Euler's Formula

Q.

A cone has ___ edge(s).

Q.

A polyhedron can have 8 faces, 12 vertices and 18 edges.

1. Yes

2. No

Q.

Let $\stackrel{\rightharpoonup }{a}$ and $\stackrel{\rightharpoonup }{b}$be two non-zero vectors perpendicular to each other and$\left|\stackrel{\rightharpoonup }{a}\right|=\left|\stackrel{\rightharpoonup }{b}\right|$. If $\left|\stackrel{\rightharpoonup }{a}\times \stackrel{\rightharpoonup }{b}\right|=\left|\stackrel{\rightharpoonup }{a}\right|$, then the angle between the vectors $\left(\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\left(\stackrel{\rightharpoonup }{a}\times \stackrel{\rightharpoonup }{b}\right)\right)$ and $\stackrel{\rightharpoonup }{a}$ is equal to.

1. ${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

2. ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

3. ${\sin }^{-1}\left(\frac{1}{\sqrt{6}}\right)$

4. ${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

Q.

If a polyhedron has 6 faces and 8 vertices, then it has ___ edges.

1. 10

2. 9

3. 11

4. 12

Q. A square pyramid has 5 faces, 5 vertices and edges.

1. 8
2. 2
3. 4

Q.

If a polyhedron has 18 edges, 12 vertices, then it has ___ faces.

1. 10

2. 9

3. 8

4. 6

Q.

Fill in the blanks:
(i) A cuboid has ...... rectangular faces ...... edges and ...... vertices.
(ii) A cylinder has ...... curved face and ...... flat faces.
(iii) A cone has one ...... face and one ...... face.
(iv) A sphere has a ...... surface.

Q. If a polyhedron has six faces eight vertices find the number of edges

Q.

A polyhedron with 8 edges and 5 faces has ___ vertices.

Q.

For the above figure,
F + V - E = ___?
where,
F is the number of faces,
V is the number of vertices and
E is the number of edges.

1. 2

2. 3

3. 4

4. 0

Q. Can a polyhedron have 10 faces, 20 edges and 15 vertices? Justify you answer. [4 MARKS]

Q.

If a polyhedron have 8 faces, 26 edges , then find the number of vertices

1. 20

2. 16

3. 24

4. 25

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+2\overrightarrow{b}$ and $5\overrightarrow{a}-4\overrightarrow{b},$ are perpendicular to each other, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

1. ${45}^{\circ }$
2. ${60}^{\circ }$
3. $co{s}^{-1}\left(\frac{1}{3}\right)$
4. $co{s}^{-1}\left(\frac{2}{7}\right)$

Q.

For the above figure,
F + V - E = ___
where,
F is the number of faces,
V is the number of vertices and
E is the number of edges.