(a) Let r₁ and r₂ be the radius of the given cones and h be their height.

Ratio of radii, r₁:r₂ = 3:4

Volume of cone, V₁ = (1/3)r₁²h

Volume of cone, V₂ = (1/3)r₂²h

V₁ /V₁ = (1/3)r₁²h/ (1/3)r₂²h

= r₁²/ r₂²

= 3²/4²

= 9/16

Hence the ratio of the volumes is 9:16.

(b) Let h₁ and h₂ be the heights of the given cones and r₁ and r₂ be their radii.

Ratio of heights, h₁:h₂ = 5:2

Ratio of radii, r₁:r₂ = 2:5

Volume of cone, V₁ = (1/3)r₁²h₁

Volume of cone, V₂ = (1/3)r₂²h₂

V₁ /V₁ = (1/3)r₁²h₁/ (1/3)r₂²h₂

= r₁² h₁/ r₂² h₂

= 2²×5/5²×2

= 4×5/25×2

= 20/50 = 2/5

Hence the ratio of the volumes is 2:5.

(c) Let r be the radius of bigger cone. Then the radius of smaller cone is r/2.

Let h be the height of bigger cone. Then the height of smaller cone is h/2.

(i)Volume of bigger cone, V₁ = (1/3)r²h

Volume of smaller cone, V₂ = (1/3)(r/2)² (h/2) = (1/3)r²h/8

Ratio of volume of smaller cone to bigger cone, V₂/V₁ = ( 1/3)r² h/8÷ (1/3)r²h

= (1/24) r² h ×(3/r² h)

= 1/8

Hence the ratio of their volumes is 1:8.

(ii)slant height of bigger cone = √(h²+r²)

slant height of smaller cone = √((h/2)²+(r/2)²) = √(h²/4+r²/4) = ½ √(h²+r²)

Curved surface area of bigger cone, s1 = rl

= r√(h²+r²)

Curved surface area of smaller cone, s2 = rl

= ×(r/2)× ½ √(h²+r²)

= ¼ r√(h²+r²)

ratio of curved surface area of smaller cone to bigger cone, s2/s1 = ¼ r√(h²+r²) ÷ r√(h²+r²)

= ¼ r√(h²+r²) ×1/(r√(h²+r²))

= 1/4

Hence the ratio of curved surface area of smaller cone to bigger cone is 1:4