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100 Questions 80 Marks 90 Mins
Given:
A cube that has been applied 13 cuts.
Concept:
By cutting the cube for the first time, it will be divided into 2 parts.
By cutting the cube along the perpendicular, it will be divided into 4 parts.
Again, by cutting the cube along the perpendicular, it will be divided into 8 parts.
⇒ With 3 cuts, there will be a total of 8 cubes.
⇒ It can be concluded that 'n' cuts made along one plane will have 'n + 1' pieces.
Calculation:
To obtain equal pieces, cuts must be made parallel to 3 planes.
Let's assume, a, b, and c cut made along three planes such that there will be (a + 1)(b + 1)(c + 1) = m pieces & a + b + c = 13.
⇒ Several combinations can be made but 'm' will be maximum in (4, 4, 5) cuts.
∴ The required number of pieces = m = (4 + 1)(4 + 1)(5 + 1)
⇒ 5 × 5 × 6
⇒ 150
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1
Cubes
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If total number of cuts is 10 then find the minimum number of pieces that can be obtained.
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If total number of cuts is 10 then minimum number of pieces is 11 when cut is made in one plane only.
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If total number of cuts is 10 then find the maximum number of pieces that can be obtained.
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View Hint View Answer Discuss in Forum
If total number of cut is 10 then for maximum number of pieces these cuts have to be well distributed in three planes. For 10 cuts,
3,3 and 4 is the distribution of cuts.
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If total number of cuts is 20 then find the ratio of maximum and minimum of pieces that can be obtained.
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View Hint View Answer Discuss in Forum
For maximum number of pieces cuts has to be 6, 7 and 7 and maximum number of pieces is (6 + 1)(7 + 1)(7 + 1) = 7 x 8 x 8 = 448.
Minimum number of pieces is 20 + 1 = 21.
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If total number of pieces (Smaller cubes/cuboids) is 45 then find the possible number of cuts.
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View Hint View Answer Discuss in Forum
If 45 = 1 x 1 x 45 then we require only 44 cuts in one plane.If 1 x 3 x 15 then we require 2 cuts in one plane and 14 cuts in other plane so total number of cuts is 2 + 14 = 16.If 1 x 5 x 9 the we require 4 cuts in one plane and 8 cuts in other plane so total number of cuts is 4 + 8 = 12
If 3 x 3 x 5 then we require 2 cuts in one plane, 2 cuts in 2nd plane and 4 cuts in 3rd plane so total number of cuts is 2 + 2 + 4 = 8.
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Find the maximum number of cuts required to get 50 pieces.
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View Hint View Answer Discuss in Forum
For maximum number of cuts it has to be in one cut only, so number of cuts is 49