${\left({\mathrm{CH}}_3\right)}_2\mathrm{NH}\stackrel{{\mathrm{KMnO}}_4}{\to }$ ${\left({\mathrm{CH}}_3\right)}_2\mathrm{NH} \stackrel{{\mathrm{H}}_2{\mathrm{SO}}_5}{\to } \mathrm{b}$

Here $\mathrm{a}$ and b are

• Secondary amines react with potassium permanganate to give tetra alkyl hydrazine. Thus, a is Tetramethylhydrazine.
• Caro's acid or ${\mathrm{H}}_2{\mathrm{SO}}_5$ removes the hydrogen attached to the secondary amine and replaces it with a $\mathrm{OH}$. Thus, b will be dimethyl hydroxyl amine.

Hence, the correct answer is option A) Tetramethylhydrazine, dimethyl hydroxyl amine.