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The pair of amphoteric hydroxides is:
a.) \[Al{\left( {OH} \right)_3},LiOH\]
b.) \[Be{\left( {OH} \right)_2},Mg{\left( {OH} \right)_2}\]
c.) \[B{\left( {OH} \right)_3},Be{\left( {OH} \right)_2}\]
d.) \[Be{\left( {OH} \right)_2},Zn{\left( {OH} \right)_2}\].

Answer
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Hint: Try to recall that amphoteric oxides are those substances which react with both acid and bases to give salt and water. They are generally metal oxide which can act as an acid as well as basic oxide. Now, by using this you can easily find the correct option.

Complete step by step solution:
It is known to you that electropositive elements form basic oxides while electronegative elements form acidic oxides.
Elements which are neither high electropositive nor high electronegative form amphoteric hydroxides.
Amphoteric hydroxides can act as either an acid or a base, depending on reaction conditions.
It is seen that metal hydroxide with a highly-charged central metal atom can be amphoteric. In addition to aluminium, metals such as zinc, tin, lead and beryllium can also form amphoteric oxide or hydroxides.
These hydroxides will behave as an acid or a base depends on the surrounding solution’s pH. So, all hydroxides are not amphoteric.
Now, coming to the question to find the pair of amphoteric oxides from the given options, you can easily find it by inspecting each option of the given option:
In option A, only \[Al{\left( {OH} \right)_3}\] is amphoteric whereas \[LiOH\] is basic hydroxide.
In option B, only \[Be{\left( {OH} \right)_2}\] is amphoteric whereas \[Mg{\left( {OH} \right)_2}\] is basic hydroxide.
In option C, only \[Be{\left( {OH} \right)_2}\] is amphoteric whereas \[B{\left( {OH} \right)_3}\] is Lewis base.
In option D, both \[Be{\left( {OH} \right)_2}\] and \[Zn{\left( {OH} \right)_2}\] are amphoteric hydroxides.
Therefore, from above we can easily conclude that option D is the correct option to the given question.

Note: It should be remembered that amphoteric oxide like \[Zn{\left( {OH} \right)_2}\] are used in calamine creams, baby powder and also as oral drugs.
Also, you should remember that \[Be{\left( {OH} \right)_2}\] is used in the processing of Beryllium-Metal alloys, Copper beryllium metal alloys etc.