Mathematics Questions

Browse 1,240 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

1. Q681
   A ​solid ​ball is exactly fitted inside the cubical box of ​side $a$. ​The volume ​of the ball is $\frac{4}{3}\pi a^3$. ​Write 'True' or 'False' and justify ​your answer.
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2. Q682
   State ​whether the following statement is true or false and ​justify: The volume of the frustum of a ​cone ​is $\frac{1}{3}\pi h[r_1^2+r_2^2-r_1{r}_2]$, where $h$ is the ​vertical height ​of the ​frustum and $r_1,r_2$ are the radii of the ​ends.
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3. Q683
   The ​capacity of a ​cylindrical vessel with ​a hemispherical portion raised ​upward ​at the bottom as shown in ​the ​Fig. 12.7 is $\frac{\pi r^2}{3}[3h-2r]$.

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4. Q684
   The curved surface area of a frustum of a cone ​is $\pi l(r_1+r_2)$, where $l=\sqrt{h^2+(r_1+r_2{)}^2}$, $r_1$ ​and $r_2$ are ​the radii of the two ends of the frustum and $h$ is ​the vertical ​height.
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5. Q685
   An open metallic bucket is in the shape of a frustum of a cone, ​mounted on a hollow cylindrical base made ​of the same ​metallic ​sheet. The surface area of the ​metallic sheet used is equal ​to curved surface area of frustum of a ​cone + area of circular base + curved surface area of cylinder.
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6. Q686
   Three metallic solid ​cubes whose edges are 3 cm, ​4 ​cm and 5 ​cm ​are melted and formed into a single cube. Find the edge ​of ​the cube ​so formed.
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7. Q687
   How many shots each ​having diameter ​3 ​cm ​can be ​made from a cuboidal lead solid of dimensions $9\text{ cm}\times 11\text{ cm}\times 12\text{ cm}$?
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8. Q688
   A bucket is in ​the form ​of ​a frustum ​of a cone ​and holds $28.490$ litres of water. The radii of the top and bottom are $28$ cm and $21$ cm, ​respectively. Find the height of the bucket.
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9. Q689
   A ​cone of radius 8 cm and height 12 cm ​is divided into two parts by a plane through the ​mid-point of its ​axis ​parallel to its base. Find the ​ratio of the ​volumes of two parts.
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10. Q690
    Two identical cubes ​each of volume $64{\text{cm}}^3$ are joined ​together end to ​end. ​What ​is the surface ​area of the resulting ​cuboid?
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11. Q691
    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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12. Q692
    Two cones with same base radius ​8 cm and height 15 cm ​are ​joined together ​along ​their bases. ​Find the surface area of the shape so ​formed.
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13. Q693
    Two solid ​cones A and ​B are placed in ​a ​cylindrical tube as shown in Fig. 12.9. The ratio ​of their capacities is 2:1. Find the heights and capacities of the cones. Also, find the volume ​of the remaining portion of the cylinder. ​(The cylindrical tube has length 21 cm and diameter 6 cm.)

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14. Q694
    An ice cream cone full of ice cream having radius 5 ​cm and ​height 10 cm as ​shown in Fig. 12.10. Calculate the volume of ​ice cream, provided that its $\frac{1}{6}$ part ​is left unfilled ​with ​ice cream.

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15. Q695
    Marbles of diameter 1.4 ​cm ​are ​dropped into a cylindrical beaker of ​diameter 7 cm containing some water. Find the number of marbles ​that should be dropped ​into the beaker so that the water level rises by 5.6 cm.
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16. Q696
    How many spherical lead shots each ​of ​diameter 4.2 cm can be obtained from a solid rectangular lead ​piece with dimensions 66 ​cm, 42 cm ​and 21 ​cm.
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17. Q697
    How ​many ​spherical ​lead shots of diameter 4 cm can be made out of ​a ​solid cube of lead whose edge ​measures 44 cm?
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18. Q698
    A wall 24 ​m long, 0.4 m thick and 6 m high is constructed ​with ​the bricks ​each of dimensions $25\text{ cm}\times 16\text{ cm}\times 10\text{ cm}$. ​If ​the mortar ​occupies $\frac{1}{10}$th of the volume of the wall, then find the number ​of bricks used in constructing the wall.
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19. Q699
    Find the number ​of metallic circular discs ​with 1.5 cm base diameter and ​of height 0.2 cm to be melted to form a ​right circular cylinder ​of height 10 cm ​and diameter 4.5 cm.
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20. Q700
    In ​the formula $\bar{x}=a+\frac{\sum f_i{d}_i}{\sum f_i}$, ​for finding ​the mean of grouped data $d_i$'s are ​deviations ​from $a$ of

    1. (a)lower limits of the classes
    2. (b)upper limits of the classes
    3. (c)mid points of the classes
    4. (d)frequencies of the class marks
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