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For \[\left[ X \right]\] and \[\left[ Y \right]\] , the incorrect choice is:
\[M{g_3}{B_2}\xrightarrow{{HC{l_{\left( {aq} \right)}}}}\left[ X \right] + MgC{l_2}\]
A.\[\left[ X \right] + {H_2}O\xrightarrow{{HC{l_{\left( {aq} \right)}}}}\left[ Y \right] + {H_2}\]
B.\[\left[ X \right]\] is \[BC{l_3}\] and \[\left[ Y \right]\] is \[{H_3}B{O_3}\]
C.\[\left[ X \right]\] with air and \[\left[ Y \right]\] on strong heating (red heat) give same compound
D.In \[\left[ Y \right]\], Boron completes its octet by removing \[{H^ + }\] from water molecule

Answer
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Hint: Magnesium boride on treating with hydrochloric acid forms diborane and magnesium chloride. When magnesium boride treated with water forms ortho boric acid and hydrogen gas. Diborane treated with oxygen and ortho boric acid on heating forms boron oxide.

Complete answer:
Magnesium and boron react with each other and form magnesium boride. Magnesium boride on treatment with hydrochloric acid forms diborane and magnesium chloride. The chemical reaction involved will be as follows:
\[2M{g_3}{B_2} + 12HCl \to 2{B_2}{H_6} + 6MgC{l_2}\]
Thus, \[\left[ X \right]\] is a diborane with the molecular formula of \[{B_2}{H_6}\]
When diborane reacts with water, boric acid will be formed. The chemical reaction will be as follows:
\[{B_2}{H_6} + 3{H_2}O\xrightarrow{{HC{l_{\left( {aq} \right)}}}}{H_3}B{O_3} + {H_2}\]
Thus, \[\left[ Y \right]\] is boric acid with molecular formula of \[{H_3}B{O_3}\]
When diborane is treated with air, it reacts with oxygen and forms boric oxide along with water.
The chemical reaction will be as follows:
\[{B_2}{H_6} + 3{O_2} \to {B_2}{O_3} + 3{H_2}O\]
The compound formed in the above chemical reaction is boric oxide with molecular formula of \[{B_2}{O_3}\]
When boric acid is heated it forms boric oxide with molecular formula of \[{B_2}{O_3}\]
The chemical reaction will be as follows:
\[{H_3}B{O_3}\xrightarrow{\Delta }{B_2}{O_3} + 3{H_2}O\]
Thus, in the above two reactions the products are the same.
When boric acid is treated with water it removes \[{H^ + }\] from water molecules.

Thus, option (B) is the correct one.

Note:
All the options except B are the reactions shown above. \[\left[ X \right]\] is a diborane with the molecular formula of \[{B_2}{H_6}\], but in the given option B, \[\left[ X \right]\] is boron trichloride with molecular formula of \[BC{l_3}\]. Thus, it is not the correct option though \[\left[ Y \right]\]is \[{H_3}B{O_3}\].