Cubes Test

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Cubes Test - 6

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Weekly Quiz Competition

Question 1
1 / -0

Directions: Read the following information and answer the given question.

A cuboid of dimensions 4 cm x 3 cm x 3 cm is painted yellow on the pair of opposite faces of dimensions 4 cm x 3 cm. The two opposite faces of dimensions 4 cm x 3 cm are painted red and the two remaining faces of dimensions 3 cm x 3 cm are painted green. Now, the cuboid is cut into small cubes, each of dimensions 1 cm x 1 cm x 1 cm.

How many small cubes will have only one painted face?

A
10

B
8

C
12

D
6

Solution

The sides painted red have 2 cubes in between which are painted with single colour. Hence, for both the sides, total number of cubes which are painted with single colour = 2 x 2 = 4 cubes
Same is the case with the yellow sides. Hence, for both the sides, total number of cubes which are painted with single colour = 2 x 2 = 4 cubes
In case of green sides, only one cube in the middle has one side painted.
So, for both sides, total number of cubes which are painted with single colour = 1 x 2 = 2 cubes
Required total number of cubes = 4 + 4 + 2 = 10
Question 2
1 / -0

Directions: Read the following information and answer the given question.

A cuboid of dimensions 4 cm 3 cm 3 cm is painted yellow on the pair of opposite faces of dimensions 4 cm 3 cm. The two opposite faces of dimensions 4 cm 3 cm are painted red and the two remaining faces of dimensions 3 cm 3 cm are painted with green colour. Now, the cuboid is divided into small cubes, each of dimensions 1 cm 1 cm 1 cm.

How many cubes will have no face painted?

A
1

B
2

C
4

D
8

Solution

Only two cubes inside the block will have no colour on any of their faces.

Number of small cubes having no face coloured = (4 - 2) × (3 - 2) = 2 × 1 = 2
Question 3
1 / -0

Directions: Read the following information and answer the given question.

A cuboid of dimensions 4 cm 3 cm 3 cm is painted yellow on the pair of opposite faces of dimensions 4 cm 3 cm. The two opposite faces of dimensions 4 cm 3 cm are painted red and the two remaining faces of dimensions 3 cm 3 cm are painted with green colour. Now, the cuboid is divided into small cubes, each of dimensions 1 cm 1 cm 1 cm.

On how many cubes do all the three colours appear?

A
24

B
20

C
16

D
8

Solution

All the three colours appear on eight cubes at the corners.
Question 4
1 / -0

Directions: Read the following information and answer the given question.

A cuboid of dimensions 4 cm 3 cm 3 cm is painted yellow on the pair of opposite surfaces of dimensions 4 cm 3 cm. The remaining two opposite surfaces of dimensions 4 cm 3 cm are painted red and two surfaces of dimensions 3 cm 3 cm are painted with green colour. Now, the block is divided into small cubes, each of dimensions 1 cm 1 cm 1 cm.

How many cubes will have only two painted surfaces?

A
32

B
24

C
16

D
12

Solution

Two cubes each on four edges and one cube each on eight edges will have only two surfaces painted. That is, a total of 16 such cubes will be obtained.
Question 5
1 / -0

Directions: Read the following information and answer the given question.

A cuboid of dimensions 4 cm 3 cm 3 cm is painted yellow on the pair of opposite surfaces of dimensions 4 cm 3 cm. The remaining two opposite surfaces of dimensions 4 cm 3 cm are painted red and two opposite surfaces of dimensions 3 cm 3 cm are painted green. Now the cuboid is divided into small cubes of dimensions 1 cm 1 cm 1 cm.

How many cubes will have at least one painted surface?

A
32

B
24

C
18

D
None of these

Solution

Number of cubes having at least one surface painted = Number of cubes having three surfaces painted + Number of cubes having two surfaces painted + Number of cubes having one surface painted = 8 + 16 + 10 = 34
Question 6
1 / -0

Directions: Read the following information and answer the question given below.

A cube of side 10 cm is coloured red, with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 smaller cubes of equal sizes.

How many cubes have three green faces each?

A
0

B
4

C
8

D
16

Solution

Clearly, upon colouring the cube as stated and then cutting it into 125 smaller cubes of equal sizes, we get a stack of cubes as shown in the figure.

The figure can be analysed by assuming the stack to be composed of 5 horizontal layers.
There are 4 corner cubes in layer 1, and 4 corner cubes in layer 5, having three green faces each. Thus, there are 8 such cubes.
Question 7
1 / -0

Directions: A cube of side 10 cm is coloured red with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 small cubes of equal size.

How many cubes have one red face and an adjacent green face?

A
0

B
6

C
8

D
16

Solution

There is no cube having one face red and an adjacent face green.
Question 8
1 / -0

Directions: Read the following information and answer the given question.

A cube of side 10 cm is coloured red with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 small cubes of equal size.

How many cubes have at least one coloured face?

