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The higher homologue of dimethylamine $ \left( C{{H}_{3}}-NH-C{{H}_{3}} \right) $ has the structure:
(A) $ C{{H}_{3}}-N\left( C{{H}_{3}} \right)-C{{H}_{3}} $
(B) $ \left( C{{H}_{3}}-C{{H}_{2}}-C{{H}_{2}}-N{{H}_{2}} \right) $
(C) $ \left( C{{H}_{3}}-NH-C{{H}_{2}}-C{{H}_{3}} \right) $
(D) $ \left( C{{H}_{3}}-CH\left( N{{H}_{2}} \right)-C{{H}_{3}} \right) $

Answer
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Hint: We need to find the higher homologue of dimethylamine. Homologues have the same general formula but have higher or lower numbers of carbon atoms in them.

Complete step by step answer
We already know that homologues have the same general formula.
Also, they belong to the same functional group and differ in the order of $ C{{H}_{2}} $ .
Thus, we add $ C{{H}_{2}} $ to methylamine to get the higher homologue.
 $ \left( C{{H}_{3}}-NH-C{{H}_{3}} \right)+C{{H}_{2}}\to $
So, we need to see from the above options, and select the correct value.
Thus, the correct answer is option C.

Additional Information
Such organic compounds that vary from one another by a repeating unit and have the same general formula form a series of compounds. Alkanes with general formula $ {{C}_{n}}{{H}_{2n+2}} $ , alkenes with general formula $ {{C}_{n}}{{H}_{2n}} $ and alkynes with general formula $ {{C}_{n}}{{H}_{2n-2}} $ form the most basic homologous series in organic chemistry. The chemical formula of methane is $ C{{H}_{4}} $ . It is the first member of alkane homologous series.
Methylamine is an organic chemical compound, the simplest of primary amines. It is a derivative of ammonia, wherein one H atom is replaced by a methyl group. Methylamine is used to produce a variety of important chemical products: pharmaceuticals; pesticides; fuel additives; explosives; solvents and cleaning agents; photographic processing chemicals; and chemicals for the tanning of leather and dye processes. Methylamine, appears as a colourless gas or a liquid. It has a pungent fishy odour resembling the odour of ammonia.

Note
In organic chemistry, a homologous series is a sequence of compounds with the same functional group and similar chemical properties in which the members of the series can be branched or unbranched, or differ by $ C{{H}_{2}} $ . Also, they have a common general method of preparation.