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In a right circular cone, the cross-section made by a plane parallel to the base is a

(A) circle                                                         (B) frustum of a cone

(C) sphere                                                       (D) hemisphere

According to the question if a right circular cone is cut by a plane parallel to its base the figure formed is

 Here BECD is not a circle, not a sphere not a hemisphere but it is a frustum of a cone.

 Hence in a right circular cone, the cross-section made by a plane parallel to the base is a frustum of a cone.

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If two solid hemispheres of same base radius r are joined together along their bases, then curved surface area of this new solid is

(A) 4 \pi r^{2}                      (B) 6 \pi r^{2}                  (C) 3 \pi r^{2}                      (D) 8 \pi r^{2}

           

         

 

  The radius of the hemisphere = r

      \text{ Curved surface area} =2 \pi r^{2}

            The curved surface area of two solid hemisphere

                        =2 \times 2 \pi r^{2}

                        =4 \pi r^{2}

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A medicine capsule is in the shape of a cylinder of diameter 0.5 cm with two hemispheres stuck to each of its ends. The length of the entire capsule is 2 cm. The capacity of the capsule is

(A) 0.36 cm3               (B) 0.35 cm3                (C) 0.34 cm3                (D) 0.33 cm3

 

            Diameter of hemisphere = 0.5 cm

        Radius =\frac{0.5}{2}=\frac{5}{20}=\frac{1}{4}=0.25cm

       \text{ Volume of hemisphere }=\frac{2}{3}\pi r^{3}

                        = \frac{2}{3}\times\frac{22}{7}\times \frac{1}{4}\times\frac{1}{4}\times \frac{1}{4}=\frac{11}{336}

        \text{ Volume of two hemispheres} =\frac{2 \times11}{336} =\frac{11}{168}

            Similarly radius of the cylinder = 0.25

            Height  = 2- 0.25 -0.25

                        = 2- 0.5

                        =1.5 cm

        Volume = \pi r ^{2}h

                        = \frac{22}{7}\times \frac{1}{4}\times\frac{1}{4}\times\frac{15}{10}\Rightarrow \frac{33}{112}

            The total volume of the capsule = volume of two hemispheres + volume of the cylinder

                        =\frac{11}{168}+\frac{33}{112}

                        =0.065 +0.294

                        = 0.359cm3

                        =0.36cm3   (approximate)

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A shuttle cock used for playing badminton has the shape of the combination of

(A) a cylinder and a sphere

(B) a cylinder and a hemisphere

(C) a sphere and a cone

(D) frustum of a cone and a hemisphere

Answer (D) frustum of a cone and a hemisphere

            Cylinder – A cylinder is a three dimensional solid that has two parallel based joined by a curved surface at a fixed distance.

            Sphere – The set of all points in the dimensional space lying the same distance from a given point.

            Similarly hemisphere

            Cone – A cone is a three-dimensional geometrical shape that takes smoothly from a flat base to a point called the apex or vertex.

 

            Similarly, the frustum of cone

            Hence we conclude that shuttle cock which is used for playing badminton has the shape of a combination of the frustum of a cone and hemisphere.

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The shape of a gilli, in the gilli-danda game (See figure), is a combination of

(A) two cylinders                                            (B) a cone and a cylinder

(C) two cones and a cylinder                          (D) two cylinders and a cone

Answer

  1. Option (A): Two cylinders

    • This option is incorrect because the gilli does not have two cylindrical parts. It has one cylindrical section.

  2. Option (B): A cone and a cylinder

    • This option is not fully accurate because there are two cones (one on each end) along with the cylinder.

  3. Option (C): Two cones and a cylinder

    • This option correctly describes the shape of the gilli as it has a cylindrical middle with a cone at each end.

  4. Option (D): Two cylinders and a cone

    • This option does not fit because the gilli does not have two cylindrical sections.

Correct Answer

The correct option is:

(C) Two cones and a cylinder

 

 

 

 

Answer(C) two cones and a cylinder

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The shape of a glass (tumbler) (see figure) is usually in the form of

(A) a cone                                                       (B) frustum of a cone

(C) a cylinder                                                  (D) a sphere

Answer (B) frustum of a cone

Solution

A cone – A cone is a three-dimensional geometrical shape that tapers smoothly from a flat base to a point called the apex or vertex.

