Cube Questions For NMAT:
Download Cube Questions for NMAT PDF. Top 10 very important Cube Questions for NMAT based on asked questions in previous exam papers.
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Question 1: A cuboid dimensions 10x11x12 is cut into unit cubes and then painted with red colour. Find the number of unit cubes that have 0 faces painted?
a) 990
b) 1320
c) 0
d) None of the above
Question 2: A cube of dimensions 8x8x8 is painted with 3 different colours such that opposite faces have the same colour. What is the number of unit cubes that have exactly 2 faces painted and with more than 1 colour?
a) 32
b) 48
c) 60
d) 72
Question 3: A painted cuboid of dimensions 5x6x7 is cut into unit cubes. What is the number of unit cubes that have 0 faces painted?
a) 50
b) 60
c) 40
d) 210
Question 4: A 3x3x3 cube is placed on top of a 5x5x5 cube, which is placed on a table. On top of the 3x3x3 cube is placed a unit cube such that the axes of all three cubes coincide. All the visible faces are now painted red. What is the number of unit cubes that have exactly 2 faces painted?
a) 32
b) 48
c) 49
d) None of the above
Question 5: A painted 6 x 6 x 6 cube is cut into 216 unit cubes. What is the number of unit cubes that have exactly 2 faces painted?
a) 24
b) 36
c) 48
d) 60
Question 6: A cube made up of 1000 unit cubes is placed on a table. On the top face of this cube is placed another cube made up of 512 unit cubes. On the top face of this cube is placed a cube made up of 216 unit cubes. The centers of all the big cubes lie in a straight line. If the entire structure is now painted, find the number of unit cubes that have no face painted.
a) 216
b) 532
c) 1008
d) 764
Question 7: A cube is painted green and cut into 343 identical cubes. Now, for all the cubes, the faces that have no paint on them are painted red. Find the ratio of the number of cubes that have one colour painted on them to the number of cubes that have two colours painted on them.
a) 216:125
b) 1:1
c) 125:218
d) 125: 216
Question 8: A cube is painted and then cut into 125 cubes of equal volume. Find the number of small cubes that have paint on at least one face.
a) 81
b) 64
c) 27
d) 98
Question 9: A cuboid of dimensions 10X11X12 is painted on all its faces. The cuboid is then entirely cut into smaller cubes of unit volume. Find the number of such unit cubes which have paint on exactly two of their faces.
a) 216
b) 108
c) 724
d) 484
Question 10: Opposite faces of a 5X5X5 cube are painted with the colours green, red and yellow. The cube is then cut into smaller cubes of unit volume. What is the number of cubes which are painted with exactly 2 colours – red and yellow?
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Answers & Solutions:
1) Answer (C)
Since the cubes are painted after the 10x11x12 cuboid is cut, all the unit cubes will have their all faces covered with paint.
So, there will be no cube that will have none of its faces covered by paint.
2) Answer (D)
All the unit cubes residing on the edges of the 8x8x8 cube have exactly 2 of their faces covered by paint. Among these, all have 2 different colours since no two adjacent sides of the big cube have the same colour. So, the number of unit cubes that have exactly 2 faces covered by 2 different colours = 6*12 = 72
3) Answer (B)
The number of unit cubes that have no face painted = number of unit cubes that are hidden = (5-2)*(6-2)*(7-2) = 3*4*5 = 60
4) Answer (D)
Usually the 3 cubes in the middle along the 12 edges would be painted on two sides. However, as the cube rests on the table, the bottom face would not painted and as a result the bottom 4 edges would not be counted.
Also, the bottom 4 corners would be painted on only 2 faces.
So, the number of unit cubes in the 5x5x5 cube that have exactly 2 faces painted = 3*8 + 4 = 28
Similarly for the 3x3x3 cube, the number of cubes with exactly 2 faces painted = 1*8 + 4 = 12
The unit cube placed at the top has 5 faces covered in paint.
So, the total number of unit cubes that have exactly 2 faces painted = 28+12 = 40
5) Answer (C)
The number of cubes that have exactly 2 faces painted = 4*12 (4 faces on each edge and 12 edges) = 48
6) Answer (C)
The number of unit cubes in the 10X10 cube that are not covered by paint = ( Number of unit cubes on the bottom surface of 10X10 cube ,64 + Number of unit cubes in the interior of the 10X10 cube, 512 + Number of unit cubes on the top surface of 10X10,64) + (Number of unit cubes on the bottom surface of 8X8 cube,36 + Number of unit cubes in the interior of 8X8 cube ,216 + Number of unit cubes on the top surface of 8X8 cube,36) + (Number of unit cubes on the bottom surface of 6X6 cube,16 + Number of unit cubes in the interior of 6X6 cube ,64) = 1008.
7) Answer (C)
Let us assume our original cube to be a 7x7x7 one.
If we paint this cube in green, and cut the cube into 343 identical cubes i,e, into 1x1x1 cubes,
The number of smaller cubes with 1 face painted = 5*5*6 = 150
The number of smaller cubes with 2 faces painted = (5*4*6)/2 = 60
The number of smaller cubes with 3 faces painted = (4*6)/3 = 8
The total number of cubes painted in Green = 150+60+8 = 218.
Observe that no small cube has all its faces painted in Green.
Now all these cubes will be painted Red on unpainted faces and will essentially be painted in two colours.
So the number of cubes painted in two colours = 218.
The remaining 125 cubes are painted with Red colour only as no face of them is painted before cutting the 7x7x7 cube.
So the number of cubes painted in one colour = 125.
Therefore the required ratio = 125 : 218.
8) Answer (D)
Let the side of the original cube be 5 cm. Therefore, the side of the smaller cubes is 1 cm.
Note that in the original cube (5x5x5), the central cube of sides 3x3x3 has no sides painted (as it is not exposed).
The sides that are painted are only in the external layer of width 1 cm.
Hence, the number of smaller cubes with paint on at least one side = total number of unit cubes – number of unit cubes with no face covered by paint = 125 – 3*3*3 = 125-27 = 98.
9) Answer (B)
The unit cubes with paint on exactly 2 faces reside on the edges of the cuboid. The cuboid has 12 edges. There are 4 edges of length 12, 4 edges of length 11 and 4 edges of length 10. On the edge with length 12 unit cubes, the number of cubes which have exactly 2 faces covered by paint = 12-2 = 10. Similarly, on the edge with 11 unit cubes, the number of cubes covered by paint on exactly 2 faces = 11-2 = 9. The number of cubes with exactly 2 faces covered by paint on the 10 unit cube edge is 10-2 = 8. Therefore, the total number of unit cubes that have exactly 2 faces covered by paint = 4*(10+9+8) = 4*27 = 108.
10) Answer: 12
The unit cubes with paint on exactly 2 faces reside on the edges of the big cube. Among the 12 edges, 8 edges have a touch of green. So, all these 8 can be ruled out. Each of the other 4 edges has 5 cubes, out of which 3 have exactly two faces painted. And all such cubes have only red and yellow paints on them. Therefore, the total number of cubes that are painted with exactly 2 colours – red and yellow is 4*3 = 12.
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