CAT Question Bank  Cubes and Cuboids

Q7) A cube has all its 6 faces painted in six different color being red, blue, green, violet, yellow, black such that each face is painted with only one color. The cube is place on the table with the violet face touching the table top. The cube is cut into 60 identical pieces by making the least number of cuts as possible, where all the cuts are parallel to the faces of the cube. The least number of cuts are made parallel to the red face, while the maximum number of cuts are made parallel to the black face. Green and blue faces are opposite each other. Red face is not opposite to the violet face.
How many small pieces have black color on their faces?
How many small pieces have at least two different colors on the faces?
How many small pieces have only one face painted?
How many small pieces have no color on their faces?

Q8) Each face of a cube is to be painted with red, blue or yellow colour. In how many different ways the cube can be painted?

Q9) A cube has edges of length 1 cm and has a dot marked in the center of the top face. The cube is sitting on a ﬂat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face. The length, in centimeters, of the path traveled by the dot is
a) π
b) 2π
c) (√2)π
d) (√5)π
e) (1 + √5)/2π

Q10) A cube of edge 8 cm is cut into identical and smaller cubes with edge 2 cm each. How many smaller cubes are possible ?

Q11) A cube has 7cm * 7cm * 7cm is kept in the corner of a room and painted in three different colours, each face in one colour. The cube cut into 343 smaller but identical cubes. How many smaller cubes have at the most two faces painted?

Q12) A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to:
a) 10
b) 15
c) 17
d) 21
e) 25

Q13) In how many different ways can a cube be painted if each face has to be painted either red or blue?

Q14) A cube has been cut into 150 smaller cubes of equal volume. What is the minimum number of straight cuts required to do this?

Q15) Two opposite faces of a cube are painted in blue, another pair of opposite faces are painted green and the remaining faces are painted in red. The cube is now cut into 210 smaller but identical pieces using minimum possible number of cuts. How many smaller pieces have exactly two colours on them?

Q16) A solid cube of side 1 cm is cut into two identical solid parts by a plane surface. The total surface area of both the solid parts is maximum possible. Two points are then selected on one of these solid parts. What is the maximum possible distance between these two points

Q17) What is the least possible number of cuts required to divide a cube into 34 identical pieces ?

Q18) A big cube is formed of 125 smaller cubes of which m are such that all their faces are coloured. If 60% of the area of the big cube is coloured, what is the maximum possible value of m

Q19) How many distinct cuboids can be formed using 64 cubes of area 1cm^2

Q20) A cube of dimension 4 cm × 4 cm × 4 cm is painted red on all six faces. Now this cube is cut to form 1 cm × 1 cm × 1 cm identical cubes. What is the ratio of total area of painted surfaces to the total area of unpainted surfaces

Q21) A cube is painted red and all its faces and is then divided into 343 equal cubes.
 How many cubes will have three faces painted ?
 How may cubes will have two faces painted?
 How many cubes will have one face painted?
 How many cubes have no face painted ?
 How many cubes will have at least one face painted?

Q22) A cuboid of size 5.2 m, 13 m & 39 m is completely cut into n smaller identical cubes. If n has the minimum possible value, what is the total surface area of all the small cubes?

Q23) The areas of three mutually adjacent faces of a cuboid C1 were found to be 200 sq. cm, 100 sq. cm and 50 sq. cm. Another cuboid C2 has the same volume as C1. Find the minimum value of the sum of the length, the breadth and the height of C2 (in cm).
a) 15
b) 30
c) 40
d) 50

Q24) There is a rectangular wooden block of length 4 cm, height 3 cm and breadth 3 cm. The two opposite surfaces of 4 cm x 3 cms are painted yellow on the outside. The other two opposite surfaces of 4 cm x 3 cm are painted red on the outside and the remaining two surfaces of 3 cm x 3 cm are painted green on the outside. Now, the block is cut in such a way that cubes of 1cm x 1cm x 1cm are created.
 How many cubes will have only one colour?
 How many cubes will have no colour?
 How many cubes will have any two colours?

Q25) A square has a perimeter of 108 cm. Four small squares are cut from each of its corners and the side of each square is x cm. The sheet remaining after the squares are cut is folded to form a cuboid. Find the value of x which maximizes the volume of the cuboid formed.

Q26) A set of 120 small 1 × 1 × 1 cubes are arranged to form a 6 × 5 × 4 cuboid. How many cuboids (including the original) can be formed with vertices chosen from these points such that their faces are parallel to those of the original cuboid?