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Cubes Test - 5

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Cubes Test - 5
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  • Question 1
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is coloured red on all of its faces. It is then cut into 64 smaller cubes of equal size. The smaller cubes obtained are now separated.

    How many smaller cubes have no coloured face?
    Solution
    Number of smaller cubes with no face coloured = (n – 2)3 = (4 – 2)3 = 8
  • Question 2
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on all of its faces. It is then cut into 64 smaller cubes of equal size. The smaller cubes obtained are now separated.

    How many smaller cubes will have at least two faces painted red?
    Solution
    Concept: If a cube is cut in to n3 smaller cubes then-
    Number of cubes with 0 side painted= (n - 2)3
    Number of cubes with 1 side painted =6(n - 2)2
    Number of cubes with 2 sides painted= 12(n - 2)
    Number of cubes with 3 sides painted= 8(always)

    In the given question a red color painted cube is cut into 43 smaller cubes.
    So, here n = 4.

    Number of smaller cubes with at least two faces painted red = Number of cubes with three faces painted red + Number of cubes with two faces painted red
    Number of cubes with three faces painted red = 8 (all corners of the cubes)
    Number of cubes with two faces painted red = 12(n - 2) = 12 x 2 = 24
    Number of smaller cubes with at least two faces painted red = 24 + 8 = 32
  • Question 3
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is coloured red on all of its faces. It is then cut into 64 smaller cubes of equal size. The smaller cubes so obtained are now separated.

    How many smaller cubes have only two surfaces painted with red colour?
    Solution
    Number of smaller cubes with two surfaces painted red = (n – 2) 12 = (4 – 2) 12 = 24
  • Question 4
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is coloured red on all of its faces. It is then cut into 64 smaller cubes of equal size. The smaller cubes so obtained are now separated.

    How many smaller cubes have only three surfaces painted with red colour?
    Solution
    Number of smaller cubes with three surfaces painted red = 8
    All the small cubes on the vertices of large cube have three faces coloured.
  • Question 5
    1 / -0
    Directions: One hundred and twenty five cubes of the same size are arranged in the form of a cube on a table. Then, a column of five cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are coloured red. Now, answer the question based on the above statements.

    How many small cubes are there in the solid after the removal of the columns?
    Solution






    Since out of 125 total number of cubes, we removed 4 columns of 5 cubes each, the remaining number of cubes = 125 – (4 5) = 125 – 20 = 105
  • Question 6
    1 / -0
    Directions: One hundred and twenty five cubes of the same size are arranged in the form of a cube on a table. Then, a column of five cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are coloured red. Now, answer the question based on the above statement.

    How many cubes do not have any coloured face?
    Solution

    There are (9 4) = 36 such cubes
  • Question 7
    1 / -0
    Directions: Read the following information and answer the given question.

    One hundred and twenty five cubes of the same size are arranged in the form of a cube on a table. Then, a column of five cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are coloured red.

    How many cubes have only one red face?
    Solution


    There are 9 + (4 × 4) = 25 such cubes
  • Question 8
    1 / -0
    Directions: One hundred and twenty five cubes of the same size are arranged in the form of a cube on a table. Then, a column of five cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are coloured red. Now, answer the question based on the above statements.

    How many cubes have more than two coloured faces each?
    Solution


    None of the cubes in any of the layers 2, 3, 4 or 5 has 3 coloured faces.
    Thus, there are 8 such cubes.
  • Question 9
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent surfaces, black on the surface opposite to the red surfaces and green on the remaining faces. Now, the cube is cut into 64 smaller cubes of equal size.

    How many smaller cubes have only one surface painted?
    Solution
    Since the surfaces of the big cube are painted with some colours, so the number of smaller cubes with only one surface painted is (n – 2) 12 = (4 – 2) × 12 = 2 × 12 = 24.

  • Question 10
    1 / -0
    Directions: Read the following information and answer the question given below.

    A cube is painted red on two adjacent faces, black on the faces opposite to the red faces, and green on the remaining faces. Now, the cube is cut into 64 smaller cubes of equal size.

    How many smaller cubes will have no painted face?
    Solution
    Number of smaller cubes with no painted face = (4 - 2)3 = (2)3 = 8.
    The 8 cubes at the core will have no painted face.
  • Question 11
    1 / -0
    A cube is painted red on two adjacent surfaces, black on the surface opposite to red surfaces and green on the remaining faces. Now, the cube is cut into 64 smaller cubes of equal size.

