Maths Class X Question Bank

Question 1 of 50

1. Question
1 point(s)
Find the radius of the sphere whose surface area is $154\ cm^2 $.
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Hint

$4\pi r^2=154\\ \\ \Rightarrow r^2=\dfrac{49}{4}=3.5\ cm\\ \\ \Rightarrow r=\sqrt{\dfrac{49}{4}}=\dfrac{7}{2}=3.5\ cm$
Question 2 of 50

2. Question
1 point(s)
The dimensions of a box are $12\ cm \times 4\ cm \times 3\ cm$. Find the length of the longest rod which can be placed in this box.
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Hint

length of longest rod $=\sqrt{12^2+4^2+3^2}$
$=\sqrt{144+16+9}\\ \\ =\sqrt{144+25}=13\ cm$
Question 3 of 50

3. Question
1 point(s)
How many cubes of sides $3\ cm $ can be cut from a cuboid measuring $18\ cm \times 12\ cm \times 9\ cm$?
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Hint

Number of cubes $=\dfrac{18\times12\times9}{3\times3\times}$
$=\dfrac{1944}{27}=72$
Question 4 of 50

4. Question
1 point(s)
Find the height of the cone whose volume is $1570\ cm^3$ and area of base is $314\ cm^2$.
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Hint

$\dfrac{1}{3}\pi r^2h=1570\\ \\ \Rightarrow \dfrac{1}{3}\times314\times h=1570\\ \\ \Rightarrow h=\dfrac{4710}{314}=15\ cm$
Question 5 of 50

5. Question
1 point(s)
The circumference of the edge of a hemispherical bowl is 132 cm. Find the capacity of the bowl.
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Hint

$2\pi r=132\ \Rightarrow r=\dfrac{132\times7}{44}=21\\ \\ V=\dfrac{2}{3}\pi r^3=\dfrac{2}{3}\pi\ (21)^3=19404\ cm^3$
Question 6 of 50

6. Question
1 point(s)
If the lateral surface area of a cylinder is $94.2\ cm^2$, and its height is $5\ cm$, then find the radius of its base.
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Hint

$2\pi rh=94.2\\ \\ \Rightarrow 2\times3.14\times r\times 5=94.2\\ \\ \Rightarrow r=3\ cm$
Question 7 of 50

7. Question
1 point(s)
Two cubes each of edge 4 cm are joined face to face. Find the surface area of the resulting cuboid.
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Hint

$l=8\ cm,\ b=4\ cm,\ h=4\ cm$
Surface area $=2(b+bh+lh)$
$=2(8\times4+4\times4+4\times4)=160\ cm^2$
Question 8 of 50

8. Question
1 point(s)
The diameter of a garden roller is 1.4 m and it is 2 m long. How much area will it cover in 5 revolutions?
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Hint

Area = CSA $\times$ no. of revolution
$\Rightarrow r=\dfrac{1.4}{2}=0.7\ m\\ \\ \Rightarrow h=2\ m$
CSA $=2\pi rh=2\times\dfrac{22}{7}\times0.7\times2=8.8\ m^2$
Area covered $=8.8\times5=44\ m^2$
Question 9 of 50

9. Question
1 point(s)
The largest possible sphere is carved out from a solid cube of side 7 cm. Find the volume of the sphere.
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Hint

Radius of the sphere $=\dfrac{7}{2}\ cm$
Volume $=\dfrac{4}{3}\times\dfrac{22}{7}\times\dfrac{7}{2}\times\dfrac{7}{2}\times\dfrac{7}{2}$
$=179.67\ cm^3$
Question 10 of 50

10. Question
1 point(s)
A right circular cone is 3.6 cm high and has base radius 1.6 cm. It is melted and recast into a right circular cone with radius of its base as 1.2 cm. Find its height.
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Hint

$\dfrac{1}{3}\pi\ (1.6)^2\times3.6=\dfrac{1}{3}\pi\ (1.2)^2\times h\\ \\ \Rightarrow h=6.4\ cm$
Question 11 of 50

