RD Sharma MCQs: Surface Area and Volume of a Right Circular Cone | Mathematics (Maths) Class 9 PDF Download

 Page 1


          
                    
  
    
 
               
                      
      
 
 
          
Q u e s t i o n : 4 7
Mark the correct alternative in each of the following:
The number of surfaces of a cone has, is
a
1
b
2
c
3
d
4
S o l u t i o n :
The surfaces or faces that a cone has are :
1 Base
2 Slanted Surface
So, the number of surfaces that a cone has is 2.
Hence the correct choice is
b.
Q u e s t i o n : 4 8
The area of the curved surface of a cone of radius 2r and slant height 
l
2
, is
a
prl
b
2 prl
Page 2


          
                    
  
    
 
               
                      
      
 
 
          
Q u e s t i o n : 4 7
Mark the correct alternative in each of the following:
The number of surfaces of a cone has, is
a
1
b
2
c
3
d
4
S o l u t i o n :
The surfaces or faces that a cone has are :
1 Base
2 Slanted Surface
So, the number of surfaces that a cone has is 2.
Hence the correct choice is
b.
Q u e s t i o n : 4 8
The area of the curved surface of a cone of radius 2r and slant height 
l
2
, is
a
prl
b
2 prl
c
1
2
prl
d
p(r+l)r
 
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Here the base radius is given as ‘2r’ and the slant height is given as ‘ ’
Substituting these values in the above equation we have
Curved Surface Area = 
= prl
Hence the correct choice is
a.
Q u e s t i o n : 4 9
The total surface area of a cone of radius 
r
2
and length 2l, is
a
2 pr (1 +r)
b
pr 1 +
r
4
c
pr (1 +r)
d
2 prl
 
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
Here it is given that the base radius is ‘ ’ and that the slant height is ‘2l’.
Substituting these values in the above equation we have
Total Surface Area = 
( )
Page 3


          
                    
  
    
 
               
                      
      
 
 
          
Q u e s t i o n : 4 7
Mark the correct alternative in each of the following:
The number of surfaces of a cone has, is
a
1
b
2
c
3
d
4
S o l u t i o n :
The surfaces or faces that a cone has are :
1 Base
2 Slanted Surface
So, the number of surfaces that a cone has is 2.
Hence the correct choice is
b.
Q u e s t i o n : 4 8
The area of the curved surface of a cone of radius 2r and slant height 
l
2
, is
a
prl
b
2 prl
c
1
2
prl
d
p(r+l)r
 
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Here the base radius is given as ‘2r’ and the slant height is given as ‘ ’
Substituting these values in the above equation we have
Curved Surface Area = 
= prl
Hence the correct choice is
a.
Q u e s t i o n : 4 9
The total surface area of a cone of radius 
r
2
and length 2l, is
a
2 pr (1 +r)
b
pr 1 +
r
4
c
pr (1 +r)
d
2 prl
 
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
Here it is given that the base radius is ‘ ’ and that the slant height is ‘2l’.
Substituting these values in the above equation we have
Total Surface Area = 
( )
= 
Hence the correct choice is
b.
Q u e s t i o n : 5 0
A solid cylinder is melted and cast into a cone of same radius. The heights of the cone and cylinder are in the ratio
a
9 : 1
b
1 : 9
c
3 : 1
d
1 : 3
S o l u t i o n :
Since the cylinder is re cast into a cone both their volumes should be equal.
So, let Volume of the cylinder = Volume of the cone
= V
It is also given that their base radii are the same.
So, let Radius of the cylinder = Radius of the cone
= r
Let the height of the cylinder and the cone be  and  respectively.
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone = 
The formula of the volume of a cylinder with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cylinder = 
So we have
 
Volume of cone
Volume of cylinder
=
1
3
p r
2
h
cone
pr
2
h
cylinder
?
V
V
=
1
3
h
c one
h
cylinder
? 1 =
1
3
h
c one
h
cylinder
Hence the correct choice is option
c.
Q u e s t i o n : 5 1
If the radius of the base of a right circular cone is 3r and its height is equal to the radius of the base, then its volume is
a
1
3
pr
3
b
Page 4


          
                    
  
    
 
               
                      
      
 
 
          
Q u e s t i o n : 4 7
Mark the correct alternative in each of the following:
The number of surfaces of a cone has, is
a
1
b
2
c
3
d
4
S o l u t i o n :
The surfaces or faces that a cone has are :
1 Base
2 Slanted Surface
So, the number of surfaces that a cone has is 2.
Hence the correct choice is
b.
Q u e s t i o n : 4 8
The area of the curved surface of a cone of radius 2r and slant height 
l
2
, is
a
prl
b
2 prl
c
1
2
prl
d
p(r+l)r
 
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Here the base radius is given as ‘2r’ and the slant height is given as ‘ ’
Substituting these values in the above equation we have
Curved Surface Area = 
= prl
Hence the correct choice is
a.
Q u e s t i o n : 4 9
The total surface area of a cone of radius 
r
2
and length 2l, is
a
2 pr (1 +r)
b
pr 1 +
r
4
c
pr (1 +r)
d
2 prl
 
