) \(y^2\propto x\)

4. If \( x \propto y^n\), then the correct relationship will be _____. (a) \( x^n \propto y^n\) (b) \( x^{2n} \propto y^n\) (c) \( x^{2n} \propto y^{2n}\) (d) \( x^{n} \propto y^{2n}\)

5. If \(x ∝ \cfrac{1}{y}\) and \(y ∝ \cfrac{1}{z}\), then – (a) \(x ∝ z\) (b) \(x∝ \cfrac{1}{z}\) (c) \(x∝z^2\) (d) \(x∝√z\)

6. If \( x∝y\), then—? (a) \(x^2∝y^3\) (b) \(x^3∝y^2\) (c) \(x∝y^2\) (d) \(x^2∝y^2\)

7. A is in joint variation with B and C². If A = 144 when B = 4 and C = 3, then what is the value of the constant of variation? (a) \(\frac{1}{4}\) (b) \(\frac{1}{2}\) (c) \(\frac{1}{3}\) (d) \(\frac{1}{5}\)

8. If \((x + y) ∝ (x - y)\), then \((x^2 + y^2) ∝ xy\). True / False

9. If \(a\) and \(b\) vary directly, then \(\cfrac{a}{b}\) will be constant. True / False

10. If \( x \propto yz \) and \( y \propto zx \), then show that \( z \ne 0 \) is a constant.

11. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2\) is constant.

12. If \(\cfrac{x}{y} \propto (x + y)\) and \(\cfrac{y}{x} \propto (x - y)\), then show that \((x^2 - y^2)\) is constant.

13. The volume of a sphere is directly proportional to the cube of its radius. If three solid spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form a new solid sphere, and there is no loss in volume during melting, then find the radius of the new sphere.

14. If \(a \propto b\), \(b \propto \frac{1}{c}\), and \(c \propto d\), then \(a \propto \frac{1}{d}\).

15. If \((\sqrt{a} + \sqrt{b}) \propto (\sqrt{a} - \sqrt{b})\), then show that \((a + b) \propto \sqrt{ab}\).

16. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

17. If \((a + b) \propto \sqrt{ab}\), then prove that \((\sqrt{a} + \sqrt{b}) \propto (\sqrt{a} - \sqrt{b})\).

18. \(x \propto \sqrt{y}\) and \(y = a^2\), if \(x = 2a\), then find the value of \(x^2 : y\).

19. Given that \(a \propto b\) and \(b \propto c\), show that \(a^3 + b^3 + c^3 \propto 5abc\).

20. If \(x ∝ y\) and \(y ∝ z\), then prove that \((x^2 + y^2 + z^2) ∝ (xy + yz + zx)\).

21. The volume of a sphere is directly proportional to the cube of its radius. A lead sphere has a radius of 14 cm. Using the concept of proportion, prove that this sphere can be melted to form four spheres, each with a radius of 7 cm. (Assume that the volume remains constant during melting.)

22. \(A \propto B\); when \(A = 2\), then \(B = 14\). Find the value of \(B\) when \(A = 5\).

23. It took the villagers 18 days to dig a pond. If the pond is to be dug in 15 days, how many additional people need to be employed? Use the concept of inverse proportion to calculate.

24. 15 farmers can cultivate 18 bighas of land in 5 days. Using the concept of inverse variation, find how many days 10 farmers will take to cultivate 12 bighas of land.

25. If the radius of a sphere is \(r\) and its volume is \(v\), then \(v \propto\) _____.

26. If \(x ∝ y^2\) and \(y = 2a , x = a\), then show that \(y^2 = 4ax\).

27. If \((u^2 + v^2) \propto (x^2 + y^2)\) and\(uv \propto xy\), then show that\((u + v) \propto (x + y)\) when \(\frac{u}{v} + \frac{v}{u} = \frac{x}{y} + \frac{y}{x}\)."

28. If \(x + y \propto x - y\), then show that \(x^3 + y^3 \propto x^3 - y^3\).

29. If \(\left(\frac{1}{x} - \frac{1}{y}\right) \propto \frac{1}{x - y}\), then show that \((x^2 + y^2) \propto xy\).

30. A hostel's expenses are partly fixed and partly dependent on the number of residents. When the number of residents is 120, the total expense is ₹2000. When the number of residents is 100, the expense is ₹1700. What will be the number of residents if the total expense is ₹1880?

31. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2 =\) constant.

32. If \(a ∝ \frac{1}{c}\) and \(c ∝ \frac{1}{b}\), then determine the relationship between \(a\) and \(b\).

33. If \(b ∝ a^2\) and \(a\) increases in the ratio \(2:3\), then determine the ratio in which \(b\) increases.

34. If \(\cfrac{x}{y} \propto (x+y)\) and \(\cfrac{y}{x} \propto (x-y)\), then show that \(x^2 - y^2\) is a constant.

35. If \( \cfrac{1}{y} - \cfrac{1}{x} \propto \cfrac{1}{x - y} \), then show that \( x \propto y \).

36. If \(2x^2 + 3y^2 \propto xy\), then show that \(x \propto y\).

37. If \(b\propto a^3\) and \(a\) increases in the ratio \(2:3\), determine the ratio in which \(b\) will increase.

38. If \(x \propto y\) and \(y \propto z\), then show that \[ \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]

39. Given: \(a \propto b\) and \(b \propto c\) Show that: \(a^3 + b^3 + c^3 \propto abc\)

40. If \(x ∝ y\), \(y ∝ z\), and \(z ∝ x\), then find the product of the three constants of proportionality.

41. If \(a \propto b\), \(b \propto \cfrac{1}{c}\), and \(c \propto d\), then what will be the proportional relationship between \(a\) and \(d\)?

42. If \(x ∝ y\) and \(x ∝ z\), then show that \(x ∝ (y - z)\).

43. If \(a ∝ b\) and \(b ∝ c\), then prove that \(a^3 + b^3 + c^3 ∝ 3abc\).

44. If \( a \propto b \), \( b \propto \cfrac{1}{c} \), and \( c \propto d \), determine the relation between \( a \) and \( d \).

45. 15 farmers can cultivate 18 bighas of land in 5 days. Using the theory of proportion, determine how many days 10 farmers will take to cultivate 12 bighas of land.

46. If \( \cfrac{1}{y} - \cfrac{1}{x} \propto \cfrac{1}{x - y} \), prove that \( x \propto y \).

47. If \( x ∝ y \) and \( y ∝ z \), prove that \[ x^2 + y^2 + z^2 ∝ xy + yz + zx \]

48. If \(b ∝ a^3\) and \(a\) increases in the ratio \(2 : 3\), then find the ratio in which \(b\) increases.

49. If \[ a + b \propto a - b \] then prove that \[ a^2 + b^2 \propto ab \]

50. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When x = 1, y = -1; and when x = 3, y = 5. Determine the relationship between x and y.

51. Given \(x \propto \frac{1}{y}\), When \(y = 10\), \(x = 5\) So, \(x = \frac{k}{y} = \frac{50}{y}\) When \(y = 5\), \(x = \frac{50}{5} = 10\)