11. Question
1 point(s)
Find the ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height.
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Hint

Volume of sphere $=\dfrac{4}{3}\pi\left(\dfrac{d}{2}\right)^2=\dfrac{4}{3}\pi\dfrac{d^3}{8}$
Volume of cylinder $=\pi\left(\dfrac{d}{2}\right)^2d=\pi\dfrac{d^3}{4}$
Volume of cone $=\dfrac{1}{3}\pi\left(\dfrac{d}{2}\right)^2d$
$=\dfrac{1}{3}\pi\dfrac{d^3}{4}$
Ratio $=\dfrac{4}{3}\times\dfrac{1}{3}=2:3:1$
Question 12 of 50

12. Question
1 point(s)
A cone and a sphere have equal radii and equal volume. What is the ratio of the diameter of the sphere to the height of the cone?
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Hint

$\dfrac{1}{3}\pi r^2h=\dfrac{4}{3}\pi R^3\\ \\ \Rightarrow h=4R=4.\dfrac{D}{2}=2D\\ \\ \Rightarrow \dfrac{D}{h}=\dfrac{1}{2}$
Question 13 of 50

13. Question
1 point(s)
What is the ratio of the volume of a cube to that of a sphere which will fit exactly inside the cube?
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Hint

$\dfrac{Volume\ of\ cube}{Volume\ of\ sphere\ inside\ cube}=\dfrac{a^3}{\dfrac{4}{3}\pi\dfrac{a^3}{8}}=\dfrac{6}{11}$
Question 14 of 50

14. Question
1 point(s)
A solid cylinder of radius r and height h is placed over other cylinder of same height and radius. Find the total surface area of the shape so formed.
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Hint

Height of new cylinder $=2h$
$TSA=2\pi r(2h)+2\pi r^2\\ \\ =4\pi rh+2\pi r^2$
Question 15 of 50

15. Question
1 point(s)
A solid cone of base radius r and height h is placed over a solid cylinder having same base radius and height as that of the cone. Find the curved surface area and the total surface area of the shape thus formed.
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Hint

$TSA=\pi rl+2\pi rh+\pi r^2\\ \\ =\pi r(l+2h+r)\\ \\ =\pi r\ (\sqrt r^2+h^3+2h+r)$
Question 16 of 50

16. Question
1 point(s)
A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have ? Find the surface area of the solid.
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Hint

Greater diameter = side of cube = 7 cm
Surface area of cube $=6\times7\times7=294\ cm^2$
Greater area of base of hemisphere $=\pi r^2$
$=\dfrac{22}{7}\times3.5^2=38.5\ cm^2$
CSA of hemisphere $=2\times38.5=77\ cm^2$
$TSA=294-38.5+77=332.5\ cm^2$
Question 17 of 50

17. Question
1 point(s)
A solid cube is cut into two cuboids of equal volumes. Find the ratio of the total surface area of the given cube and that of one of the cuboids.
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Hint

Length of each resulting cuboids $=\dfrac{a}{2}$
$TSA=2\ \left(\dfrac{a}{2}\times a+a\times a+\dfrac{a}{2}\times a\right)=4a^2$
Let the side of the cube $=a$
$TSA=6a^2\\ \\ =\dfrac{6a^2}{4a^2}=\dfrac{3}{2}$
Question 18 of 50

18. Question
1 point(s)
Find the capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom as shown in the figure.
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Hint

Capacity of vessel = Capacity of cylinder – capacity of hemispher
$=\pi r^2h-\dfrac{2}{3}\pi r^3=\pi\dfrac{r^2}{3}\ (3h-20)\ cm^3$
Question 19 of 50

19. Question
1 point(s)
A spherical iron ball of radius r is melted to make 8 new idential balls. Find the radius of each new ball.
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Hint

