Let h be the height,
Volume of the cone = V
and curved surface area of the cone = c
∴ 1/3 πr²h = V, πrl = c
i.e., πr√r² + h² = c
π²r²(r² + h²) = c²
Consider 3πVh³ - c²h² + 9V²
= 3π x 1/3πr²h x h³ - π²r²(r² + h²)h + 9 x 1/9 x π²r⁴ h²
= π²r²h⁴ - π²r⁴h² - π²r²h⁴ + π²r⁴ = 0

Correct Option: A

Let h be the height,
Volume of the cone = V
and curved surface area of the cone = c
∴ 1/3 πr²h = V, πrl = c
i.e., πr√r² + h² = c
π²r²(r² + h²) = c²
Consider 3πVh³ - c²h² + 9V²
= 3π x 1/3πr²h x h³ - π²r²(r² + h²)h + 9 x 1/9 x π²r⁴ h²
= π²r²h⁴ - π²r⁴h² - π²r²h⁴ + π²r⁴ = 0

Let the side of the cube be 3k, 4k and 5k, respectively.
So, volumes of these cubes are 27 k³, 64k³, 125 k³ respectively.
⇒ Volume of the new bigger cube = 27 k³ + 64k³ + 125 k³
= 216k³
So, side of the new cube = 6k

Correct Option: A

Let the side of the cube be 3k, 4k and 5k, respectively.
So, volumes of these cubes are 27 k³, 64k³, 125 k³ respectively.
⇒ Volume of the new bigger cube = 27 k³ + 64k³ + 125 k³
= 216k³
So, side of the new cube = 6k
Since, diagonal of the cube = √(6k)² + (6k)² + (6k)²
= 12√3
⇒ 108k² = 432
⇒k = 2
So, sides of the three cubes were 6 cm, 8 cm and 10 cm respectively.

External radius = 2.5 cm,
length = 100 cm
∴ External volume = [π x (2.25)² x 100] cm²
internal radius = 1.5 cm
∴ internal volume = [π x (1.5)² x 100]cm³
Volume of metal
= [π x (2.5)² x 100 - π x (1.5)² x 100] cm³
= π x 100 x [(2.5)² - (1.5)²] cm³
= (22/7 x 100 x 4 x 1 x 21/1000) kg
= 26.4 kg.

Correct Option: C

External radius = 2.5 cm,
length = 100 cm
∴ External volume = [π x (2.25)² x 100] cm²
internal radius = 1.5 cm
∴ internal volume = [π x (1.5)² x 100]cm³
Volume of metal
= [π x (2.5)² x 100 - π x (1.5)² x 100] cm³
= π x 100 x [(2.5)² - (1.5)²] cm³
= (22/7 x 100 x 4 x 1 x 21/1000) kg
= 26.4 kg.

Curved surface area of cylinder = 2πrh cm²
Total surface area of cylinder = 2πr(h + r) cm²
Now, According to the question
2πrh/2πr(h + r) = 1/2
⇒ h/(h + r) = 1/2
⇒ 2h = h + r
∴ Now, Total surface area = 616 cm²
⇒ 2π (h + r) = 616

Correct Option: B

Curved surface area of cylinder = 2πrh cm²
Total surface area of cylinder = 2πr(h + r) cm²
Now, According to the question
2πrh/2πr(h + r) = 1/2
⇒ h/(h + r) = 1/2
⇒ 2h = h + r
∴ Now, Total surface area = 616 cm²
⇒ 2π (h + r) = 616
⇒ 2πr (r + r) = 616 (∵ h = r)
⇒ 2πr(2r) = 616
⇒ 4πr² = 616
⇒ r = √style="text-decoration:overline">616/4π
= √616 x 7/4 x 22
= √7 x 7 =
∴ r = h = 7 cm
∴ Volume = π r² h
= (22/7) x (7 x 7) x 7 = 1078 cm³

If 10 circular plates each of thickness 3 cm, each are placed one above the other, then it forms a cylinder with height (3 x 10 = 30 cm) and a hemisphere of radius 6 cm is placed on the top just to cover the cylinder that means its radius is 6 cm also
∴ Radius of hemisphere (R) = 6 cm
Radius of cylinder (r) = 6 cm
and height of cylinder (h) = 30 cm
∴ Volume of the solid = Volume of cylinder + Volume of hemisphere
= πr²h + (2/3)πR³
= π(6)² x 30 + (2/3)π(6)³

Correct Option: D

If 10 circular plates each of thickness 3 cm, each are placed one above the other, then it forms a cylinder with height (3 x 10 = 30 cm) and a hemisphere of radius 6 cm is placed on the top just to cover the cylinder that means its radius is 6 cm also
∴ Radius of hemisphere (R) = 6 cm
Radius of cylinder (r) = 6 cm
and height of cylinder (h) = 30 cm
∴ Volume of the solid = Volume of cylinder + Volume of hemisphere
= πr²h + (2/3)πR³
= π(6)² x 30 + (2/3)π(6)³
= π x 36 x 30 + (2/3)π x 216
= 1080π + 2π x 72 = 1080π + 144π
= 1224π cm³
Which is the required volume of solid.