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In electrolysis of sodium hydroxide solution;
(A). Hydrogen is discharged at cathode.
(B). Hydrogen is discharged at anode.
(C). Sodium is liberated at anode
(D). No hydrogen is liberated

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Answer
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Hint:
Electrolysis is a process in which electric current is passed through a substance to effect a chemical change. This process is carried out in an electrolytic cell. An electrolytic cell is an apparatus which consists of positive and negative electrodes in a salt solution.

Complete step by step answer:
In water, sodium hydroxide splits into $N{a^ + }$ions and $O{H^ - }$ions. When electrolyzing a basic solution, the$O{H^ - }$ions undergo the reaction at anode.
At Anode
$4O{H^ - } \to {O_2} + 2{H_2}O + 4{e^ - }$
          (Oxidation)
So, we can say that, at anode we get ${O_2}$ and ${H_2}O$ both. The reason behind it not producing only ${H_2}O$ is because of not enough protons around to do that. It bubbling hydrogen gas through the solution you get following reaction
$2O{H^ - } + {H_2} \to 2{H_2}O + 2{e^ - }$
At same time, the reaction happened at the cathode of the reaction
$2{H_2}O + 2{e^ - } \to {H_2} + 2O{H^ - }$
This replenishes the \[O{H^ - }\] ions lost at anode. Now that this also means that if for some reason the reactions at your two electrons are not balanced, then pH of solution will shift.
Therefore, option (A) is correct.

Additional Information:
→ Sodium hydroxide is also known as caustic soda.
→$NaOH$ is commonly used as a base.
→ Electrolysis of concentrated sodium chloride $\left( {NaCl} \right)$ solution (brin) produces chlorine gas, hydrogen gas and aqueous sodium hydroxide.
 $2NaCl\left( {a.q} \right) + 2{H_2}O\left( l \right) \to {H_2}\left( g \right) + C{l_2}\left( g \right) + 2NaOH\left( {a.q} \right)$
→$C{l_2}$ is produced at anode (positive electrode).
→${H_2}\left( g \right)$and $NaOH\left( {a.q} \right)$ are produced at cathode (negative electrode)
→There types of electrolytic cells are used produce sodium hydroxide from brine
(1) Caster-Kellner cell (Mercury process)
(2) Nelson Diaphragm cell.
(3) Membrane cell.

Note:Electrodes are commonly made of metals such as platinum and zinc. AS great conductors of electricity, they’re found in electrical devices such as batteries and electrolytic cells.