$\dfrac{4}{3}\pi R^3=8\times\dfrac{4}{3}\pi r^3\\ \\ \Rightarrow R^3=8\times r^3\\ \\ \Rightarrow R=2r\\ \\ \Rightarrow r=\dfrac{R}{2}$
Radius of new ball is half of radius of spherical ball
Question 20 of 50

20. Question
1 point(s)
The circumference of the base of a 9 m high wooden solid cone is 44 m. Find the volume of the cone.
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Hint

$2\pi r=44\ m\\ \\ \Rightarrow r=\dfrac{44\times7}{44}=7\ m$
Volume of cone $=\dfrac{1}{3}\pi r^3$
$=\dfrac{1}{3}\times\dfrac{22}{7}\times7^3=359.33\ m^3$
Question 21 of 50

21. Question
1 point(s)
Radius of a cone is 4 cm. Find the height of the cone so that its volume may be equal to that of a sphere of radius 2 cm.
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Hint

Volume of sphere $=\dfrac{4}{3}\times\dfrac{22}{7}\times2\times2\times2$
$=\dfrac{704}{21}=33.52$
A/Q $\dfrac{1}{3}\pi r^2h=33.52\Rightarrow \dfrac{1}{3}\times3.14\times16\times h=33.52$
$\Rightarrow h=\dfrac{33.52\times3}{3.14\times16}=\dfrac{100.56}{50.24} =2.001=2\ cm$
Question 22 of 50

22. Question
1 point(s)
The height of a cone is 5 cm. Find the height of another cone whose volume is sixteen times its volume and radius equal to its diameter.
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Hint

$h=5\ cm,\ r=2\ cm$
Volume $=\dfrac{1}{3}\times\dfrac{22}{7}\times2\times2\times5$
Volume is 16 times
$=16\times\dfrac{1}{3}\times\dfrac{22}{7}\times2\times2\times5$
Radius is half $=\dfrac{1}{2}\times2=1\ cm$
A/Q
$=16\times\dfrac{1}{3}\times\dfrac{22}{7}\times2\times2\times5\\ \\ =\dfrac{1}{3}\times\dfrac{22}{7}\times1\times 1\times h\\ \\ \Rightarrow h=320\ cm$
Question 23 of 50

23. Question
1 point(s)
If the radii of the ends of a bucket 24 m high are 5 cm and 15 cm, find the surface area of the bucket.
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Hint

$l=\sqrt{(R-r)^2+h^2}=\sqrt{(15-5)^2+24^2}=26\ cm$
Surface area of bucket $=\pi (R+r)l+\pi r^2$
$=\pi(5+5)\times26+\pi\times5^2\\ \\ =\pi\ (520+25)=545\pi \ cm^2$
Question 24 of 50

24. Question
1 point(s)
The sum of length, breadth and height of a cuboid is 19 cm and its diagonal is $5\sqrt5\ cm $ What is its surface area?
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Hint

$l+b+h=19$ and $\sqrt{l^2+b^2+h^2}=5\sqrt5\\ \\ \Rightarrow l^2+b^2+h^2=(5\sqrt5)=125$
Surface area of cuboid $=2(lb+bh+lh)$
$=19^2-125=236\ cm^2$
Question 25 of 50

25. Question
1 point(s)
A cylindrical vessel of radius 4 cm contains water. A solid sphere of radius 3 cm is covered into the water untill it is completely immersed. Find the rise in the water level in the vessel.
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Hint

$\pi r^2h=\dfrac{4}{3}\pi r^3\Rightarrow 16\pi rh=\dfrac{4}{3}\pi r^3\\ \\ \Rightarrow h=\dfrac{4}{3}\times\dfrac{27}{16}=\dfrac{36}{16}=2.25\ cm$
Question 26 of 50

26. Question
1 point(s)
A hemisphere of iron of radius 12 cm is melted and cast into a right circular cone of height 54 cm. Find the radius of the base of the cone.
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Hint

$\dfrac{2}{3}\pi R^3=\dfrac{1}{3}\pi r^2h$
Substituting the values $R=12\ cm,\ h=5\ x \ cm$
we get $r=8\ cm$
Radius of the base of cone $8\ cm$
Question 27 of 50

