Question

the valency of nitrogen in nitrogen dioxide,O2 is:
a) 1
b)2
c)3
d)4

Answer 1

the valency of nitrogen in nitrogen dioxide,O2 is:
c) 3
Nitrogen has 7 electrons with 5 electrons on its valence shell. 
That means it can give away 5 and take 3. 
Oxygen has 8 electrons with 6 electrons on its valence shell. 
That means it can give away 6 and take 2. 
The valence of Nitrogen however is 3 and Oxygen is 2. 

Just a little explanation. Valence can be tricky and this is quick, dirty and frequently correct perspective. In 1916 a guy by the name of G. N. Lewis has proposed an idea of an outer valence shell. It is always to contain less or equal to 8 electrons. these electrons will be responsible for chemical bonding between the atoms. Oxygen has 8 electrons (2+6). it needs 2 electrons to complete its valence shell so we say that Oxygen is -2. 
Similarly Nitrogen is -3. If you look at the same row where Nitrogen and Oxygen are at it end we find Neon(Ne). Neon has 10 electrons(2+8). Its our shell is full and this is why it is an inert gas. It reacts only with its own atoms. 

Oh yes. 
In Nitrogen Dioxide case two Oxygen atoms need 4 electrons so they share 3 Nitrogen electrons and one among themselves.