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Mensuration · Tamil Nadu Board STD 10 (Tamil - English Medium). 15 questions.

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Choose the most appropriate answer from the given alternatives and write the option code and the corresponding answer.

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15 questions · timed · auto-graded

MCQ 11 Mark
A solid sphere of radius x cm is melted and cast into a shape of a solid cone of the same radius. The height of the cone is …………
  • A
    3x cm
  • B
    x cm
  • ✓
    4x cm
  • D
    2x cm
Answer
Correct option: C.
4x cm
$4 x cm$

Explanation :
Hint:
Radius of a sphere $=$ Radius of a cone $=x cm$

Volume of a cone $=$ Volume of a sphere
$
\begin{aligned}
& \frac{1}{3} \pi x^2 h =\frac{4}{3} \pi x^3 \\
& x ^2 h =4 x ^3 \\
& x ^2 h =\frac{4 x^3}{x^2} \\
& =4 xcm .
\end{aligned}
$
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MCQ 21 Mark
The total surface area of a hemisphere is how many times the square of its radius
  • A
    π
  • B
    4π
  • ✓
    3π
  • D
    2π
Answer
Correct option: C.
3π
3π
Explanation;
Hint:
T.S.A of the hemisphere $=3 \pi r^2$
The square of the radius is $3 \pi$ times.
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MCQ 31 Mark
If the radius of the base of a cone is tripled and the height is doubled then the volume is …………..
  • A
    made 6 times
  • ✓
    made 18 times
  • C
    made 12 times
  • D
    unchanged
Answer
Correct option: B.
made 18 times
made 18 times

Explanation :
Hint:
Radius of a cone $=r$
Height of a cone $=h$
Volume of the cone $=\frac{1}{3} \pi r^2 h$ cu.units
When the radius is increased three-time (tripled) and the height is doubled
Radius is $3 r$ and the height is $2 h$

Volume of the new cone
$
\begin{aligned}
& =\frac{1}{3} \pi(3 \pi)^2 \times 2 h \\
& =\frac{1}{3} \pi \times 9 \pi^2 \times 2 h
\end{aligned}
$
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MCQ 41 Mark
In a hollow cylinder, the sum of the external and internal radii is 14 cm and the width is 4 cm. If its height is 20 cm, the volume of the material in it is ………
  • A
    $5600 \pi \mathrm{cm}^3$
  • ✓
    $11200 \pi \mathrm{cm}^3$
  • C
    $56 \pi \mathrm{cm}^3$
  • D
    $3600 \pi \mathrm{cm}^3$
Answer
Correct option: B.
$11200 \pi \mathrm{cm}^3$
$11200 \pi \mathrm{cm}^3$
Explanation;
Hint:
Here, let the external radius be "R" and the internal radius be "r"
R + r = 14 ...(1)
Width (R – r) = 4 ...(2)
Height of the hollow cylinder = 20 cm
Volume of the hollow cylinder $= πh × (R^2– r^2)$
= πh(R + r) (R – r)
= π × 20 (14) × 4
= π × 1120
$= 1120π cm^3$
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MCQ 51 Mark
The total surface area of a cylinder whose radius is $\frac{1}{3}$ of its height is …….
  • A
    $\frac{9 \pi h ^2}{8}$ sq.units
  • B
    $24 \pi h^2$ sq.units
  • ✓
    $\frac{8 \pi h ^2}{9}$ sq.units
  • D
    $\frac{56 \pi h ^2}{9}$ sq.units
Answer
Correct option: C.
$\frac{8 \pi h ^2}{9}$ sq.units
$\frac{8 \pi h ^2}{9}$ sq.units

Explanation :
Hint:
Let the height of the cylinder be " $h$ "
Radius of the cylinder $=\frac{1}{3} h$
T.S.A of the cylinder $=2 \pi r(h+r)$
$
\begin{aligned}
& =2 \pi \times \frac{ h }{3}\left( h +\frac{ h }{3}\right) \\
& =2 \pi \times \frac{ h }{3}\left(\frac{4 h }{3}\right) \\
& =\frac{8 \pi h ^2}{9} \text { sq.units }
\end{aligned}
$
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MCQ 61 Mark
If the radius of the base of a right circular cylinder is halved keeping the same height, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is ……..
  • A
    (1:2)
  • ✓
    (1:4)
  • C
    (1:6)
  • D
    (1:8)
Answer
Correct option: B.
(1:4)
$1: 4$

