Nitrogen Dioxide ($N{{O}_{2}}$) cannot be obtained by heating which of the following chemical compounds?
$\begin{align}
 & a)KN{{O}_{3}} \\
 & b)Pb{{(N{{O}_{3}})}_{2}} \\
 & c)Cu{{(N{{O}_{3}})}_{2}} \\
 & d)AgN{{O}_{3}} \\
\end{align}$

Hint: The heating of alkali metal nitrates, excluding Lithium, results in the production of alkali metal nitrates and Oxygen gas as a result of the initial decomposition.
With this information, now try and solve the given question.

Complete step-by-step answer:
Since this question is related to the properties of certain elements and their corresponding nitrates, let us try and break this down by going through the decomposition reactions that take place upon heating of each of the given nitrates.

Let us start with Lead Nitrate:
$2Pb{{(N{{O}_{3}})}_{2}}\xrightarrow{873K}2PbO\downarrow +4N{{O}_{2}}+{{O}_{2}}\uparrow $
As we can observe, the decomposition of Lead Nitrate does lead to the production of Nitrogen Dioxide along with the formation of Lead Oxide (which is a white precipitate) and Oxygen gas which is immediately released into the atmosphere.

Now, let us observe the decomposition of Cupric Nitrate:
$2Cu{{(N{{O}_{3}})}_{2}}\xrightarrow{453K}2CuO\downarrow +4N{{O}_{2}}+{{O}_{2}}\uparrow $
As we can observe, the decomposition of Cupric Nitrate does lead to the production of Nitrogen Dioxide along with the formation of Copper (II) Oxide (which is a black precipitate) and Oxygen gas which is immediately released into the atmosphere.

Similarly, let us now observe the decomposition of Silver Nitrate:
$2AgN{{O}_{3}}\xrightarrow{713K}2Ag+2N{{O}_{2}}+{{O}_{2}}$
As we can observe, the decomposition of Silver Nitrate does lead to the production of Nitrogen Dioxide along with the production of a silver mirror and Oxygen gas which is immediately released into the atmosphere.

Now, let us observe the decomposition of Potassium Nitrate:
$2KN{{O}_{3}} \longrightarrow 2KN{{O}_{2}}+{{O}_{2}}$
As we can observe, the decomposition of Potassium (an alkali metal) Nitrate does not result in the production of Nitrogen Dioxide.

Therefore, the answer to this question is a)

NOTE: A thorough understanding of the properties of alkali metals is required to help solve this question. Remember that, apart from lithium, all alkali metal nitrates decompose in a reversible reaction to give the corresponding nitrite and release oxygen into the atmosphere.