27. Question
1 point(s)
The radius and slant height of a right circular cone are in the ratio of 7 : 13 and its curved surface area is $ 286\ cm^2$. Find its radius $\left(use\ \pi=\dfrac{22}{7}\right) $
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Hint

$r=7a,\ l=13a\\ \\ CSA=\pi\ (7a)\ (13a)=91\pi a^2\\ \\ 9\pi a^2=28.6\Rightarrow a^2=\dfrac{28.6}{91}\times\dfrac{7}{22}\\ \\ \Rightarrow a^2=1\Rightarrow a=1$
Radius $=7\times1=7\ cm$
Question 28 of 50

28. Question
1 point(s)
A toy is in the form of a cone mounted on a hemisphere of common base radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.
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Hint

Height of cone $=31-7=24\ cm$
$r=7\ cm\\ \\ l=\sqrt{r^2+h^2}=\sqrt{7^2+24^2}=25\ cm$
Surface area $=2\pi r^2+\pi rl$
$=\dfrac{22}{7}\times7\times(14+25)=858\ cm^2$
Question 29 of 50

29. Question
1 point(s)
A solid cone of radius 4 cm and vertical height 3 cm has to be painted from outside except the base. Find the surface area to be painted.
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Hint

$LSA=\pi rl\\ \\ L=\sqrt{3^2+4^2}=5\\ \\ LSA=\dfrac{22}{7}\times4\times5\\ \\ =62.831\ cm^2$
Question 30 of 50

30. Question
1 point(s)
Three cubes of volume $64\ cm^3 $ each are joined end to end to form a solid. Find the surface area of the cuboid so formed.
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Hint

Side of each cube $=\sqrt[3]{64}=4\ cm$
Dimensions of cuboid $=4\times4\times12$
Surface area $=2(4\times4+4\times12+12\times4)=224\ cm^2$
Question 31 of 50

31. Question
1 point(s)
The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, find the height of the frustum.
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Hint

$R-r=4,\ l=5\\ \\ l^2=(R-r)^2+h^2\Rightarrow 5^2=4^2+h^2\\ \\ \Rightarrow 25-16=h^2\\ \\ \Rightarrow h=3\ cm$
Question 32 of 50

32. Question
1 point(s)
The volume of a right circular cylinder of height $7\ cm $ is $567\ \pi\ cm^3 $. Find its curved surface area. $\left(Take\ \pi=\dfrac{22}{7}\right) $.
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Hint

$\pi r^2h=567\ \pi\\ \\ \Rightarrow \pi r^2\times7=567\pi\\ \\ r^2=81\ \Rightarrow r=9\ cm\\ \\ CSA=2\pi rh=2\times\dfrac{22}{7}\times9\times7=396\ cm^2$
Question 33 of 50

33. Question
1 point(s)
Find the number of solid cylindrical structures of radius 7 cm and height 10 cm which can be made from a solid cylinder of radius 7 m and height 10 m.
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Hint

$n\times \pi r^2_1h_1=\pi r^2_2\ h_2\\ \\ \Rightarrow n\ r^2_1h_1=r^2_2\ h_2\\ \\ \Rightarrow n=\dfrac{7^2\times10000\times1000}{7^2\times10}=10^6$
Question 34 of 50

34. Question
1 point(s)
A solid cuboidal slab of iron of dimensions $66\ cm \times 20\ cm \times 27\ cm$ is used to cast an iron pipe. If the outer diameter of the pipe is 10 cm and thickness is 1 cm, then calculate the length of the pipe.
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Hint

Volume $=66\times20\times27$
outer radius of pipe $=\dfrac{10}{2}=5\ cm$
Inner radius $=(5-1)=4\ cm$
volume of pipe $=\pi(R^2-r^2)h$
$\Rightarrow \pi(5^2-4^2)h=\dfrac{22}{7}(25-16)h\\ \\ =\dfrac{22}{7}\times9\times h$
A/Q $\dfrac{22}{7}\times 9\times h=66\times 20\times27\\ \\ \Rightarrow h=20\times9\times7=1260\ cm$
Question 35 of 50

