Answer: Option A. ->
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Solution: 1 Mark each
A polyhedron is a solid with flat polygonal faces, straight edges and vertices.
(i) This is not true. The minimum number of faces a polyhedron can have is 4. Hence 3 triangles cannot be the faces of a polyhedron.
(ii) This is true. A polyhedron with 4 triangles as faces is known as a regular tetrahedron.
(iii) This is true. A polyhedron with 1 square and 4 triangular faces is known as a square pyramid.

Answer: Option A. ->
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Formula: 1 Mark
Application: 1 Mark each
Euler's formula = F + V - E = 2
$\Rightarrow$ F + 6 - 12 = 2
F - 6 = 2
F = 8
$\Rightarrow$ 5 + V - 9 = 2
V - 4 = 2
V = 6
$\Rightarrow$ 20 + 12 - E = 2
32 - 2 = E
E = 30

Answer: Option A. ->
:

Each drawing: 1 Mark
The three isometric sketches of the given cuboid can be drawn as follows:

Answer: Option A. ->
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Each part: 1.5 Marks
(a) As it is clearly evident from the figure, when we place two cubes side by side, their length gets added, but there is no change in the other dimensions.

 

Length = 2 + 2 = 4 cm
Breadth = 2 cm
Height = 2 cm

                   
(b) 

Number of Faces = 5
Number of Vertices = 6
Number of Edges = 9

Answer: Option A. ->
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Each part: 2 Marks
(i)

(a) Left view
(b) Top view
(c) Right view
(d) Front view
(ii)
  
(a) Front View
(b) Top View
   
(c) Side View

Answer: Option A. ->
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Each part: 2 Marks
(i) Faces = 7, Vertices = 10, Edges = 15
By Euler's Formula, F + V - E = 2
7 + 10 - 15 = 2
17 - 15 = 2
Therefore, Euler's Theorem is verified.
(ii) Faces = 9, Vertices = 9, Edges = 16
By Euler's Formula, F + V - E = 2
9 + 9 - 16 = 2
18 - 16 = 2
Therefore, Euler's Theorem is verified.

Answer: Option A. ->
:

Formula: 1.5 Marks
Application: 0.5 Mark each
Euler’s Formula:
Faces + Vertices – Edges = 2

1)  3 + 4 - 6 = 1
1 $\ne$  2
2) 3 + 8 -12 = -1
1 $\ne$  2
3) 4 + 6 - 12 = -2
-2 $\ne$ 2
4) 3 + 20 - 30 = -7
-7 $\ne$ 2
5)  5 + 12 - 30 = -13  
-13 $\ne$ 2

Answer: Option A. ->
:
Each part: 2 Marks
(a) There are 3 vertical columns of sugar bags in this box.
The first column has 9 bags of which 5 can be seen and the other 4 are completely hidden.
The second column has 6 bags of which 4 can be seen, and similarly, the third column has 3 bags.
So, the total number of bags = 9 + 6 + 3 = 18.
(b) Now, in the second column, we can see that there is space for 3 more bags and the third column has space for 6 more bags.
The remaining space can be filled with 2 more vertical columns, each having 9 bags.
So, there is a space in the box for $3+6+(9\times 2)=9+18=27$ more bags.

Answer: Option A. ->
:
Formula: 1 Mark
Answers: 3 Marks
Euler's formula: F + V - E = 2
$\Rightarrow$ 1) 10 + 10 - E = 2
E = 20 - 2
E = 18
$\Rightarrow$ 2) 8 + V - 18 = 2
V = 10 + 2
V = 12
$\Rightarrow$ 3) 6 + 8 - E = 2
E = 14 - 2
E = 12
$\Rightarrow$ 4) F + 5 - 8 = 2
F = 2 + 3
F = 5
$\Rightarrow$ 5) 12 + V - 19 = 2
V = 2 + 7
V = 9