Step 1: Split the question into three tasks.
We must finish two equations and give one reason. Each equation needs both sides to have equal atoms.
Step 2: Equation (i) using electron bookkeeping.
Heating solid permanganate at 513 K is the usual way to get a little oxygen in the lab.
The products are potassium manganate, manganese dioxide and oxygen.
Count the electrons. Two Mn atoms start at +7. One becomes +6 (gains 1 electron) and one becomes +4 (gains 3 electrons). Total gain is 4 electrons.
Two oxide ions change to $(\$mathrm{O_2}$) and lose 4 electrons in all. Gain equals loss, so the coefficients are right.
\$[ 2$mathrm{KMnO_4} \$xrightarrow{$Delta,\$ 513$ \$mathrm{K}} $mathrm{K_2MnO_4} + \$mathrm{MnO_2} + $mathrm{O_2} \$]
Step 3: Equation (ii) using solubility.
No oxidation number changes here. The two salts only swap partners.
In the solution, $(\$mathrm{K_2Cr_2O_7}$) is the least soluble of the four possible salts, so it crystallises out.
Write the swap: \$($mathrm{Na^+}\$) leaves with $(\$mathrm{Cl^-}$), and \$($mathrm{K^+}\$) joins the dichromate ion.
$[ \$mathrm{Na_2Cr_2O_7} + 2$mathrm{KCl} \$longrightarrow $mathrm{K_2Cr_2O_7} + 2\$mathrm{NaCl} $]
Atom count: Na 2, K 2, Cl 2, Cr 2, O 7 on both sides.
Step 4: Reason for part (b).
Lanthanoids differ only in the number of 4f electrons, and these are buried deep inside the atom.
As the nuclear charge goes up by one unit at each step, the 4f electrons fail to pull the outer shell in fully. The size still falls a little, step by step.
This steady, tiny fall is the lanthanoid contraction.
Neighbouring ions have almost the same radius, charge (+3) and solubility of salts. So chemical tricks like precipitation do not split them cleanly.
Only repeated ion exchange or solvent extraction gives the pure elements.
Final Answer:
The balanced equations are \$(2$mathrm{KMnO_4} \$to $mathrm{K_2MnO_4} + \$mathrm{MnO_2} + $mathrm{O_2}\$) and $(\$mathrm{Na_2Cr_2O_7} + 2$mathrm{KCl} \$to $mathrm{K_2Cr_2O_7} + 2\$mathrm{NaCl}$). Lanthanoids are hard to separate because of the very small change in size and the same +3 chemistry. \$[ $boxed{\$text{Lanthanoid contraction}} \]