Explanation;
Hint:
Let the radius of the cylinder be " $r$ " and the height be " $h$ " Radius of the new cylinder $=\frac{ r }{2} \ldots($ Height will be same $)$

Volume of the new cylinder : Volume of the original cylinder
$
\begin{aligned}
& =\pi r _1^2 h : \pi r _2^2 h \ldots(\pi h \text { is same }) \\
& =\pi_1^2: r _2^2 \\
& =\left(\frac{ r }{2}\right)^2: r ^2 \\
& =\frac{ r ^2}{4}: r ^2=\frac{1}{4}: 1 \\
& =1: 4
\end{aligned}
$
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MCQ 71 Mark
The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be
  • ✓
    12 cm
  • B
    10 cm
  • C
    13 cm
  • D
    5 cm
Answer
Correct option: A.
12 cm
$12 cm$

Explanation :
Hint:

Here $r=5 cm$ and $I =13 cm$
$
\begin{aligned}
& h=\sqrt{1^2-r^2} \\
& =\sqrt{13^2-5^2} \\
& =\sqrt{169-25} \\
& h=\sqrt{144} \\
& =12 cm
\end{aligned}
$
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MCQ 81 Mark
If two solid hemispheres of same base radius r units are joined together along with their bases, then the curved surface area of this new solid is ______.
  • ✓
    $4πr^2$ sq.units
  • B
    $6πr^2$ sq.units
  • C
    $3πr^2$ sq.units
  • D
    $8πr^2$ sq.units
Answer
Correct option: A.
$4πr^2$ sq.units
If two solid hemispheres of same base radius r units are joined together along with their bases, then the curved surface area of this new solid is $4πr^2$ sq.units.
Explanation:
Because a hemisphere's curved surface area is, we unite two solid hemispheres along their bottoms of radii r to form a solid sphere.
Hence the curved surface area of new solid $ = 2πr^2+ 2πr^2= 4πr^2$
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MCQ 91 Mark
The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is ………..
  • A
    60π cm$^2$
  • B
    68π cm$^2$
  • C
    120π cm$^2$
  • ✓
    136π cm$^2$
Answer
Correct option: D.
136π cm$^2$
$136 cm ^2$

Explanation :
Hint:
Here, $h=15 cm , r=8 cm$
$
\begin{aligned}
& I =\sqrt{ h ^2+ r ^2} \\
& =\sqrt{15^2+8^2} \\
& =\sqrt{225+64} \\
& =\sqrt{289} \\
& =17
\end{aligned}
$
C.S.A of a cone $=\pi r l$ sq.units.
$
=\pi \times 8 \times 17
$
= 136π cm$^2$
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MCQ 101 Mark
The ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height is
  • A
    (1 : 2 : 3)
  • B
    (2 : 1 : 3)
  • C
    (1 : 3 : 2)
  • ✓
    (3 : 1 : 2)
Answer
Correct option: D.
(3 : 1 : 2)
$3: 1: 2$

Explanation;
Hint:
Volume of (cylinder: cone : sphere)
$
=\pi r ^2 h : \frac{1}{3} \pi r ^2 h : \frac{4}{3} \pi r^3
$
$\left(\right.$ Dlvidend by $\left.\pi r ^2\right)= h : \frac{ h }{3}: \frac{4 r }{3}$
But $r =\frac{ h }{2}= h : \frac{ h }{3}: \frac{4 \times h }{3 \times 2}$
$
\begin{aligned}
& = h : \frac{ h }{3}: \frac{2 h }{3} \\
& =3: 1: 2
\end{aligned}
$
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MCQ 111 Mark
The height and radius of the cone of which the frustum is a part are $h_1$ units and $r_1$ units respectively. Height of the frustum is $h_2$ units and the radius of the smaller base is $r_2$ units. If $h_2: h_1= 1 : 2$ then $r_2: r_1$ is
  • A
    (1:3)
  • ✓
    (1:2)
  • C
    (2:1)
  • D
    (3:1)
Answer
Correct option: B.
(1:2)
$1: 2$