35. Question
1 point(s)
The circumference of the circular end of a hemispherical bowl is 132 cm. Find the capacity of the bowl.
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Hint

$2\pi r=132\ \Rightarrow r=\dfrac{132\times7}{44}=21\\ \\ V=\dfrac{2}{3}\pi r^3=\dfrac{2}{3}\pi\ (21)^3=19404\ cm^3$
Question 36 of 50

36. Question
1 point(s)
A solid cylinder of radius r and height h is placed over other cylinder of the same height and radius. Find the total surface area of the shape so formed.
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Hint

Height of new cylinder $=2h$
Radius of new cylinder $=r$
Total surface area $=2\pi rh+2\pi r^2$
$=2\pi r(2h)+2\pi r^2$
$=4\pi rh+2\pi r^2$
Question 37 of 50

37. Question
1 point(s)
How many coins $1.75\ cm $ in diameter and 2 mm thick must be melted to form a cuboid of dimensions $11\ cm \times 10\ cm \times 7\ cm$? $\left(Take\ \pi=\dfrac{22}{7}\right) $
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Hint

Diameter $= 1.75\ cm$ , radius $=\dfrac{1.75}{2}$
Volume of coin $=\pi r^2h=\dfrac{22}{7}\times\left(\dfrac{1.75}{2}\right)^2\times0.2$
Volume of cuboid $=11\times 10\times7$
Number of coins $=\dfrac{11\times10\times7}{\dfrac{22}{7}\times\left(\dfrac{1.75}{2}\right)^2}\times0.2=1600$
Question 38 of 50

38. Question
1 point(s)
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 m of standing water is needed?
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Hint

Volume of canal $=5000\times6\times1.5$
$5000\times6\times1.5=$ Area irrigated $\times\dfrac{8}{100}$
$\Rightarrow$ Area irrigated $=\dfrac{5000\times6\times1.5\times100}{8}=562500\ m^2\\ \\ =\dfrac{562500}{10000}=56.25$ hectares
Question 39 of 50

39. Question
1 point(s)
The total surface area of a right circular cone is $90\ \pi\ cm^2 $. If the radius of base of the cone is $5\ cm $, find the height of the cone.
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Hint

$90\pi=\pi\ (rl+r^2)\\ \\ 90-r^2=rl\Rightarrow l=\dfrac{90-25}{5}=13\\ \\ l^2=13^2\\ \\ \Rightarrow r^2+h^2=169\\ \\ \Rightarrow h^2=144\Rightarrow h=12\ cm$
Question 40 of 50

40. Question
1 point(s)
The radius of the base and the height of a right circular cylinder are in the ratio of 2 : 3 and its volume is 1617 cu.cm. Find the curved surface area of the cylinder $\left(use\ \pi=\dfrac{22}{7}\right) $.
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Hint

$\pi\ (2r)^2\ (3x)=1617\\ \\ \Rightarrow x^3=\left(\dfrac{7}{2}\right)^3\Rightarrow x=3.5\ cm\\ \\ r=7\ cm, h=10.3\ cm$
Surface area $=2\pi rh=\dfrac{22}{7}\times2\times7=462\ cm^2$
Question 41 of 50

41. Question
1 point(s)
A solid is hemispherical at the bottom and conical above. If the curved surface area of the two parts are equal, then find the ratio of the radius and height of the conical part.
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Hint

$2\pi r^2=\pi rl\Rightarrow l=2r\\ \\ h=\sqrt{l^2-r^2}=\sqrt{4r^2-r^2}=\sqrt{3r}\\ \\ \Rightarrow \dfrac{h}{r}=\dfrac{\sqrt3}{1}\ \Rightarrow \dfrac{r}{h}=\dfrac{1}{\sqrt3}$
Question 42 of 50

