426. A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of sphere. What is the ratio of the height and radius of the cone so formed?
     

1. 1. 4 : 3
      

   
   
   

   2. 2 : 3
      

   
   
   

   3. 3 : 4
      

   
   
   

   4. 4 : 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   In this case volume remains same.
   

   ∴	4	πR³ =	1	πR³H
   	3		3

   
   ⇒ 4 R = H
   ⇒ H : R = 4 : 1

   Correct Option: D

   In this case volume remains same.
   

   ∴	4	πR³ =	1	πR³H
   	3		3

   
   ⇒ 4 R = H
   ⇒ H : R = 4 : 1

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427. A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is just completely submerged in water, then the rise of water level in the cylindrical vessel is
     

1. 1. 2 cm
      

   
   
   

   2. 1 cm
      

   
   
   

   3. 3 cm
      

   
   
   

   4. 4 cm

   
   
   

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1. View Hint View Answer Discuss in Forum

   Volume of sphere =	4	πr³
   	3

   
   

   =	4	π × 3 × 3 × 3
   	3

   
   = 36π cu.cm.
   If the water level rises by h cm, then
   πR²h = = 36π
   ⇒ 6 × 6 × h = 36
   ⇒ h = 1 cm

   Correct Option: B

   Volume of sphere =	4	πr³
   	3

   
   

   =	4	π × 3 × 3 × 3
   	3

   
   = 36π cu.cm.
   If the water level rises by h cm, then
   πR²h = = 36π
   ⇒ 6 × 6 × h = 36
   ⇒ h = 1 cm

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428. The length, breadth and height of a cuboid are in the ratio 1 : 2 : 3. If they are increased by 100%, 200% and 200% respectively, then compared to the original volume the increase in the volume of the cuboid will be
     

1. 1. 5 times
      

   
   
   

   2. 18 times
      

   
   
   

   3. 12 times
      

   
   
   

   4. 17 times

   
   
   

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1. View Hint View Answer Discuss in Forum

   Let length, breadth and height of a cuboid are increased by x%, y% and z% respectively, then its volume is increased by
   

   		x + y + z +	xy + yz + zx	+	xyz		%
   			100		10000

   
   

   ∴ Effective increase =		100 + 200 + 20000 + 40000 +	20000	+	4000000		%
   			100		10000

   
   = 500 + 800 + 400 = 1700 %
   

   = 1700 ×	1	= 17 times
   	100

   
   Method : 2
   Original volume = x × 2x × 3x = 6x³ cubic units
   New volume = 2x × 6x × 9x = 108x³ cubic units
   Change in volume = 108x³ – 6x³ = 102x³ cubic units
   

   Increase =	102x³	= 17 times
   	6x³

   Correct Option: D

   Let length, breadth and height of a cuboid are increased by x%, y% and z% respectively, then its volume is increased by
   

   		x + y + z +	xy + yz + zx	+	xyz		%
   			100		10000

   
   

   ∴ Effective increase =		100 + 200 + 20000 + 40000 +	20000	+	4000000		%
   			100		10000

   
   = 500 + 800 + 400 = 1700 %
   

   = 1700 ×	1	= 17 times
   	100

   
   Method : 2
   Original volume = x × 2x × 3x = 6x³ cubic units
   New volume = 2x × 6x × 9x = 108x³ cubic units
   Change in volume = 108x³ – 6x³ = 102x³ cubic units
   

   Increase =	102x³	= 17 times
   	6x³

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429. A conical flask is full of water. The flask has base radius r and height h. This water is poured into a cylindrical flask of base radius m. The height of water in the cylindrical flask is
     

1. 1. 	m
      	2h

      
      

      	h	m²
      	2

   
   
   

   2. 	2h
      	m

   
   
   

   3. 	2h
      	m

   
   
   

   4. 	h
      	3m²

   
   
   

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1. View Hint View Answer Discuss in Forum

   Volume of water in conical flask =	1	πr²h
   	3

   
   If the height of water level in cylindrical flask be H units, then
   

   πm²r²H =	1	πr²h
   	3

   
   

   ⇒ H =	1	.	π² h	=	h
   	3		πm²r²		3m²

   Correct Option: D

   Volume of water in conical flask =	1	πr²h
   	3

   
   If the height of water level in cylindrical flask be H units, then
   

   πm²r²H =	1	πr²h
   	3

   
   

   ⇒ H =	1	.	π² h	=	h
   	3		πm²r²		3m²

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430. The volume of a cylinder and a cone are in the ratio 3 : 1. Find their diameters and then compare them when their heights are equal.
     

1. 1. Diameter of cylinder = 2 times of diameter of cone
      

   
   
   

   2. Diameter of cylinder = Diameter of cone
      

   
   
   

   3. Diameter of cylinder > Diameter of cone
      

   
   
   

   4. Diameter of cylinder < Diameter of cone

   
   
   

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1. View Hint View Answer Discuss in Forum

   =	Volume of cylinder	=	3
   	Volume of cone		1

   
   

   =	πr1²h		=	3	⇒ r1 = r2
   	1	πr2²h		1
   	3

   
   ∴Diameter of cylinder = Diameter of cone

   Correct Option: B

   =	Volume of cylinder	=	3
   	Volume of cone		1

   
   

   =	πr1²h		=	3	⇒ r1 = r2
   	1	πr2²h		1
   	3

   
   ∴Diameter of cylinder = Diameter of cone

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