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Scale of map = Size drawn/Actual size 

= 9 cm/ 900 cm (because 9 m = 900 cm) 

= 1/100 

Thus, scale is 1:100.

(d) 1.1 

Given, scale 1 cm = 0-5 km ,

The distance between school and the book shop shown in map is equal to 2.2 cm. So, the actual distance between them will be = 2.2 x 0.5 km
= 1.1 km

(d) V = 4, F = 6, E = 6

We have,

Euler’s formula for any polyhedron is, F + V – E = 2

Where, face (F) = 6, Vertex (V) = 4, Edge (E) =6

Then,

6 + 4 – 6 = 2

LHS 6 + 4 -6

10 – 6

4

RHS = 2

By comparing LHS and RHS

LHS ≠ RHS

(b) 1: 10

From the question it is given that,

An architect has shown the height of the room as 33 cm

The actual height of the room is 330 cm

Then, the scale used by an architect is = Drawn size/actual size

= 33/330 … [divide both by 33]

= 1/10

= 1: 10

A pyramid on an n sided polygon has n + 1 faces.

We know that, in a pyramid, the number of faces is 1 more than the number of sides of the polygohal base.

(d) 3: 2

By observing the given map,

The number of general stores = 6

The number of ground = 4

Then,

The ratio of the number of general stores and that of the ground is = 6/4

= 3/2

= 3: 2

(b) 2

The number of hospitals in the town is 2.

If a solid shape has 12 faces and 20 vertices, then the number of edges in this solid is 30.

We have,

Euler’s formula for any polyhedron is, F + V – E = 2

Given, F = 12, V = 20

Where, face (F) = 12, Vertex (V) = 20, Edge (E) =?

Then,

12 + 20 – E = 2

32 – E = 2

32 – 2 = E

Edges (E) = 30

(i) The cube can cast a shadow in the shape of a rectangle.True

(ii) The cube can cast a shadow in the shape of a hexagon.False.

(c) Octagon

A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles.

(i) True

(ii) False

(i) (1) → Top, (2)→ Side (3) → Front

(ii) (1) → Top, (2)→ Side (3) → Front

(iii) (1)→ Side, (2)→ Front (3) → Top

(iv) (1)→ Side, (2)→ Top (3) → Front

A regular polyhedron is a solid made up of congruent faces.
[according to the definition of regular polyhedron]

(c) 5

According to the map, the number of schools in the town is 5.

(a) 3 triangles

3 triangles will not form a polyhedron because it must have more than four faces. So, it is not possible in 3 triangles which have 3 faces only.

(a) Cone has one edge.

(b) Cylinder has two edges.

(c) Sphere has no edge.

(d) Octagonal pyramid has 16 edges.

(e) Hexagonal prism has 18 edges.

(f) Kaleidoscope has 9 edges.

False

A net is a flat figure that can be folded to form a closed, three-dimensional object. So, for an object, more than one net is possible but it is not true for the objects of all shapes.

For finding the number of cubes in the given shapes you have to count all cubes which are visible or not. Hence, we have the total number of cubes for the given figures as:

(a) 10 cubes (b)10 cubes
(c) 10 cubes (d) 9 cubes
(e) 11 cubes (f) 9 cubes
(g) 11 cubes (h) 110 cubes
(i) 113 cubes(j)66 cubes
(k) 15 cubes (l)14 cubes

(a) Rectangle  

A two dimensional figure have two dimensions (measurements) like length and breadth. In the given options, only rectangle has two dimensions, i.e. length and breadth.

Square prism is called a cube.

We know that, a square prism has a square base, a congruent square top and the sides are parallelograms. So, it is also a cube.

(c) Cube

Because, a cube is a platonic solid because all six of its faces are congruent squares.

True

A cube is a prism because it has a square base, a congruent square top and the lateral sides are parallelograms.

First figure is made by using a hemisphere and cylinder. In this figure, cylinder is mounted by hemisphere.

The second figure is made by using a cone and hexagonal prism. In this figure, hexagonal prism is mounted by a cone.

Equidistance figures are drawn on dotted line paper.

(i) V + F = E + 2

6 + F = 12 + 2

F = 14 - 6

F = 8

Therefore number of faces are 8

(ii) V + F = E + 2

V + 5 = 9 + 2

V = 11 - 5

V = 6

Therefore number of vertices are 6

(iii) V + F = E + 2

12 + 20 = E + 2

E = 32 - 2

E = 30

Therefore number of edges are 8

Yes, a square is a three dimensional shape with six rectangular shaped sides, at least two of which are squares. Cubes are rectangular prisms length, width and height of same measurement.

For any polyhedron, 

F + V – E = 2 

Here, F = 7, V = 10, E = ? 

Using above formula, 

⇒7 + 10 – E = 2 

⇒17 – E = 2 

⇒17 – 2 = E 

⇒ Ε = 15

(a) Polyhedron

A polyhedron is formed by four or more polygons that intersect only at their edges. The faces of a regular polyhedron are all congruent regular polygons and the same number of faces intersect at each vertex.