Explanation;
Hint:
$
\begin{aligned}
& h_2: h_1=1: 2 \\
& h_1: h_2=2: 1
\end{aligned}
$

Ratio of their volumes
$
\begin{aligned}
& =\frac{1}{3} \pi h _1\left( r _1^2+ r _2^2+ r _1 r _2\right): \frac{1}{3} \pi h _2\left( r _1^2+ r _2^2+ r _1 r _2\right) \\
& =2\left( r _1^2+ r _2^2+ r _1 r _2\right): 1\left( r _1^2+ r _2^2+ r _1 r _2\right)
\end{aligned}
$

Volume is $2: 1$ the ratio of their radius also $2: 1$
$
r_1: r_2=2: 1 \text { But } r_2: r_1=1: 2
$
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MCQ 121 Mark
The volume (in $cm ^3$ ) of the greatest sphere that can be cut off from a cylindrical log of wood of base radius $1 cm$ and height $5 cm$ is
  • ✓
    $\frac{4}{3} \pi$
  • B
    $\frac{10}{3} \pi$
  • C
    $5 \pi$
  • D
    $\frac{20}{3} \pi$
Answer
Correct option: A.
$\frac{4}{3} \pi$
$
\frac{4}{3} \pi
$

Explanation;
Hint:
Radius of the sphere $=1 cm$
Volume of the Sphere $=\frac{4}{3} \pi r^3$ cu.units
$
\begin{aligned}
& =\frac{4}{3} \times \pi \times 1 \times 1 \times 1 cm ^3 \\
& =\frac{4}{3} \pi cm ^3
\end{aligned}
$
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MCQ 131 Mark
A spherical ball of radius $r_1$ units is melted to make 8 new identical balls each of radius $r_2$ units. Then $r_1: r_2$ is
  • ✓
    (2:1)
  • B
    (1:2)
  • C
    (4:1)
  • D
    (1:4)
Answer
Correct option: A.
(2:1)
2:1

Explanation :
Hint:

Volume of the first sphere: Volume of second sphere $=8: 1$
$
\begin{aligned}
& \frac{4}{3} \pi r_1^3: \frac{4}{3} \pi r_2^3=8: 1 \\
& r_1^3: r_2^3=2^3: 1^3 \\
& r_1: r_2=2: 1
\end{aligned}
$
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MCQ 141 Mark
A shuttlecock used for playing badminton has the shape of the combination of ______.
  • A
    a cylinder and a sphere
  • B
    a hemisphere and a cone
  • C
    a sphere and a cone
  • ✓
    frustum of a cone and a hemisphere
Answer
Correct option: D.
frustum of a cone and a hemisphere
A shuttlecock used for playing badminton has the shape of the combination of frustum of a cone and a hemisphere.
Explanation:
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MCQ 151 Mark
A frustum of a right circular cone is of height 16 cm with radius of its ends as 8 cm and 20 cm. Then, the volume of the frustum is
  • ✓
    3328π cm$^3$
  • B
    3228π cm$^3$
  • C
    3240π cm$^3$
  • D
    3340π cm$^3$
Answer
Correct option: A.
3328π cm$^3$
$3328 \pi cm ^3$

Explanation :
Hint:
Here, $h=16 cm , r =8 cm , R =20 cm$

Volume of the frustum
$
\begin{aligned}
& =\frac{1}{3} \pi h \left( R ^2+ r ^2+ Rr \right) \\
& =\frac{1}{3} \pi \times 16\left(20^2+8^2+20 \times 8\right) \\
& =\frac{1}{3} \pi \times 16(400+64+160) \\
& =\frac{1}{3} \pi \times 16 \times 624 \\
& =\pi \times 16 \times 208 \\
& =3328 \pi cm ^3
\end{aligned}
$
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Choose the most appropriate answer from the given alternativ - MATHS STD 10 Questions - Vidyadip