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D Divya Prakash Singh
Checking using Euler's Formula: F + V = E +2; Given that F = 10, V = 15 and E = 20 so, Putting in equation: 10 + 15 = 35  and 20 +2 = 22 Hence, given polyhedron having 10 faces, 20 edges and 15 vertices is not possible.

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D Divya Prakash Singh
We know the Euler Formula: F + V = E + 2: hence we can easily find the unknown let say 'F, V, and E' So, For Ist case:  F + 6  = 12 + 2  or  F = 8, hence the number of faces in Ist case is 8. For IInd Case; 5 + V = 9 + 2   or  V = 6, hence the number of vertices in IInd case is 6. For IIIrd case; 20 + 12 = E + 2     or   E = 30, hence the number of Edges in the IIIrd case is...

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D Divya Prakash Singh
(ii) Euler formula F+V = E + 2. So, Number of faces (F) = 9, Number of vertices (V)  = 9 and number of edges (E)  = 16. Hence putting in the formula; we get, 9 + 9 = 16 + 2 18 = 18 hence, it satisfies the Euler formula.

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D Divya Prakash Singh
(i) Euler formula F+V = E + 2. So, Number of faces (F) = 7, Number of vertices (V)  = 10 and number of edges (E)  = 15. Hence putting in the formula; we get, 7 + 10 = 15 + 2 17 = 17 hence, it satisfies the Euler formula.  

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D Divya Prakash Singh
No, not always, because it can be a cuboid also when extended.

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D Divya Prakash Singh
(i) A prism is a polyhedron whose base and top are congruent polygons and whose other faces, i.e., lateral faces are parallelograms in shape. Also, the cylinder has base on top and bottom and faces are parallel.

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D Divya Prakash Singh
(iv) Given box is a prism.

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D Divya Prakash Singh
(iii) Table weight is not a prism.  

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D Divya Prakash Singh
(ii) The unsharpened pencil is a prism.

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D Divya Prakash Singh
(i) a nail is not a prism.  

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D Divya Prakash Singh
Yes, it is possible if the number of faces is four or more than four.

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D Divya Prakash Singh
(iii) Yes, a polyhedron can have faces of a square and four triangles which makes a pyramid on square base.

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D Divya Prakash Singh
(ii) Yes, a polyhedron having 4 triangles is pyramid on triangular base.

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D Divya Prakash Singh
(i) No, a polyhedron cannot have 3 triangles for its faces. 
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