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Maths MCQ(1M)

Surface Area And Volume of Sphere · Maharashtra Board STD 9 (MSBSHSE - English Medium). 15 questions.

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MCQ(1M)

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15 questions · auto-graded multiple-choice test.

MCQ 11 Mark
If a solid sphere of radius $10\ cm$ is moulded into $8$ spherical solid balls of equal radius, then the surface area of each ball $($in sq.$cm)$ is:
  • ✓
    $100\pi$
  • B
    $75\pi$
  • C
    $60\pi$
  • D
    $50\pi$
Answer
Correct option: A.
$100\pi$
Volume of solid sphere $=\frac{4}{3}\pi(10)^3=\frac{4000\pi}{3}\text{ cm}^3$
Vomule $8$ solid sphere of radius $($say$)\ \text{r}=8\times\frac{4}{3}\pi\text{r}^3=\frac{32\pi\text{r}^3}{3}\text{ cm}^3$
Now, $\frac{32\pi\text{r}^3}{3}=\frac{4000\pi}{3}$
$\Rightarrow\text{r}=\Big(\frac{1000}{8}\Big)^\frac{1}{3}$
$=\frac{10}{2}=5\text{ cm}$
Surface Area of each small ball $=4\pi\text{r}^2$
$=4\pi(5)^2=100\pi\text{ cm}^2 $
Hence, correct option is $(a)$.
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MCQ 21 Mark
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is:
  • ✓
    $1 : 2 : 3$
  • B
    $2 : 1 : 3$
  • C
    $2 : 3 : 1$
  • D
    $3 : 2 : 1$
Answer
Correct option: A.
$1 : 2 : 3$

If all of these have equal bases, then their radii are equal.
Their heights are same. $($given$)$
$\text{r}=\text{h}_1=\text{h}_2$
$\text{V}_\text{cone}=\frac{1}{3}\pi\text{r}^2\text{h}_1$
$=\frac{1}{3}\pi\text{r}^2(\text{r})=\frac{1}{3}\pi\text{r}^3$
$\text{V}_\text{hemisphere}=\frac{2}{3}\pi\text{r}^3$
$\text{V}_\text{cylinder}=\pi\text{r}^2\text{h}_2=\pi\text{r}^2(\text{r})=\pi\text{r}^3$
$\text{V}_\text{cone}:\text{V}_\text{hemisphere}:\text{V}_\text{cylinder}=\frac{1}{2}:\frac{2}{3}:1$
$=1:2:3$
Hence, correct option is $(a)$.
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MCQ 31 Mark
The largest sphere is cut off from a cube of side $6\ cm$. The volume of the sphere will be:
  • A
    $27\pi\ \text{cm}^2$
  • ✓
    $36\pi\ \text{cm}^3$
  • C
    $108\pi\ \text{cm}^3$
  • D
    $12\pi\ \text{cm}^3$
Answer
Correct option: B.
$36\pi\ \text{cm}^3$
The largest sphere that can be cut from a cube of side $6\ cm$ will have its diameter $=$ side of cube.
i. e. $2r = 6\ cm $
$\Rightarrow r = 3\ cm$
Volume of that sphere $=\frac{4}{3}\pi\text{r}^3$
$=\frac{4}{3}\pi\times3\times3\times3=36\pi\text{ cm}^3$
Hence, correct option is $(b)$
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MCQ 41 Mark
A sphere is placed inside a right circular cylinder so as to touch the top, base and lateral surface of the cylinder. If the radius of the sphere is $r,$ then the volume of the cylinder is
  • A
    $4\pi\text{r}^3$
  • B
    $\frac{8}{3}\pi\text{r}^3$
  • ✓
    $2\pi\text{r}^3$
  • D
    $8\pi\text{r}^3$
Answer
Correct option: C.
$2\pi\text{r}^3$

