276. If the volume of a sphere is numerically equal to its surface area then its diameter is
     

1. 1. 6 cm
      

   
   
   

   2. 4 cm
      

   
   
   

   3. 2 cm
      

   
   
   

   4. 3 cm

   
   
   

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1. View Hint View Answer Discuss in Forum

   Radius of sphere = r units According to the question,
   

   	4	πr³ = 4πr² ⇒ r = 3 units
   	3

   
   ∴ Diameter = 2 × 3 = 6 units

   Correct Option: A

   Radius of sphere = r units According to the question,
   

   	4	πr³ = 4πr² ⇒ r = 3 units
   	3

   
   ∴ Diameter = 2 × 3 = 6 units

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277. A conical iron piece having diameter 28 cm and height 30cm is totally immersed into the water of a cylindrical vessel, resulting in the rise of water level by 6.4 cm. The diameter, in cm, of the vessel is :
     

1. 1. 3.5
      

   
   
   

   2. 35/2
      

   
   
   

   3. 32
      

   
   
   

   4. 35

   
   
   

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1. View Hint View Answer Discuss in Forum

   Radius of cylindrical vessel = r cm. (let).
   

   Volume of conical piece of iron =	1	πR²h
   	3

   
   

   =			1	π × 1 × 14 × 30		cubic metre
   			3

   
   Volume of raised water = πr² × 6.4 cu. cm.
   ∴ πr² × 6.4
   

   =	1	π × 14 × 14 × 30
   	3

   
   

   ⇒ r² =	14 × 14 × 10
   	6.4

   
   

   ⇒ r² =	14² × 10²
   	8²

   
   

   ⇒ r =	14 × 10
   	8

   
   

   ⇒ 2r =	2 × 14 × 10
   	8

   
   = 35 cm = diameter

   Correct Option: D

   Radius of cylindrical vessel = r cm. (let).
   

   Volume of conical piece of iron =	1	πR²h
   	3

   
   

   =			1	π × 1 × 14 × 30		cubic metre
   			3

   
   Volume of raised water = πr² × 6.4 cu. cm.
   ∴ πr² × 6.4
   

   =	1	π × 14 × 14 × 30
   	3

   
   

   ⇒ r² =	14 × 14 × 10
   	6.4

   
   

   ⇒ r² =	14² × 10²
   	8²

   
   

   ⇒ r =	14 × 10
   	8

   
   

   ⇒ 2r =	2 × 14 × 10
   	8

   
   = 35 cm = diameter

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278. A solid right prism made of iron has cross section of a triangle of sides 5cm, 10cm, 13cm and of height 10 cm. If one cubic cm of iron weights 7g, then the weight of the prism is (approximately)
     

1. 1. 1570.8 gram
      

   
   
   

   2. 1371.32 gram
      

   
   
   

   3. 1470.8 gram
      

   
   
   

   4. 1100.68 gram

   
   
   

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1. View Hint View Answer Discuss in Forum

   The base of a prism is a triangular.
   Semi-perimeter of triangle(s)
   

   =	5 + 10 + 13	=	28	= 14cm.
   	2		2

   
   ∴ Area of triangle = √s(s - a)(s - b)(s - c)
   = √14(14 - 5)(14 - 10 (14 - 13)
   = √14 × 9 × 4 × 1
   = 6 √14 cu.cm.
   = ∴ Volume of prism = Area of base × height
   = 6 √14 × 10 = 60√14cu.cm.
   = 60 × 3.742 = 224.52 cu.cm.
   ∴ Weight of the prism = (224.52 × 7) gram = 1571.64 gram

   Correct Option: A

   The base of a prism is a triangular.
   Semi-perimeter of triangle(s)
   

   =	5 + 10 + 13	=	28	= 14cm.
   	2		2

   
   ∴ Area of triangle = √s(s - a)(s - b)(s - c)
   = √14(14 - 5)(14 - 10 (14 - 13)
   = √14 × 9 × 4 × 1
   = 6 √14 cu.cm.
   = ∴ Volume of prism = Area of base × height
   = 6 √14 × 10 = 60√14cu.cm.
   = 60 × 3.742 = 224.52 cu.cm.
   ∴ Weight of the prism = (224.52 × 7) gram = 1571.64 gram

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279. A right circular cone of height 20 cm and base radius 15 cm is melted and cast into smaller cones of equal sizes of height 5 cm and base radius 1.5 cm. The number of cones cast are
     

1. 1. 300
      

   
   
   

   2. 150
      

   
   
   

   3. 400
      

   
   
   

   4. 100

   
   
   

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1. View Hint View Answer Discuss in Forum

   Volume of cone =	1	πr²h
   	3

   
   

   =	1	× π × 15 × 15 × 20
   	3

   
   = 1500π cu.cm.
   

   ∴ Volume of a smaller cone =	1	× π × 1.5 × 1.5 × 5
   	3

   
   = 3.75π cu.cm.
   

   ∴ Number of smaller cones =	1500π	= 400
   	3.75π

   Correct Option: C

   Volume of cone =	1	πr²h
   	3

   
   

   =	1	× π × 15 × 15 × 20
   	3

   
   = 1500π cu.cm.
   

   ∴ Volume of a smaller cone =	1	× π × 1.5 × 1.5 × 5
   	3

   
   = 3.75π cu.cm.
   

   ∴ Number of smaller cones =	1500π	= 400
   	3.75π

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280. A sector is formed by opening out a cone of base radius 8 cm and height 6 cm. Then the radius of the sector is (in cm)
     

1. 1. 4
      

   
   
   

   2. 8
      

   
   
   

   3. 10
      

   
   
   

   4. 6

   
   
   

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1. View Hint View Answer Discuss in Forum

   Radius of sector = Slant height of cone = √r² + h²
   = √6² + 8²
   = √36 + 64 = √100
   = 10 cm

   Correct Option: C

   Radius of sector = Slant height of cone = √r² + h²
   = √6² + 8²
   = √36 + 64 = √100
   = 10 cm

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