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The compound 'X' reacts with ammoniacal \[AgN{O_3}\] to give a white precipitate and on oxidation with hot alkaline \[KMn{O_4}\] gives the acid ${(C{H_3})_2}CHCOOH$ . Therefore, ' \[X\] ' is:
A.$C{H_2} = CH - CH = CH - C{H_2}$
B.$C{H_3}{(C{H_2})_2}C \equiv CH$
C.${(C{H_3})_2}CH - C \equiv CH$
D.${(C{H_3})_2}C = C = C{H_2}$

Answer
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Hint:In the given question firstly we have to get the condition for both the reactions that are performed in the question and when we will get that we just have to apply the conditions in order to get the reaction and then we can get the correct answer for the given problem. Like for the reaction with the ammonical \[AgN{O_3}\] , the reaction would be $C{H_3}{(C{H_2})_2}C \equiv CH \to white{\text{ }}precipitate$ .

Complete step by step answer:
Here according to the question statement we get that the supposed compound ' \[X\] ' in the problem have subsequent reaction that gives a specific kind of products under the given conditions :
-The compound it reacts to is : \[AgN{O_3}\] also categorized as the tollens reagent in organic chemistry.
-The colour of the precipitate obtained in the process of this reaction is : White
-The product obtained when this same compound is oxidised in the presence of hot alkaline \[KMn{O_4}\] : the carboxylic acid ${(C{H_3})_2}CHCOOH$
In the question we can see and observe that there is the presence of the two reactions of the same compound in the two different cases. So we have to take them case wise :
Case 1 : In this situation there is the reaction of the compound $X$ with the \[AgN{O_3}\]
So the reaction for the given reaction would be :
$C{H_3}{(C{H_2})_2}C \equiv CH \to white{\text{ }}precipitate$ under the presence of the \[AgN{O_3}\]
As the generation of the white precipitate gives the output that the given compound should have an unsaturated triple bond.
Case 2 : In this situation there is the reaction of the compound $X$ is in the presence of hot alkaline \[KMn{O_4}\]
So the reaction for the given reaction would be :
$C{H_3}{(C{H_2})_2}C \equiv CH \to {(C{H_3})_2}CH COOH$ oxidation under the presence of the hot alkaline \[KMn{O_4}\]
So we can clearly say that the correct option would be option C, ${(C{H_3})_2}CH - C \equiv CH$.

Note:
Tollens' reagent is a chemical reagent used to determine the presence of aldehyde and aromatic aldehyde functional groups along with some alpha-hydroxy ketone which can tautomerize into aldehyde. The reagent consists of a solution of silver nitrate, ammonia and some sodium hydroxide.