Radius of sphere $= r$
Sphere touches cylinder at
Top, Base and Lateral Surface.
Then,
$2r =$ height of cylinder $= h$
$r =$ Radius of cylinder
Volume of cylinder $=\pi\text{r}^2\text{h}$
$=\pi\text{r}^2(2\text{r})$
$=2\pi\text{r}^3$
Hence, correct option is $(c)$.
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MCQ 51 Mark
The total surface area of a hemisphere of radius $r$ is:
  • A
    $\pi\text{r}^2$
  • B
    $2\pi\text{r}^2$
  • ✓
    $3\pi\text{r}^2$
  • D
    $4\pi\text{r}^2$
Answer
Correct option: C.
$3\pi\text{r}^2$
A hemisphere has two surfaces : one top surface and other curved surface.
$\text{T.S.A} =2\pi\text{r}^2+(\pi\text{r}^2)$
$\{$Area of Top $-$ face $= \pi\text{r}^2\}$
$=3\pi\text{r}^2$
Hence, correct option is $(c).$
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MCQ 61 Mark
If the ratio of volumes of two spheres is $1 : 8,$ then the ratio of their surface areas is
  • A
    $1 : 2$
  • ✓
    $1 : 4$
  • C
    $1 : 8$
  • D
    $1 : 16$
Answer
Correct option: B.
$1 : 4$
Volume of sphere $=\frac{4}{3}\pi\text{r}^3=\text{v}$
$\frac{\text{V}_1}{\text{V}_1}=\frac{\frac{4}{3}\pi\text{r}^3}{\frac{4}{3}\pi\text{r}^3_2}$
$=\frac{\text{r}^3_1}{\text{r}^3_2}=\frac{1}{8}$
$\Rightarrow\frac{\text{r}_1}{\text{r}_2}=\frac{1}{2}$
now, Surface Area of Sphere $=4\pi\text{r}^2=\text{S}$
$\frac{\text{S}_1}{\text{S}_2}=\frac{4\pi\text{r}^2_1}{4\pi\text{r}^2_2}$
$=\Big(\frac{\text{r}_1}{\text{r}_2}\Big)^2=\frac{1}{4}=1:4$
Hence, correct option is $(b)$.
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MCQ 71 Mark
In a sphere the number of faces is:
  • ✓
    $1$
  • B
    $2$
  • C
    $3$
  • D
    $4$
Answer
Correct option: A.
$1$
Sphere has only one surface
i.e. curved surface, so number of faces $= 1$
Hence, correct option is $(a)$.
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MCQ 81 Mark
The ratio between the volume of a sphere and volume of a circumscribing right circular cylinder is:
  • A
    $2 : 1$
  • B
    $1 : 1$
  • ✓
    $2 : 3$
  • D
    $1 : 2$
Answer
Correct option: C.
$2 : 3$
Volume of sphere of radius radius $ r =\frac{4}{3}\pi\text{r}^3=\text{v}_1\ ...(1)$
If a cylinder is circumscibibing the sphre, then
diameter of cylinder $=$ diameter of sphere
height of cylinder $=$ Radius of sphere
Height of cylinder $= 2r$
Volume of cylinder $= \text{V}_2=\pi\text{r}^2\text{h}$
$=\pi\text{r}^2(2\text{r})$
$\Rightarrow \text{V}_2=2\pi\text{r}^3\ ....(2)$
dividing equation $(1)$ and $(2)$
$\frac{\text{V}_1}{\text{V}_2}=\frac{\frac{4}{3}\pi\text{r}^3}{2\pi\text{r}^3}$
$\Rightarrow\frac{\text{V}_1}{\text{V}_2}=\frac{2}{3}$
hence, correct option is $(c)$.
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MCQ 91 Mark
If a solid sphere of radius $r$ is melted and cast into the shape of a solid cone of height $r,$ then the radius of the base of the cone is
  • ✓
    $2r$
  • B
    $3r$
  • C
    $r$
  • D
    $4r$
Answer
Correct option: A.
$2r$
Volume of sphere $=\frac{4}{3}\pi\text{r}^3$
sphere costed into a cone of height $r$.
Let the radius of cone $= R$
$\therefore$ Volume of cone $=\frac{1}{3}\pi\text{R}^2(\text{r})$
Volume of cone $=$ volume of sphere
$\Rightarrow\frac{1}{3}\pi\text{R}^2\text{r}=\frac{4}{3}\pi\text{r}^3$
$\Rightarrow\text{R}^2=4\text{r}^2$
$\Rightarrow\text{R}=2\text{r}$
Hence, correct option is $(a)$.
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MCQ 101 Mark
If a sphere is inscribed in a cube, then the ratio of the volume of the sphere to the volume of the cube is:
  • A
    $\pi:2$
  • B
    $\pi:3$
  • C
    $\pi:4$
  • ✓
    $\pi:6$
Answer
Correct option: D.
$\pi:6$


