M.C.Q - Maths STD 9 Questions - Vidyadip

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42 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. $2r = 6cm \Rightarrow r = 3\ cm$
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.
$T.S.A$ $=2\pi\text{r}^2+(\pi\text{r}^2)$ $\{$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 $= hcm$
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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MCQ 161 Mark

A largest sphere of radius 3.5 cm is carved out from a cubical solid. The difference between their surface areas is

A
$224 cm^2$
✓
$140 cm^2$
C
$176 cm^2$
D
$80.5 cm^2$

Answer

Correct option: B.

$140 cm^2$

(B) $140 cm^2$
We have, $r=$ radius of the sphere $=3.5 cm$
$\therefore \quad a=\text { Length of an edge of the cubical solid }=2 r=7 cm$
Hence, required difference
$=6 a^2-\frac{4}{3} \pi r^3=6 \times 7^2-4 \times \frac{22}{7} \times\left(\frac{7}{2}\right)^2=(294-154) cm ^2 = 140 cm^2$

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MCQ 171 Mark

The surface area of a football is $100 \pi cm^2$. The volume of air in it is

A
$\frac{200 \pi}{3} cm^3$
B
$\frac{350 \pi}{3} cm^3$
✓
$\frac{500 \pi}{3} cm^3$
D
$\frac{400 \pi}{3} cm^3$

Answer

Correct option: C.

$\frac{500 \pi}{3} cm^3$

(C) $\frac{500 \pi}{3} cm^3$
Let the radius of the football be $r cm$. It is given that
$\begin{array}{ll}& 4 \pi r^2=100 \pi \Rightarrow r^2=25 \Rightarrow r=5 \\
\therefore \quad & \text { Volume of air }=\frac{4}{3} \pi \times 5^3 cm^3=\frac{500 \pi}{3} cm^3\end{array}$

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MCQ 181 Mark

The volume of a sphere which is exactly inserted in a cube of edge 6 cm , is

✓
$36 \pi cm^3$
B
$64 \sqrt{3} \pi cm^3$
C
$14 \pi cm^3$
D
$288 \pi cm^3$

Answer

Correct option: A.

$36 \pi cm^3$

(A) $36 \pi cm^3$
Let the $r$ be the radius of the sphere. Then,
Diameter of the sphere $=$ Length of an edge of the cube $\Rightarrow 2 r=6 \Rightarrow r=3 cm$
$\therefore \quad$ Volume of the sphere $=\frac{4}{3} \pi r^3=\frac{4}{3} \pi \times 3^3 cm^3=36 \pi cm^3$

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MCQ 191 Mark

The magnitude of surface area of a sphere is half the magnitude of the volume of the sphere. The diameter of the sphere is

A
3 units
B
6 units
✓
12 units
D
18 units

Answer

Correct option: C.

12 units

(C) 12 unirts
Let $r$ be the radius of the sphere. It is given that
$4 \pi r^2=\frac{1}{2}\left(\frac{4}{3} \pi r^3\right) \Rightarrow r=6 \Rightarrow \text { Diameter }=2 r=12 cm$

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MCQ 201 Mark

A solid metal sphere is cut through the centre in two equal parts. If the radius of the sphere is 7 cm, then the total surface area of each part is

A
$924 cm^2$
B
$231 cm^2$
✓
$462 cm^2$
D
$308 cm^2$

Answer

Correct option: C.

$462 cm^2$

(C) $462 cm^2$
Total surface area of each part $=3 \pi r^2=3 \times \frac{22}{7} \times 7^2 cm^2=462 cm^2$

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MCQ 211 Mark

A solid metallic come of radius 10 cm and height $\frac{2}{5} m$ is melted and recast into a sphere. The radius of the sphere is

A
$10 \sqrt{10} cm$
B
$(10)^{1 / 3} cm$
✓
10 cm
D
8 cm

Answer

Correct option: C.

10 cm

(C) 10 cm
We have, $r=$ radius of the cone $=10 cm, h=$ height of the cone $=\frac{2}{5} m=40 cm$
Let the radius of the sphere be $r cm$. Then,
$\frac{4}{3} \pi r^3=\frac{1}{3} \pi× 10^2×40 \Rightarrow r^3=10^3 \Rightarrow r=10 cm$

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MCQ 221 Mark

A metallic sphere of radius 12 cm is melted and cast into a cone whose base radius is 16 cm. The height of the cone is

✓
27 cm
B
18 cm
C
90 cm
D
270 cm

Answer

Correct option: A.

27 cm

(A) 27 cm
Let h cm be the height of the cone.
Clearly, Volume of the cone $=$ Volume of the sphere
$\Rightarrow \quad \frac{1}{3} \pi \times 16^2, h=\frac{4}{3} \pi \times 12^3 \Rightarrow h=27 cm$.

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MCQ 231 Mark

The volume of a cube which can be inserted exactly in a sphere of radius $\frac{3 \sqrt{3}}{2} cm$ is

A
$24 cm^3$
✓
$27 cm^3$
C
$18 cm^3$
D
$30 cm^3$

Answer

Correct option: B.

