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Cone cut by a plane parallel to its base: similar cones, volumes as cubes of heights

This type of question, asked 5 times: CDS 2 2016, CDS 2 2017, CDS 1 2018, CDS 2 2022, CDS 2 2023. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2016

    58
    The height of a cone is 60 cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 164\frac{1}{64} the volume of original cone. What is the height from the base at which the section is made ?

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  2. Question 2

    CDS 2 2017

    73
    The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone ?

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  3. Question 3

    CDS 1 2018

    69
    A right circular cone is sliced into a smaller cone and a frustum of a cone by a plane perpendicular to its axis. The volume of the smaller cone and the frustum of the cone are in the ratio 64 : 61. Then their curved surface areas are in the ratio

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  4. Question 4

    CDS 2 2022

    81
    The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is 127\frac{1}{27} of the volume of the given cone, then what is the height of the frustum of the cone ?

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  5. Question 5

    CDS 2 2023

    52
    A lamp shade is in the shape of a part of a cone and its top and bottom ends are circles whose circumferences are respectively 30 cm and 40 cm. The perpendicular distance between the ends is 6 cm. If the cone were to be completed, then how far would its vertex be from the top end ?

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