Edge of cube $= a$
$\Rightarrow$ Volume of cube $=a^3$
If Sphere is inscribed inside cube then a $=2\text{r}\Rightarrow\text{r}=\frac{\text{a}}{2}$
Volume of sphere $=\frac{4}{3}\pi\text{r}^3=\frac{4}{3}\pi\Big(\frac{\text{a}}{2}\Big)^3=\frac{\pi}{6}\text{a}^3$
Ratio of volume of sphere to volume of cube $=\frac{\frac{\pi}{6}\text{a}^3}{\text{a}^3}\frac{\pi}{6}$
Hence, correct option is $(d).$
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MCQ 111 Mark
A cylindrical rod whose height is $8$ times of its radius is melted and recast into spherical balls of same radius. The number of balls will be
  • A
    $4$
  • B
    $3$
  • ✓
    $6$
  • D
    $8$
Answer
Correct option: C.
$6$
Volume of cylindrical rod $=\pi\text{r}^\text{h}$
$=\pi\text{r}^2(8\text{r}) [h = 8r \ ($given$)]$
$=8\pi\text{r}^3$
Now, if spherical balls have same radius, then the volume of one ball $=\frac{4}{3}\pi\text{r}^3$
$\therefore$ No. of balls $=\frac{\text{Volume of Cylindrical Rod}}{\text{Volume of one Rod}}=\frac{8\pi\text{r}^3}{\frac{4}{3}\pi\text{r}^3}=6$
Hence, correct option is $(c).$
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MCQ 121 Mark
If the surface area of a sphere is $144\pi\text{ m}^2,$ then its volume $($ in $m^3)$ is:
  • ✓
    $288\pi$
  • B
    $316\pi$
  • C
    $300\pi$
  • D
    $188\pi$
Answer
Correct option: A.
$288\pi$
Surface Area of Sphere $\Rightarrow 4\pi\text{r}^2=144\pi$
$\Rightarrow\text{r}^2=36\Rightarrow\text{r}=6$
Volume of Sphere $=\frac{4}{3}\pi\text{r}^3=\frac{4}{3}\pi(6)^3=288\pi\text{ m}^3$
Hence, correct option is $(a).$
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MCQ 131 Mark
A sphere and a cube are of the same height. The ratio of their volumes is
  • A
    $3 : 4$
  • B
    $21 : 11$
  • C
    $4 : 3$
  • ✓
    $11 : 21$
Answer
Correct option: D.
$11 : 21$
Height of sphere $=$ diameter $= 2r$
Height of cube $=$ Side of cube $=$ Height of sphere $= 2r$
Volume of sphere $=\frac{4}{3}\pi\text{r}^3$
Volume of cube $(2\text{r})^3=8\text{r}^3$
Ratio of their volumes $=\frac{\frac{4}{3}\pi\text{r}^3}{8\text{r}^3}$
$=\frac{\pi}{6}=\frac{22^{11}}{7\times6_3}=\frac{11}{21}$
$=11:21$
Hence, correct option is $(d)$.
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MCQ 141 Mark
A cone and a hemisphere have equal bases and equal volume the ratio of their heights is:
  • A
    $1:2$
  • ✓
    $2:1$
  • C
    $4:1$
  • D
    $\sqrt{2}:1$
Answer
Correct option: B.
$2:1$
In the given problem, we are given a cone and a hemisphere which have equal bases equal volumes.
We need to find the ratio of their heights.
So,
Let the radius of the cone and hemisphere be $x \ cm.$
Also, height of the hemisphere is equal to the radius of the hemisphere.
Now, let the height of the cone $= h\ \ cm$
So, the ratio of the height of cone to the height of the hemisphere $=\frac{\text{h}}{\text{x}}$
Here Volume of the hemisphere $=$ volume of the cone
$\Big(\frac{2}{3}\Big)\pi\text{r}^3_\text{h}=\Big(\frac{1}{3}\Big)\pi\text{r}^2_\text{c}\text{h}$
$\Big(\frac{2}{3}\Big)\pi(\text{x})^3=\Big(\frac{1}{3}\Big)\pi(\text{x})^2\text{h}$
$\Big(\frac{2}{3}\Big)(\text{x})=\Big(\frac{1}{3}\Big)\text{h}$
$2\text{x}=\text{l}\text{h}$
$\frac{\text{h}}{\text{x}}=\frac{2}{1}$
Therefore, the ratio of the heights of the cone and the hemisphere is $2 : 1$.
So, the correct option is $(b)$.
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MCQ 151 Mark
The ratio of the total surface area of a sphere and a hemisphere of same radius is :
  • A
    $2 :1$
  • B
    $3 : 2$
  • C
    $4 : 1$
  • ✓
    $4 : 3$
Answer
Correct option: D.
$4 : 3$
Total surface area of sphere $=4\pi\text{r}^2$
Total surface area of hemisphere $3\pi\text{r}^2$
$\therefore$ Required ratio $=\frac{4\pi\text{r}^2}{3\pi\text{r}^2}=\frac{4}{3}=4:3$
Hence, correct option is $(d)$.
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