For any polyhedron, 

F + V – E = 2 

Here, F = 12, V = ?, E = 30 

Using above formula, 

12 + V – 30 = 2 

V – 18 = 2 

V = 2 + 18 

V = 20

False

Pentagonal prism has 2 pentagons, one on the top and other on the base.

False

Euler’s formula is true only for polyhedrons,
i.e. F+V-E = 2

Where F = faces, V = vertices
and E = edges

False

A regular octahedron is obtained by joining two congruent square pyramids such that the vertices of the two square pyramids coincide. It has eight congruent equilateral triangular faces.

(i) Vertices = 10

Faces = 7

Edges = 15

V + F = E + 2

10 + 7 = 15 + 2

17 = 17

(ii) Vertices = 9

Faces = 9

Edges = 16

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

(iii) Vertices = 14

Faces = 8

Edges = 20

V + F = E + 2

14 + 8 = 20 + 2

22 = 22

(iv) Vertices = 6

Faces = 8

Edges = 12

V + F = E + 2

6 + 8 = 12 + 2

14 = 14

(v) Vertices = 9

Faces = 9

Edges = 16

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

(a)-(iv) Because multiplication of numbers on adjacent faces are equal, where 6×4 = 24 and 4×4 = 16

(b)-(i) Because multiplication of numbers on adjacent faces are equal, where 3×3 = 9 and 8×3 = 24

(c)-(ii) Because multiplication of numbers on adjacent faces are equal, where 6×4 = 24 and 6×3 = 18

(d)-(iii) Because multiplication of numbers on adjacent faces are equal, where 3×3 = 9 and 3×9 = 27

(i) Vertices = 10

Faces = 7

Edges = 15

By using Euler’s formula

V + F = E + 2

10 + 7 = 15 + 2

17 = 17

Hence verified.

(ii) Vertices = 9

Faces = 9

Edges = 16

By using Euler’s formula

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

Hence verified.

(iii) Vertices = 14

Faces = 8

Edges = 20

By using Euler’s formula

V + F = E + 2

14 + 8 = 20 + 2

22 = 22

Hence verified.

(iv) Vertices = 6

Faces = 8

Edges = 12

By using Euler’s formula

V + F = E + 2

6 + 8 = 12 + 2

14 = 14

Hence verified.

(v) Vertices = 9

Faces = 9

Edges = 16

By using Euler’s formula

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

Hence verified.

a – (ii)

b – (iii)

c – (iv)

d – (i)

(a) — (iv) Because multiplication of numbers on adjacent faces are equal, i.e 6×4 = 24 and 4×4 = 16

(b) — (i) Because multiplication of numbers on adjacent faces are equal, i.e 3×3 = 9 and 8×3 = 24

(c) — (ii) Because multiplication of numbers on adjacent faces are equal, i.e 6×4 = 24 and 6×3 = 18

(d) — (iii) Because multiplication of numbers on adjacent faces are equal, i.e 3×3 = 9 and 3×9 = 27

(i) 3 triangles?

No, Because a polyhedron is a solid shape bounded by polygons.

(ii) 4 triangles?

Yes, Because four triangles will form a tetrahedron, which is a polygon.

(iii) a square and four triangles?

Yes, because a square pyramid has a square and four triangles as its faces. Since pyramid is a polyhedron whose base is a polygon of any number of sides and whose other faces are triangles with common vertex.

For any polyhedron, F + V – E = 2 

Here, F = 20, V = 12, E = ? 

Using above formula,

20 + 12 - E = 2

32 - E = 2

E = 32 - 2

E = 30

Figure (iv), (v), (vi) are the nets for a cube.

False

The cylinder has a congruent cross-section which is a circle, so it could be called as a circular prism.

(i) From first figure Square pyramid can be made

(ii) From second figure Triangular prism can be made

(iii) From third figure Triangular prism can be made

(iv) From fourth figure Hexagonal prism can be made

(iv) From fifth figure Hexagonal pyramid can be made

(v) From fifth figure Cube can be made

(i) From figure (i), a Square pyramid can be made by folding each net.

(ii) From figure (ii), a Triangular prism can be made by folding each net.

(iii) From figure (iii), a Triangular prism can be made by folding each net.

(iv) From figure (iv), a Hexagonal prism can be made by folding each net.

(iv) From figure (v), a Hexagonal pyramid can be made by folding each net.

(v) From figure (vi), a Cube can be made by folding each net.

Fig (i) is a dice because the sum of numbers on opposite faces is 7 (3 + 4 = 7 and 6 + 1 = 7).

Figure (i), is a dice. Since the sum of numbers on opposite faces is 7 (3 + 4 = 7 and 6 + 1 = 7).

Matching is such way :

(i) ↔ (b),
(ii) ↔ (a),
(iii) ↔ (e),
(iv) ↔ (c),
(v) ↔ (d)

(i) (c), (ii) (a).