:
Definition: 1 Mark each
AnOblique sketchdoes not have proportional lengths, still, it conveys all important aspects ofthe appearance of the solid.
AnIsometric Sketchis drawn on an isometric dot paper. In an isometric sketch of the solid, the measurements are kept proportional.

:
Explanation: 1 Mark each
(i) A prism becomes a cylinder whenthe number of sides increases.
(ii) Whenthe number of sides of the pyramid increases, itbecomes a cone.

:
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.

:
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

:
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. Apolyhedron with 4 triangles as faces is known as a regular tetrahedron.
(iii) This is true. Apolyhedron with 1 square and 4 triangular faces is known as a square pyramid.

:
Formula: 1.5 Marks
Application: 0.5Mark 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

:
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

Answer: Option A. -> Oblique sketches
:
A
The sketches which givean idea of how afigure looks like, when seen from the front (the exact dimensions need not be the same) are called oblique sketches.

Answer: Option B. -> False
:
B
No, it cannot. A cube can cast a shadow in the shape of a square.

Answer: Option B. -> Isometric sketches
:
B
The sketches drawn on a sheet which is divided into small equilateral triangles are called isometric sketches.