    How many smaller cubes have less than three surfaces painted?
    Solution
    Number of smaller cubes with less than three surfaces painted = Number of cubes with two surfaces painted + Number of cubes with one surface painted + Number of cubes with no surface painted
    = 12(n - 2) + 6(n - 2)2 + (n - 2)3
    = 12(4 - 2) + 6(4 - 2) 2 + (4 - 2)3
    = 24 + 24 + 8 = 56
  • Question 12
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent surfaces, black on the surface opposite to the red surfaces and green on the remaining faces. Now, the cube is cut into 64 smaller cubes of equal size.

    How many smaller cubes have three surfaces painted?
    Solution
    Number of smaller cubes with three surfaces painted = 8
  • Question 13
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent surfaces, black on the surface opposite to the red surfaces and green on the remaining faces. Now the cube is cut into 64 smaller cubes of equal size.

    How many smaller cubes with two surfaces painted have one face green and one of the adjacent faces black or red?
    Solution
    Green painted surfaces are on the upper and lower surfaces of the cube. Hence, two surfaces painted with one surface green and another black or red will be present at the edges on the upper and bottom surfaces. Their number is (n – 2) × 8 = 16.
  • Question 14
    1 / -0
    A cube is painted red on two adjacent surfaces and black on the surface opposite to red surfaces and green on the remaining faces. Now, the cube is cut into 64 smaller cubes of equal size. How many smaller cubes have at least one surface painted green?
    Solution
    All the smaller cubes present on the top and bottom of the big cube have green colour on at least one of its surfaces.
    Required answer = 16 + 16 = 32
  • Question 15
    1 / -0
    A cube, painted yellow on all faces, is cut into 27 small cubes of equal size. How many small cubes are painted on one face only?
    Solution
    The big cube can be cut into 27 small cubes are shown. Clearly, out of these 27 small cubes, the cubes having only one side painted are those which lie at the centre of each face of the big cube. Since, there are 6 faces of a cube, the required number of cubes is 6.

  • Question 16
    1 / -0
    A cube of edge length 4 inches is painted yellow on all the faces. It is cut into small cubes of equal size, each of edge length 1 inch. Find the number of small cubes with only one face painted.
    Solution
    Clearly, the original (coloured) cube is divided into 64 smaller cubes as shown in the figure.
    The four central cubes on each face of the larger cube have only one side painted.
    Since there are six faces, so total number of such cubes = 4 6 = 24

  • Question 17
    1 / -0
    A cube is painted blue on all faces and is then cut into 125 cubes of equal size. How many cubes are not painted on any face?
    Solution

    The following figure shows the cubes painted blue on all faces and divided into 125 smaller cubes.



    The figure may be analysed by dividing it into five horizontal layers:

    In layer I: The nine central cubes have only one face painted, four cubes at the corner have three faces painted and the remaining 12 cubes have two faces painted.



    In each of the layers II, III and IV: The nine central cubes have no
    face painted, the four cubes at the corner have two faces painted and the remaining 12 cubes have one face painted.



    In layer V: The 9 central cubes have only one face painted, four cubes at the corner have three faces painted and the remaining 12 cubes have two faces painted.



    There are 9 central cubes in each of the layers II, III and IV which have no face painted. Thus, there are 9 3 = 27 such cubes.
  • Question 18
    1 / -0
    A cube is painted blue on all faces and then cut into 125 cubes of equal size. How many cubes are painted on one face only?
    Solution
    There are 9 cubes in each of the layers I and V, and there are 12 cubes in each of the layers II, III and IV which are painted on one face only.



    Thus, there are (9 × 2) + (12 × 3) = 54 such cubes.
  • Question 19
    1 / -0
    A cube of white material is painted black on all its surfaces. If it is cut into 125 smaller cubes of the same size, then how many cubes will have two sides painted black?
    Solution
    The following figure shows the cubes painted black on all faces and then divided into 125 smaller cubes.



    The figure may be analysed by dividing it into five horizontal layers.
    In layer I: The 9 central cubes have only 1 face painted, 4 cubes at the corner have 3 faces painted and the remaining 12 cubes have two faces painted.



    In each of the layers II, III and IV: The 9 central cubes have no face painted, the 4 cubes at the corner have 2 faces painted and the remaining 12 cubes have 1 face painted.



    In layer V: The 9 central cubes have only 1 face painted, the 4 cubes at the corner have 3 faces painted and the remaining 12 cubes have 2 faces painted.



    Clearly, there are 12 cubes in each of the layers I and V and there are 4 cubes in each of the layers II, III and IV which have two sides painted black i.e. there are (12 2) + (4 3) = 24 + 12 = 36 such cubes.
  • Question 20
    1 / -0
    Directions: A cube is coloured red on two opposite faces, blue on two adjacent faces and yellow on two remaining faces. It is then cut into two halves along the plane parallel to the red faces. One piece is then cut into four equal cubes and the other one into 32 equal cubes.