            (B) Frustum of a cone – It is a portion of a solid that lies between one or two parallel planes cutting it.

            (C) A cylinder – A cylinder is a three-dimensional solid that has two parallel bases joined by a curved surface at a fixed distance.

            (D) A sphere – The set of all points in three-dimensional space lying the same distance from a given point.  

            Hence the shape of glass is usually in the form of the frustum of a cone.

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A plumbline (Sahul) is the combination of

                                                

 

 

 

(A) a cone and a cylinder                                (B) a hemisphere and a cone

(C) frustum of a cone and a cylinder              (D) sphere and cylinder

Answer 

The bob (the weight at the end of the string) typically combines different geometric shapes to ensure it hangs straight and minimizes air resistance. Let's analyze the options provided to see which one best matches the typical shape of a plumbline bob:

  1. Option (A): A cone and a cylinder
    This combination is plausible because many plumbline bobs have a pointed cone-like bottom to ensure accuracy and a cylindrical section for stability and weight distribution.

  2. Option (B): A hemisphere and a cone
    This combination is less common for a plumbline bob. A hemisphere would not provide the same stability or downward-pointing accuracy as a cone.

  3. Option (C): Frustum of a cone and a cylinder
    This combination is frequently used. A frustum of a cone (a cone with the top cut off) provides a stable weight, and a cylinder ensures that the bob has a streamlined shape and sufficient mass to hang steadily.

  4. Option (D): A sphere and a cylinder
    This combination is unlikely for a plumbline bob because a sphere does not offer a downward-pointing feature, making it less precise for determining vertical alignment.

Correct Answer

The best answer is:

(C) A frustum of a cone and a cylinder

This combination provides both the stability and the necessary weight distribution for accurate vertical alignment.

 

Answer(B) a hemisphere and a cone

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A Surahi is a combination of

(A) a sphere and a cylinder                 (B) a hemisphere and a cylinder

(C) two hemispheres                           (D) a cylinder and a cone.

Answer [A]

Solution

            (A) A Sphere – The set of all points in three-dimensional space lying the same distance from a given point.

            A cylinder – A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface at a fixed distance.

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A cylindrical pencil sharpened at one edge is a combination of

(A) a cone and a cylinder                                (B) frustum of a cone and a cylinder

(C) a hemisphere and a cylinder                     (D) two cylinders.

Answer(A) a cone and a cylinder      

Solution

            (A) A cone –A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

            A cylinder – A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface at a field distance.

            (B) Frustum of a cone – It is the portion of a solid that lies between one or two parallel planes cutting it.

            A cylinder – A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface at a fixed distance.

            (C) Hemisphere – In geometry, it is an exact half of a sphere and it is a three-dimensional geometric.

            A cylinder – A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface at a fixed distance.

            (D) Cylinder – A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface at a fixed distance.

(Combination of two cylinders)

            Hence from the above diagrams, we conclude that option (A) is correct.

            Therefore, A cylindrical pencil sharpened at one edge is the combination of a cone and a cylinder.

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The diameters of the two circular ends of the bucket are 44 cm and 24 cm. The height of the bucket is 35 cm. The capacity of the bucket is

(A) 32.7 litres
(B) 33.7 litres
(C) 34.7 litres
(D) 31.7 litres


\text{Volume of frustum of cone }=\frac{\pi h}{3}(r_{1}^{2}+r_{2}{^2}+r_{1}r_{2})
Diameter of first end = 44 cm
Radius

 (r_{1})=\frac{44}{2}=22cm
Diameter of second end = 24 cm
Radius (r_{2})=\frac{24}{2}=12cm
 \text{Height }(h)=35cm
\text{Volume} =\frac{\pi h}{3}(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})
             =\frac{22\times 35}{7\times 3}(22^{2}+12^{2}+22\times 12)   
     \Rightarrow \frac{110}{3}[484+144+264]
            \frac{110}{3}[892]
        
          =32706.67cm^{3}
 \text{We know that 1 liter}= 1000 cm^{3}???????
\therefore 32706.67cm^{3}=32.7litres

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