A
76

B
89

C
98

D
102

Solution

There are nine central cubes in each of the layers 2, 3 and 4 having none of their faces coloured, i.e. there are (9 3) = 27 such cubes.

Therefore, there are 125 – 27 = 98 cubes having at least one face coloured.
Question 9
1 / -0

A cube of side 10 cm is coloured red, with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 smaller cubes of equal sizes. How many cubes have at least two green faces each?

A
8

B
27

C
44

D
63

Solution

There are 12 cubes in each of the layers 1 and 5 and 4 cubes in each of the layers 2, 3 and 4 which have two faces coloured green. Also, there are 4 cubes in each of the layers 1 and 5 which have three faces coloured green.

3 faces coloured cubes - 8
2 faces coloured cubes - 24 + 12 = 36
Total cubes - 44
Question 10
1 / -0

Directions: Read the following information and answer the question given below.

A cube of side 10 cm is coloured red, with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 smaller cubes of equal sizes.

How many cubes are without any colour?

A
64

B
27

C
12

D
81

Solution

There are 9 central cubes in each of the layers 2, 3 and 4, which are without any colour.

Thus, there are 9 × 3 = 27 such cubes.
Question 11
1 / -0

Directions: A cube is cut into two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other is coloured green on two smaller adjacent faces and red on the remaining. Each is then cut into 32 cubes of the same size. The 64 cubes are then mixed up.

How many cubes have no coloured face at all?

A
16

B
8

C
4

D
0

Solution

The two stacks formed by cutting each of the two blocks (obtained from a cube) into 32 cubes of the same size and colouring them in different patterns shall be analysed separately.

We shall label each one of the horizontal layers in each stack and also each one of the columns of cubes in each stack as shown in the above figure.

In both the stacks, there is no cube which has no coloured face at all.
Question 12
1 / -0

Directions: A cube is cut into two equal parts along a plane parallel to one of its faces. One piece is then coloured green on the two larger faces and red on the remaining, while the other is coloured red on the two smaller adjacent faces and green on the remaining. Each piece is then cut into 32 cubes of the same size. The 64 cubes are then mixed up.

How many cubes have only one coloured face?

A
8

B
16

C
20

D
24

Solution

Four cubes (in columns a₃, a₄, a₅ and a₆) in each of the two layers A₂ and A₃ in stack A and similarly four cubes (in columns b₃, b₄, b₅ and b₆) in each of the two layers B₂ and B₃ in stack B have only one coloured face.

Thus, there are (4 2) + (4 2) = 8 + 8 = 16 such cubes.
Question 13
1 / -0

A cube is cut into two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other piece is coloured green on the two smaller adjacent faces and red on the remaining. Each is then cut into 32 same-sized cubes. The 64 cubes are then mixed up.

How many cubes have two red and one green face?

A
4

B
8

C
12

D
16

Solution

There is no cube in stack A which has two red faces.
Two cubes (in columns b₇ and b₈) in layer B₁ and two cubes (in columns b₁ and b₂) in layer B₄ in stack B have two red and one green face each.

Thus, there are (2 + 2) = 4 such cubes.
Question 14
1 / -0

A cube is cut into two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other piece is coloured green on the two smaller adjacent faces and red on the remaining. Each is then cut into 32 same-sized cubes. The 64 cubes are then mixed up.

How many cubes have only one red face and one green face?

A
32

B
24

C
16

D
8

Solution

Four cubes (in columns a₃, a₄, a₅ and a₆) in each of the two layers A₁ and A₄ and four cubes (in columns a₁, a₂, a₇ and a₈) in each of the two layers A₂ and A₃ in stack A have one red face and one green face each.

Four cubes (in columns b₃, b₄, b₅ and b₆) in layer B₁ and two cubes (in columns b₁ and b₂) in each of the two layers B₂ and B₃ in stack B have one red face and one green face each.
Thus, there are (4 2) + (4 2) + 4 + (2 2) = 8 + 8 + 4 + 4 = 24 such cubes.
Question 15
1 / -0

A cube is cut into two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other piece is coloured green on the two smaller adjacent faces and red on the remaining. Each is then cut into 32 same-sized cubes. The 64 cubes are then mixed up.

What is the number of cubes with at least one green face?

A
46

B
38

C
36

D
28

Solution

Eight cubes (in columns a₁, a₂, a₃, a₄, a₅, a₆, a₇ and a₈) in each of the two layers A₁ and A₄ and four cubes (in columns a₁, a₂, a₇ and a₈) in each of the two layers A₂ and A₃ in stack A have at least one green face each.

Eight cubes (in columns b₁, b₂, b₃, b₄, b₅, b₆, b₇, b₈) in layer B₁ and two cubes (in columns b₁ and b₂) in each of the three layers B₂, B₃ and B₄ in stack B have at least one green face each.
Thus, there are (8 × 2) + (4 × 2) + 8 + (2 × 3) = 16 = 8 + 8 + 6 = 38 such cubes.