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Potassium manganate (${{\text{K}}_{\text{2}}}\text{Mn}{{\text{O}}_{\text{4}}}$) is formed when
(a) Chlorine is passed into aqueous ${{\text{K}}_{\text{2}}}\text{Mn}{{\text{O}}_{\text{4}}}$ solution.
(b) Manganese dioxide is fused with potassium hydroxide in air.
(c) Formaldehyde reacts with potassium permanganate in the presence of a strong alkali.
(d) Potassium permanganate reacts with concentrated sulphuric acid.

Answer
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Hint: Potassium manganate is the inorganic compound with the chemical formula ${{\text{K}}_{\text{2}}}\text{Mn}{{\text{O}}_{\text{4}}}$. This is a green coloured salt is an intermediate in the industrial synthesis of potassium permanganate ($\text{KMn}{{\text{O}}_{\text{4}}}$), a common chemical.

Complete answer:
We will write the reaction between reagents mentioned in the options and see if the product formed is potassium manganate.
When Manganese dioxide is fused with potassium hydroxide in air, manganese dioxide gets oxidised to give potassium manganate as the product. The reaction is as follows:
$2Mn{{O}_{2}}+4KOH+{{O}_{2}}\xrightarrow{{}}\text{ }2{{K}_{2}}Mn{{O}_{4}}+2{{H}_{2}}O$
Also, when formaldehyde reacts with potassium permanganate in the presence of a strong alkali, we get potassium manganate. Here, the manganese will convert from the +7 oxidation state to +6 oxidation state thereby oxidising the formaldehyde. So, this is also a method of formation of potassium manganate.
$2KMn{{O}_{4}}+HCHO\text{+}2KOH\to \text{ }{{K}_{2}}Mn{{O}_{4}}+{{H}_{2}}O+HCOOH$

Therefore, the correct answers are options (b) and (c).

Additional Information: You can test an unknown substance for the presence of manganese by heating the sample in strong KOH in air. The production of a green colouration indicates the presence of Mn. This green colour results from an intense absorption at 610 nm.
The easy method for the preparation of potassium manganate in the laboratory involves heating crystals or powder of pure potassium permanganate. Potassium permanganate will decompose into potassium manganate, manganese dioxide and oxygen gas.

Note: Please don’t be confused with the chemical formula of potassium manganate and potassium permanganate.
The option (a) given in the question is not correct because there will be no reaction.
Also, the last option is incorrect because here potassium permanganate reacts with concentrated sulphuric acid to form $\text{M}{{\text{n}}_{\text{2}}}{{\text{O}}_{\text{7}}}$ which is highly explosive.