RD Sharma 2020 solution class 9 chapter 20 Surface Area and Volume of a Right Circular Cone MCQS

MCQS

Page-20.23

Question 1:

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

Answer 1:

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).

Question 2:

The area of the curved surface of a cone of radius 2r and slant height $\frac{l}{2}$, is

(a) $\mathrm{\pi }rl$

(b) 2$\mathrm{\pi }rl$

(c) $\frac{1}{2}\mathrm{\pi }rl$

(d) $\mathrm{\pi }(r+l)r$

Answer 2:

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = πrl
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 =
= πrl
Hence the correct choice is (a).

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Question 3:

The total surface area of a cone of radius $\frac{r}{2}$ and length 2l, is
(a) 2$\pi r (\mathit{1}+r)$

(b) $\mathrm{\pi r}\left(1+\frac{\mathrm{r}}{4}\right)$

(c) $\mathrm{\pi }r (1+r)$

(d) 2$\mathrm{\pi }r\mathrm{l}$

Answer 3:

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).

Question 4:

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

Answer 4:

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
 $$\begin{gathered} \frac{Volume of cone}{Volume of cylinder}=\frac{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2{\mathrm{h}}_{\mathrm{cone}}}{\pi r^2{\mathrm{h}}_{cylinder}} \\ \Rightarrow \frac{V}{V}=\frac{{\displaystyle \frac{1}{3}{\mathrm{h}}_{cone}}}{{\mathrm{h}}_{cylinder}} \\ \Rightarrow 1=\frac{{\displaystyle \frac{1}{3}{\mathrm{h}}_{cone}}}{{\mathrm{h}}_{cylinder}} \end{gathered}$$

Hence the correct choice is option (c).

Question 5:

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) $\frac{1}{3}{\mathrm{\pi r}}^3$

(b) $\frac{2}{3}{\mathrm{\pi r}}^3$

(c) $3{\mathrm{\pi r}}^3$

(d) 9${\mathrm{\pi r}}^3$

Answer 5:

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).

Question 6:

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

Answer 6:

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).

Question 7:

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

Answer 7:

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).

Question 8:

If the height and radius of a cone of volume V are doubled, then the volume of the cone, is

(a) 3 V

(b) 4 V

(c) 6 V

(d) 8 V

Answer 8:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone = = V
Since it is given that the radius and height are doubled we have the radius as ‘2r’ and the vertical height as ‘2h’
So now,
Volume of modified cone =
=
= 8V
Hence the correct answer is option (d).

Question 9:

The ratio of the volume of a right circular cylinder and a right circular cone of the same base and height, is

(a) 1 : 3

(b) 3 : 1

(c) 4 : 3

(d) 3 : 4

Answer 9:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone =
And, the formula of the volume of a cylinder with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cylinder=
Now, substituting these to arrive at the ratio between the volume of a cylinder and the volume of a cone, we get
=
=
Hence the correct answer is option (b).

Question 10:

A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is

(a) 3 : 5

(b) 2 : 5

(c) 3 : 1

(d) 1 : 3

Answer 10:

It is given that the volumes of both the cylinder and the cone are the same.
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

Hence the correct choice is option (d).

Question 11:

The diameters of two cones are equal. If their slant heights are in the ratio 5 : 4, the ratio of their curved surface areas, is

(a) 4 : 5

(b) 25 : 16

(c) 16 : 25

(d) 5 : 4

Answer 11:

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = πrl
Now there are two cones with base radius and slant heights as , & , respectively.
The ratio between slant heights of the two cones is given as 5 : 4, we shall use them by introducing a constant ‘k’
So, now = 5k
= 4k
Since the base diameters of both the cones are equal we get that = =
Using these values we shall evaluate the ratio between the curved surface areas of the two cones
=
=
=
Hence the correct answer is option (d).

Question 12:

If the heights of two cones are in the ratio of 1 : 4 and the radii of their bases are in the ratio 4 : 1, then the ratio of their volumes is

(a) 1 : 2

(b) 2 : 3

(c) 3 : 4

(d) 4 : 1

Answer 12:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume =
Let the base radius and the height of the two cones be and respectively.
It is given that the ratio between the heights 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 radius of the two cones is 4 : 1.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So, = 4p
= 1p
Substituting these values in the formula for volume of cone we get,
=
=
Hence the correct choice is option (d).

Question 13:

The slant height of a cone is increased by 10%. If the radius remains the same, the curved surface area is increased by

(a) 10%

(b) 12.1%

(c) 20%

(d) 21%

Answer 13:

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = πrl
Now, it is said that the slant height has increased by 10%.So the new slant height is ‘1.1l’
So, now
New Curved Surface Area = 1.1πrl
We see that the percentage increase of the Curved Surface Area is 10%
Hence the correct option is (a).

