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Answer the following questions based on the information given below.
A cube is cut in two equal parts along a plane parallel to one of its faces. One piece is then colored blue on the two larger faces and pink on the remaining, while the other is colored pink on two smaller adjacent faces and blue on the remaining. Each is then cut into 32 cubes of same size and mixed up.
Q1MULTIPLE CHOICE
How many cubes have only one colored face each?
Answer: C
The original cube is finally cut into 64 cubes. Let the original cube be 4 × 4 × 4 cm.
When the original cube is cut, the two new cubes will be 4 × 4 × 2 cm each.
When these two cubes are cut into 32 pieces each, the smaller cubes will be 1 × 1 × 1 cm each.
Considering one of these two larger cubes.
Number of cubes with exactly one colored face will come from the two larger faces i.e., total 8.
Hence, considering both these cubes, number of cubes with exactly one colored face = 16.
Hence, option (c).
Q2MULTIPLE CHOICE
What is the number of cubes with at least one pink face each?
Answer: D
Cube I:
Number of pink faces =
8 × 2 = 16 (from top and bottom faces) and
4 × 2 = 8 (from front and back faces).
∴ Total 16 + 8 = 24 cubes.
Cube II:
Number of pink faces =
8 (from top face) and
6 (from front face).
∴ Total 8 + 6 = 14 cubes.
∴ Total number of cubes with pink on them = 24 + 14 = 38.
Hence, option (d).
Q3MULTIPLE CHOICE
How many cubes have two blue and one pink face on each?
Answer: D
Cube I:
No cubes will have two blue face colored.
∴ 0 cubes. Cube II:
Out of total 8 corner cubes,
The top back corner cubes and bottom front corner cubes will have 2 blue and 1 pink face colored.
∴ 4 cubes.
Hence, option (d).
Q4TYPE IN THE ANSWER
How many cubes have no colored face at all?
Answer: 0
There is no cube which will have no face colored.
Hence, 0.
Q5MULTIPLE CHOICE
How many cubes have each exactly one blue and exactly one pink colored face?
Answer: D
Cube I:
Exactly one blue and exactly one pink colored face will come from the 8 longer edges.
∴ 8 × 2 = 16 cubes. Cube II:
Exactly one blue and exactly one pink colored face will come from the 2 longer top and 2 longer front edges.
∴ 4 × 2 = 8 cubes.
∴ Total such cubes = 16 + 8 = 24
Hence, option (d).
Answer the following questions based on the information given below.
125 small but identical cubes are labelled 1 to 125 and a larger cube is built by placing the cubes in an ascending order. In each horizontal layer the front row is laid first from left to right and then the cubes are laid in a similar pattern in the second row and so on. The bottom most layer is laid first and then the layer just above that and so on.
Q6TYPE IN THE ANSWER
What is the sum of numbers that form the diagonal connecting the left bottom corner cube of front face to the right bottom cube of the rear face?
Answer: 65
In the given pattern all the smaller cubes in any straight line/diagonal in each of the larger cube will be in arithmetic progression.
Sum of n number in AP whose first and last term is ‘a’ and ‘l’ = n/2 × (a + l).
125 = 5 × 5 × 5.
Front row … Last row
Fifth Layer: (101 – 105) … (121 - 125)
Fourth Layer: (76 - 80) … (96 - 100)
Third Layer: (51 - 75) … (71 - 75)
Second Layer: (26 - 30) … (46 - 50)
First Layer: (1 – 5) … (21 - 25)
End points of the required diagonal are the first and the last smaller cubes of the bottom layer. i.e., cube numbered 1 and 25.
The numbers on required cubes will be 1, 7, 13, 19 and 25.
∴ Sum = 1 + 7 + 13 + 19 + 25 = 65.
Alternately,
Sum = 5/2 × (1 + 25) = 65
Hence, 65.
Q7TYPE IN THE ANSWER
The sum of the numbers of the cubes forming diagonal from the bottom right of the front face of the larger cube to the top left corner of the rear face of the larger cube is:
Answer: 315
End points of required diagonal are numbers 5 and 121.
Sum = 5/2 × (5 + 121) = 315
Hence, 315.
Q8TYPE IN THE ANSWER
What is the sum of the numbers of the cube in the vertical column which has its base as the fourth cube from the left in the row just behind the front row?
Answer: 295
End points of the required diagonal are numbers 9 and 109.
Sum = 5/2 × (9 + 109) = 295.
Hence, 295.
Q9TYPE IN THE ANSWER
The sum of the numbers of the cubes forming diagonal from the bottom left of the front face of the larger cube to the top right corner of the front face of the larger cube is
Answer: 265
End points of the required diagonal are the numbers 1 and 105.