    How many cubes do not have any coloured face?
    Solution
    Only four smaller cubes from inside the cube will have none of their faces painted.

  • Question 21
    1 / -0
    Directions: A cube is coloured red on two opposite faces, blue on two adjacent faces and yellow on two remaining faces. It is then cut into two halves along the plane parallel to the red faces. One piece is then cut into four equal cubes and the other one into 32 equal cubes.

    How many cubes do not have any red face?
    Solution
    16 cubes of the second layer from the bottom will not have any red face.
  • Question 22
    1 / -0
    Directions: A cube is coloured red on two opposite faces, blue on two adjacent faces and yellow on two remaining faces. It is then cut into two halves along the plane parallel to the red faces. One piece is then cut into four equal cubes and the other one into 32 equal cubes.

    How many cubes have at least two coloured faces?
    Solution
    No. of at least two coloured faces cubes = No. of three faces coloured cubes + No. of two faces coloured cubes = 8 + 12 = 20.
  • Question 23
    1 / -0
    Directions: A cube is coloured red on two opposite faces, blue on two adjacent faces and yellow on two remaining faces. It is then cut into two halves along the plane parallel to the red faces. One piece is then cut into four equal cubes and the other one into 32 equal cubes.

    How many cubes have each a yellow face with other faces without colour?
    Solution
    Two cubes each from two yellow faces will have one face painted yellow with other faces without colour. Hence, 2 2 = 4 such cubes will be obtained.
  • Question 24
    1 / -0
    Directions: A cube is coloured red on two opposite faces, blue on two adjacent faces and yellow on two remaining faces. It is then cut into two halves along the plane parallel to the red faces. One piece is then cut into four equal cubes and the other one into 32 equal cubes.

    How many cubes have at least one face painted blue?
    Solution
    Seventeen cubes including four bigger cubes have at least one face painted blue.
  • Question 25
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent faces, yellow on the two faces opposite to the red faces and green on the remaining faces. It is then cut into 64 smaller cubes of equal size.

    How many cubes are painted on all faces?
    Solution

    Let the upper and the LHS faces of the cube be painted red. Then, the lower and the RHS faces are painted yellow, and the front and the rear faces are painted green, as indicated in the following figure, which also shows the division of this cube into 64 smaller cubes.



    We shall label the four different horizontal layers (of 16 cubes each) as layers 1 to 4. The 16 columns (of 4 cubes each) may also be labelled from a to p as shown in the figure.

    There is no cube which is painted on all faces.
  • Question 26
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent faces, yellow on the two faces opposite to the red faces and green on the remaining faces. It is then cut into 64 smaller cubes of equal size.

    How many cubes are there which have no face painted?
    Solution
    Four central cubes (in columns f, g, j and k) in each of the layers 2 and 3 have no face painted. Thus, there are (4 2) = 8 such cubes.

  • Question 27
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent faces, yellow on the two faces opposite to the red faces and green on the remaining faces. It is then cut into 64 smaller cubes of equal size.

    How many cubes have exactly one face painted and that too yellow?
    Solution
    Two cubes (in columns n and o) in each of the layers 2 and 3 are painted yellow on one face only. Thus there are only four cubes on this face and there will be four cube on the other yellow face. Hence, there are in total 8 cubes, which are painted yellow on one face only.

  • Question 28
    1 / -0
    Directions: Read the following information and answer the given question.

    A cube is painted red on two adjacent faces, yellow on the two faces opposite to the red faces, and green on the remaining faces. It is then cut into 64 smaller cubes of equal size.

    How many cubes have three painted faces?
    Solution
    Four corner cubes (in columns a, d, m and p) in each of the layers 1 and 4 have 3 faces painted. Thus, there are (4 2) = 8 such cubes.

  • Question 29
    1 / -0
    Directions: A cube is painted red on two adjacent faces, yellow on the two faces opposite to the red faces and green on the remaining faces. It is then cut into 64 smaller cubes of equal size. Answer the following question based on this information.

    How many cubes have one face green and one face red?
    Solution
    Eight cubes (in columns a, b, c, d, m, n, o and p) in each of the layers 1 and 4 and four cubes (in columns (a, d, m and p) in each of the layers 2 and 3 have one face green and one of the adjacent faces red or yellow.

  • Question 30
    1 / -0
    Directions: A cube is painted red on two adjacent faces, yellow on the two faces opposite to the red faces and green on the remaining faces. It is then cut into 64 smaller cubes of equal size. Answer the following question based on this information.

    How many cubes are painted on two faces only and with the same colour?
    Solution
    Two cubes in layer 1 and two cubes in layer 4 have only two faces painted and with the same colour.

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