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
Here it is given that the base radius is ‘ ’ and that the slant height is ‘2l’.
Substituting these values in the above equation we have
Total Surface Area = 
( )
= 
Hence the correct choice is
b.
Q u e s t i o n : 5 0
A solid cylinder is melted and cast into a cone of same radius. The heights of the cone and cylinder are in the ratio
a
9 : 1
b
1 : 9
c
3 : 1
d
1 : 3
S o l u t i o n :
Since the cylinder is re cast into a cone both their volumes should be equal.
So, let Volume of the cylinder = Volume of the cone
= V
It is also given that their base radii are the same.
So, let Radius of the cylinder = Radius of the cone
= r
Let the height of the cylinder and the cone be  and  respectively.
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone = 
The formula of the volume of a cylinder with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cylinder = 
So we have
 
Volume of cone
Volume of cylinder
=
1
3
p r
2
h
cone
pr
2
h
cylinder
?
V
V
=
1
3
h
c one
h
cylinder
? 1 =
1
3
h
c one
h
cylinder
Hence the correct choice is option
c.
Q u e s t i o n : 5 1
If the radius of the base of a right circular cone is 3r and its height is equal to the radius of the base, then its volume is
a
1
3
pr
3
b
2
3
pr
3
c
3 pr
3
d
9 pr
3
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone = 
Here it is given that the base radius is ’3r’ and that the vertical height is ‘3r’
Substituting these values in the above equation we get
Volume of cone = 
= 
Hence the correct answer is option
d.
Q u e s t i o n : 5 2
If the volume of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then the ratio of their heights, is
a
1 : 5
b
5 : 4
c
5 : 16
d
25 : 64
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
Let the volume, base radius and the height of the two cones be  and  respectively.
It is given that the ratio between the volumes of the two cones is 1 : 4.
Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.
So,  = 1k
= 4k
It is also given that the ratio between the base diameters of the two cones is 4 : 5.
Hence the ratio between the base radius will also be 4 : 5.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So,  = 4p
Page 5


          
                    
  
    
 
               
                      
      
 
 
          
Q u e s t i o n : 4 7
Mark the correct alternative in each of the following:
The number of surfaces of a cone has, is
a
1
b
2
c
3
d
4
S o l u t i o n :
The surfaces or faces that a cone has are :
1 Base
2 Slanted Surface
So, the number of surfaces that a cone has is 2.
Hence the correct choice is
b.
Q u e s t i o n : 4 8
The area of the curved surface of a cone of radius 2r and slant height 
l
2
, is
a
prl
b
2 prl
c
1
2
prl
d
p(r+l)r
 
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Here the base radius is given as ‘2r’ and the slant height is given as ‘ ’
Substituting these values in the above equation we have
Curved Surface Area = 
= prl
Hence the correct choice is
a.
Q u e s t i o n : 4 9
The total surface area of a cone of radius 
r
2
and length 2l, is
a
2 pr (1 +r)
b
pr 1 +
r
4
c
pr (1 +r)
d
2 prl
 
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
Here it is given that the base radius is ‘ ’ and that the slant height is ‘2l’.
Substituting these values in the above equation we have
Total Surface Area = 
( )
= 
Hence the correct choice is
b.
Q u e s t i o n : 5 0
A solid cylinder is melted and cast into a cone of same radius. The heights of the cone and cylinder are in the ratio
a
9 : 1
b
1 : 9
c
3 : 1
d
1 : 3
S o l u t i o n :
Since the cylinder is re cast into a cone both their volumes should be equal.
So, let Volume of the cylinder = Volume of the cone
= V
It is also given that their base radii are the same.
So, let Radius of the cylinder = Radius of the cone
= r
Let the height of the cylinder and the cone be  and  respectively.
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone = 
The formula of the volume of a cylinder with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cylinder = 
So we have
 
Volume of cone
Volume of cylinder
=
1
3
p r
2
h
cone
pr
2
h
cylinder
?
V
V
=
1
3
h
c one
h
cylinder
? 1 =
1
3
h
c one
h
cylinder
Hence the correct choice is option
c.
Q u e s t i o n : 5 1
If the radius of the base of a right circular cone is 3r and its height is equal to the radius of the base, then its volume is
a
1
3
pr
3
b
2
3
pr
3
c
3 pr
3
d
9 pr
3
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone = 
Here it is given that the base radius is ’3r’ and that the vertical height is ‘3r’
Substituting these values in the above equation we get
Volume of cone = 
= 
Hence the correct answer is option
d.
Q u e s t i o n : 5 2
If the volume of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then the ratio of their heights, is
a
1 : 5
b
5 : 4
c
5 : 16
d
25 : 64
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
Let the volume, base radius and the height of the two cones be  and  respectively.
It is given that the ratio between the volumes of the two cones is 1 : 4.
Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.
So,  = 1k
= 4k
It is also given that the ratio between the base diameters of the two cones is 4 : 5.
Hence the ratio between the base radius will also be 4 : 5.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So,  = 4p
= 5p
Substituting these values in the formula for volume of cone we get,
= 
= 
= 
= 
Hence the correct answer is option
d.
Q u e s t i o n : 5 3
The curved surface area of one cone is twice that of the other while the slant height of the latter is twice that of the former.
The ratio of their radii is
a
2 : 1
b
4 : 1
c
8 : 1
d
1 : 1
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Now there are two cones with base radius, slant height and Curved Surface Area
C. S. A as , , & , , respectively.
It is given that  = 2( ) and also that = 2( ). Or this can also be written as
= 
= 
= 
= 
= 
Hence the correct choice is option
b.