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Each correct match: 1 Mark

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(a) Diagram of each view: 0.5 Mark
(b)Diagram of each view: 0.5 Mark
(a)


(b)

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Each part: 2 Marks
(a) By Euler's formula,
F + V - E = 2
Substituting the given values
10 + 15 - 20 = 35 - 20 =15 which is not equal to 2.
A solid needs tosatisfythe Euler's formulato be called a polyhedron.
$\Rightarrow$ Apolyhedron cannot have 10 faces, 20 edges and 15 vertices.
(b) The shapes of the shadows of these figures will be as follows.
(i) A ball
The shape of the shadow of a ball will be a circle.
(ii) A cylindrical pipe
The shape of the shadow of a circular pipe will be a rectangle.
(iii) A book
The shape of the shadow of a book will be a rectangle.

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Answer: 1 Mark each
The key feature of a sphere is that every point on the surface of a sphere is equidistantfrom the centre.
The normal definition of faces, vertices and edges are not appropriate for a sphere because itis not a polyhedron.

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Each part: 2 Marks
(a) Sphere, cone and cylinder are not polyhedrons.
Sphere: Sphere has only one curved face and no polygons as faces. So it is not apolyhedron.
Cone: Cone has one circular face and the remaining part is a curved surface. So it is not apolyhedron.
Cylinder:Cylinderhas two circular faces and the remaining part is a curved surface. So it is not apolyhedron.
(b) (i) - (b)
(ii) - (a)
(iii) - (e)
(iv) - (c)
(v) - (d)

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We need a square with same side length as that of the triangle. The resultant solid is known as a square pyramid.

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Each part: 1 Mark
(a) By visual estimation, it is easy for us to see that 3 more cans can fit in the empty space. Hence, we can say that there will be 12 cans on a full shelf.
(b)The front side of the figure is a pentagon.

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Each part: 1 Mark
From visual estimation, each column contains 4 boxes.

(a) 6 columns can be fit in the shelf.
In the 5th column, there is space for 2 more boxes.
and in the 6th column, there is space for 4 boxes.
$\therefore$ Number of boxes on the full shelf = 18 + 2 + 4 = 24 boxes
(b) If the length of the shelf is doubled, the shelf can have 12 columns.
If the height of the shelf is doubled, then each column can have 8 boxes.
$\therefore$ Total number of boxes on the full new shelf:
= $12\times 8$
= 96 boxes
(c)If the length of the shelf is halved, the shelf can have 3 columns.
If the height of the shelf is doubled, then each column can have 8 boxes
$\therefore$ Total number of boxes on the full new shelf:
= $3\times 8$
= 24 boxes
(d) If the length of the shelf is doubled, the shelf can have 12 columns.
If the height of the shelf is doubled, then each column can have 4 boxes.
$\therefore$ Total number of boxes on the full new shelf:
= $12\times 4$
= 48 boxes

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Each part: 2 Marks
(a) By Euler's formula,
F + V - E = 2
Substituting the given values
10 + 15 - 20 = 35 - 20 =15 which is not equal to 2.
A solid needs tosatisfythe Euler's formulato be called a polyhedron.
$\Rightarrow$ Apolyhedron cannot have 10 faces, 20 edges and 15 vertices.
(b) The shapes of the shadows of these figures will be as follows.
(i) A ball
The shape of the shadow of a ball will be a circle.
(ii) A cylindrical pipe
The shape of the shadow of a circular pipe will be a rectangle.
(iii) A book
The shape of the shadow of a book will be a rectangle.

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Each part: 1 Mark
(a) Edge is defined as the line segment where two faces of a polyhedron meet. A cylinder has no such line segments. Therefore, it has 0 edges.
(b)There are no vertices in a sphere as there are no sharp corners in a sphere.
(c)The figure which has all the three dimensions, viz. length, breadth and height, equal is called a cube. Therefore, the given figure is a cube.