$27 cm^3$

(B) $27 cm^3$
Let the length of each edge of the cube be a cm.
Clearly, Diagonal of the cube $=$ Diameter of the sphere $\Rightarrow \sqrt{3} a=3 \sqrt{3} \Rightarrow a=3$
$\therefore$ Volume of the cube $=3^3 cm^3=27 cm^3$

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MCQ 241 Mark

If the volumes of two spheres are in the ratio $8: 27$, then the ratio of their surface areas is

A
$2: 3$
✓
$4: 9$
C
$3: 2$
D
$8: 9$

Answer

Correct option: B.

$4: 9$

(B) $4: 9$
Let $V_1, V_2$ be the volumes and $S_1, S_2$ be the surface areas of two spheres. It is given that $V_1: V_2=8: 27$. We know that$
\left(\frac{V_1}{V_2}\right)^2=\left(\frac{S_1}{S_2}\right)^3=\frac{S_1}{S_2}=\left(\frac{V_1}{V_2}\right)^{2 / 3} \Rightarrow \frac{S_1}{S_2}=\left(\frac{8}{27}\right)^{2 / 3}=\frac{4}{9}
$

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MCQ 251 Mark

If the volume and surface ares of a sphere are numerically equal, then its radius is

A
1 unit
B
2 unit
✓
3 unit
D
4 unit

Answer

Correct option: C.

3 unit

(C) 3 unit
Let $r$ be the radius of the sphere. It is given that$
\text { Volume }=\text { Surface area } \Rightarrow \frac{4}{3} \pi r^3=4 \pi r^2 \Rightarrow r=3 \text { units }
$

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MCQ 261 Mark

The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratio of the surface areas of the balloon in the two cases is

✓
$1: 4$
B
$1: 3$
C
$2: 3$
D
$2: 1$

Answer

Correct option: A.

$1: 4$

(A) $1: 4$
Let $S_1$ and $S_2$ be the surface areas of balloon in two cases. Then, $S_1=3 \pi(6)^2$ and $S_2=3 \pi(12)^2 \Rightarrow S_1: S_2=3 \pi(6)^2: 3 \pi(12)^2=1: 4$

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MCQ 271 Mark

A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. The radius of the sphere is

A
4.2 cm
✓
2.1 cm
C
2.4 cm
D
1.6 cm

Answer

Correct option: B.

2.1 cm

(B) 2.1 cm
Let the radius of the sphere be $r cm$. We find that
Volume of the sphere $=$ Volume of the cone
$\Rightarrow \quad \frac{4}{3} \pi r^3=\frac{1}{3} \pi(2.1)^2 \times 8.4 \Rightarrow r^3=(2.1)^2(2.1) \Rightarrow r=2.1$

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MCQ 281 Mark

If the surface area of a sphere is $144 \pi m^2$, then its volume is

✓
$288 \pi m^3$
B
$188 \pi m^3$
C
$300 \pi m^3$
D
$316 \pi m^3$

Answer

Correct option: A.

$288 \pi m^3$

(A) $288 \pi m^3$
Let $r$ be the radius of the sphere. It is given that$
\begin{array}{ll}
& 4 \pi r^2=144 \pi \Rightarrow r^2=36 \Rightarrow r=6 \\
\therefore \quad & \text { Volume }=\frac{4}{3} \pi r^3=\frac{4}{3} \pi \times 6^3 cm^3=288 \pi m^3
\end{array}
$

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MCQ 291 Mark

A cone and a hemisphere have equal bases and equal volumes the ratio of their heights is

✓
1:2
B
2:1
C
4:1
D
$\sqrt{2}: 1$

Answer

Correct option: A.

1:2

A

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MCQ 301 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

C

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MCQ 311 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
$4 \pi r^3$
B
$\frac{8}{3} \pi r^3$
✓
$2 \pi r^3$
D
$8 \pi r^3$

Answer

Correct option: C.

$2 \pi r^3$

C

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MCQ 321 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

A

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MCQ 331 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.

$\pi: 6$

D

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MCQ 341 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.

$100 \pi$

A

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MCQ 351 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

A

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MCQ 361 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

B

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MCQ 371 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

C

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MCQ 381 Mark

The largest sphere is cut off from a cube of side 6 cm. The volume of the sphere will be

A
$27 \pi cm^3$
✓
$36 \pi cm^3$
C
$108 \pi cm^3$
D
$12 \pi cm^3$

Answer

Correct option: B.

$36 \pi cm^3$

B

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MCQ 391 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

D

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MCQ 401 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

D

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MCQ 411 Mark

The total surface area of a hemisphere of radius r is

A
$\pi r^2$
B
$2 \pi r^2$
✓
$3 \pi r^2$
D
$4 \pi r^2$

Answer

Correct option: C.

$3 \pi r^2$

C

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MCQ 421 Mark

In a sphere the number of faces is

✓
1
B
2
C
3
D
4

Answer

Correct option: A.

1

A

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M.C.Q - Maths STD 9 Questions - Vidyadip