In the reaction:
${\text{M}}{{\text{e}}_{\text{3}}}{\text{N}}\xrightarrow{{BrCN}}{\text{A}}\xrightarrow[{(ii)\Delta }]{{(i){H_2}O}}{\text{B}}$
(A)${\text{MeN}}{{\text{H}}_{\text{2}}}$
(B)${\text{M}}{{\text{e}}_{\text{3}}}{\text{NO}}$
(C)${\text{M}}{{\text{e}}_{\text{2}}}{\text{NH}}$
(D)${\text{M}}{{\text{e}}_{\text{2}}}{\text{NCOOH}}$

Hint: Tertiary amine reacts with cyanogen bromide to give cyanamide; this reaction is named as von Braun reaction. Primary or secondary amines also react with cyanogen bromide to give mono or di substituted cyanamide.

Complete step by step solution:
The above given reaction contains the two reactants trimethylamine $\left( {{\text{M}}{{\text{e}}_{\text{3}}}{\text{N}}} \right)$and cyanogen bromide $\left( {{\text{BrCN}}} \right)$. Trimethylamine reacts with cyanogen bromide to give dimethyl cyanamide $\left( {{\text{M}}{{\text{e}}_{\text{2}}}{\text{N - CN}}} \right)$ as compound A and methyl bromide $\left( {{\text{MeBr}}} \right)$.
The compound A dimethyl cyanamide upon hydrolysis with water and heated, it gives ${\text{M}}{{\text{e}}_{\text{2}}}{\text{NCOOH}}$.
The reaction for the above question is given below mentioning the unknown compounds A and B.
\[{\text{M}}{{\text{e}}_{\text{3}}}{\text{N}}\xrightarrow{{BrCN}}{\text{M}}{{\text{e}}_{\text{2}}}{\text{NCN}}\xrightarrow[{(ii)\Delta }]{{(i){H_2}O}}{\text{M}}{{\text{e}}_{\text{2}}}{\text{NCOOH}}\].

Therefore the compound B in the above given question is option (D).

Additional information:
There is an organic named reaction called the von Braun reaction in which tertiary amines are heated with cyanogen bromide to give N,N-dimethyl cyanamide. The reaction mechanism is nucleophilic substitution reaction in which cyanide ion is substituted upon the alkyl group of the amine. The methyl group then is attacked by the bromide ion to form methyl bromide and dimethyl cyanamide, which is a bimolecular nucleophilic substitution reaction and it follows second order kinetics. The cyanamide is heated in presence of steam or water to give ${\text{M}}{{\text{e}}_{\text{2}}}{\text{NCOOH}}$. Cyanamide undergoes oxidation with oxygen as an oxidizing agent to give cyano compounds.

Note: Dimethyl cyanamide the compound A in the above reaction is an organic amide and is hazardous. It is a flammable compound and gives flame with oxygen or oxidizers and releases toxic nitrogen dioxide upon reaction with steam or water in heat.