42. Question
1 point(s)
A spherical solid ball of diameter 21 cm is melted and recast into cubes, each of side 1 cm. Find the number of cubes thus formed. $\left(use\ \pi=\dfrac{22}{7}\right) $
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Hint

Volume of sphere $=\dfrac{4}{3}\ \pi r^3$
$=\dfrac{4}{3}\times\dfrac{22}{7}\times\left(\dfrac{21}{2}\right)^3=4851\ cm^3$
Volume of cube $=a^3=1^3$
A/Q
$4851=n\times1\ \Rightarrow n=4851$
Question 43 of 50

43. Question
1 point(s)
A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. $\left(Take\ \pi=\dfrac{22}{7}\right) $
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Hint

Radius of hemisphere $=7\ cm$
Hight of cylinder $=13-7=6 \ cm$
Inner surface area $=(2\pi rh+2\pi r^2)$
$=2\times\dfrac{22}{7}\times7\times(6+7)=572\ cm^2$
Question 44 of 50

44. Question
1 point(s)
The surface area of a sphere is $616\ cm^2$. Find its radius.
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Hint

$4\times\dfrac{22}{7}\times r^2=616\\ \\ \Rightarrow r^2=\dfrac{616\times7}{4\times22}=49\\ \\ \Rightarrow r=7\ cm$
Question 45 of 50

45. Question
1 point(s)
A cylinder and a cone are of same base radius and of same height. Find the ratio of the volume of cylinder to that of the cone.
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Hint

Volume of cone $=\dfrac{1}{3}\pi r^2h$
Volume of cylinder $=\pi r^2h$
Height and radius are equal $\Rightarrow h=r$
Ratio $=3:1$
Question 46 of 50

46. Question
1 point(s)
The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, write the height of the frustum.
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Hint

$R-r=4,\ l=5\\ \\ l^2=(R-r)^2+h^2\Rightarrow 5^2=4^2+h^2\\ \\ \Rightarrow h=3\ cm$
Question 47 of 50

47. Question
1 point(s)
From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out. Find the volume of the remaining solid.
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Hint

Volume of cube $=7^3=343\ cm^3$
Volume of conical $=\dfrac{1}{3}\pi r^2h$
$=\dfrac{1}{3}\times\dfrac{22}{7}\times3^2\times7=65.94\ cm^3$
Volume $=343-65.94=277.60\ cm^3$
Question 48 of 50

48. Question
1 point(s)
The cones with same base radius 8 cm and height 15 cm are joined together their bases. Find the surface area of the shape so formed.
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Hint

Surface area of the shape = CSA of cone 1 + CSA of cone 2
$=2\times\pi rl\\ \\ l=\sqrt{r^2+h^2}=\sqrt{8^2+15^2}=17\ cm$
Surface area $=2\times(\pi\ rl)$
$=2\times\dfrac{22}{7}\times8\times17=855\ cm^2$
Question 49 of 50

49. Question
1 point(s)
A spherical bowl of internal diameter 36 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required to empty the bowl?
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Hint

Volume of hemisphere $=\dfrac{2}{3}\pi\ (18)^3$
Volume of cylindrical bottle $=\pi\times 3^2\times6=(\pi\times9\times6)\ cm^3$
A/Q $\pi\times54\times x=\dfrac{2\pi}{3}\times 18^3\\ \\ \Rightarrow x=72$
Question 50 of 50

50. Question
1 point(s)
A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the cone and of the remaining solid left out after the cone carved out.
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Hint

TSA of cone $=\pi\ rl+\pi r^2$
$=\dfrac{22}{7}\times7\times\sqrt{7^2+14^2}+\dfrac{22}{7}\times7^2$
$=\dfrac{22}{7}\times7\times\sqrt{245}+154\\ \\ =154\ (\sqrt5+1)\ cm^2$
Surface area of cube $=6\times196=1176\ cm^2$
Surface area of remainings solid
$=1176-154+154\sqrt5\\ \\ =(1022+154\sqrt5)\ cm^2$