Question 14:

The height of a solid cone is 12 cm and the area of the circular base is 64$\mathrm{\pi }$cm². A plane parallel to the base of the cone cuts through the cone 9 cm above the vertex of the cone, the areas of the base of the new cone so formed is

(a) 9$\mathrm{\pi }$ cm²

(b) 16$\mathrm{\pi }$ cm²

(c) 25$\mathrm{\pi }$ cm²

(d) 36$\mathrm{\pi }$ cm²

Answer 14:

If a cone is cut into two parts by a plane parallel to the base, the portion that contains the base is called the frustum of a cone

Let ‘r’ be the top radius
‘R’ be the radius of the base
‘h’ be the height of the frustum
‘l’ be the slant height of the frustum.
‘H’ be the height of the complete cone from which the frustum is cut
Then from similar triangles we can write the following relationship

Here it is given that the area of the base is 64π cm².
The area of the base with a base radius of ‘r’ is given by the formula
Area of base = πr²
Substituting the known values in this equation we get
64 π = πr²
r² = 64
r = 8
Hence the radius of the base of the original cone is 8 cm.
So, now let the plane cut the cone parallel to the base at 9 cm from the vertex.
Based on this we get the values as
R = 8
H = 12
H – h = 9
Substituting these values in the relationship mentioned earlier



Hence the radius of the new conical part that has been formed is 6 cm.
And the area of this base of this conical part would be
Area of the base = πr²
= 36π
Hence the correct choice is option (d).

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Question 15:

If the base radius and the height of a right circular cone are increased by 20%, then the percentage increase in volume is approximately

(a) 60

(b) 68

(c) 73

(d) 78

Answer 15:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume of cone =
= V
It is given that the base radius and the height are increased by 20%. So now the base radius is ‘1.2r’ and the height is ‘1.2h’.
So,
The volume of the modified cone =
=
= 1.728 V
Hence the percentage increase in the volume of the cone is 72.8%, which is approximately equal to 73%.
Hence the correct answer is option (c).

Question 16:

If h, S and V denote respectively  the height, curved surface area and volume of a right circular cone, then $3{\mathrm{\pi Vh}}^{3 }- {\mathrm{S}}^2{\mathrm{h}}^2 + 9{\mathrm{V}}^2$  is equal to

(a) 8

(b) 0

(c) 4$\mathrm{\pi }$

(d) 32${\mathrm{\pi }}^2$

Answer 16:

Here we are asked to find the value for a given specific equation which is in terms of V, h and S representing the volume, vertical height and the Curved Surface Area of a cone.
We know $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ and S = $\mathrm{\pi rl}$.
Also, $l=\sqrt{r^2+h^2}$
Now, the given equation is $3{\mathrm{\pi Vh}}^{3 }- {\mathrm{S}}^2{\mathrm{h}}^2 + 9{\mathrm{V}}^2$
So,
$$\begin{gathered} 3{\mathrm{\pi Vh}}^{3 }- {\mathrm{S}}^2{\mathrm{h}}^2 + 9{\mathrm{V}}^2 \\ =3\mathrm{\pi }\left(\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}\right){\mathrm{h}}^3-{\left(\mathrm{\pi rl}\right)}^2{\mathrm{h}}^2+9{\left(\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}\right)}^2 \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{l}}^2{\mathrm{h}}^2+9\left(\frac{1}{9}{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2\right) \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^2{\left(\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2}\right)}^2+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \end{gathered}$$
$$\begin{gathered} ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^2{\left(\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2}\right)}^2+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^2\left({\mathrm{r}}^2+{\mathrm{h}}^2\right)+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \\ =0 \end{gathered}$$
Hence the correct choice is option (b).

Question 17:

If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, the axis, the ratio of the volumes of upper and lower part is

(a) 1 : 2

(b) 2 : 1

(c) 1: 7

(d) 1 : 8

Answer 17:

If a cone is cut into two parts by a plane parallel to the base, the portion that contains the base is called the frustum of a cone

Let ‘r’ be the top radius
‘R’ be the radius of the base
‘h’ be the height of the frustum
‘l’ be the slant height of the frustum.
‘H’ be the height of the complete cone from which the frustum is cut
Then from similar triangles we can write the following relationship

Here, since the plane passes through the midpoint of the axis of the cone we have
H = 2 h
Substituting this in the earlier relationship we have




The volume of the entire cone with base radius ‘R’ and vertical height ‘H’ would be
Volume of the uncut cone =
Replacing and H = 2 h in the above equation we get
Volume of the uncut cone =
=
Volume of the smaller cone − the top part after the original cone is cut − with base radius ‘r’ and vertical height ‘h’ would be
Volume of the top part=
Now, the volume of the frustum − the bottom part after the original cone is cut − would be,
Volume of the bottom part= Volume of the uncut cone − Volume of the top part after the cone is cut
= -
Volume of the bottom part=
Now the ratio between the volumes of the top part and the bottom part after the cone is cut would be,


Hence the correct choice is option (c).