Among the reactions (a) - (d), the reaction(s) that does/do not occur in the blast furnace during the extraction of iron is/are:
(a) CaO + SiO$$_2$$ $$\rightarrow$$ CaSiO$$_3$$
(b) 3Fe$$_2$$O$$_3$$ + CO $$\rightarrow$$ 2Fe$$_3$$O$$_4$$ + CO$$_2$$
(c) FeO + SiO$$_2$$ $$\rightarrow$$ FeSiO$$_3$$
(d) FeO $$\rightarrow$$ Fe + $$\frac{1}{2}$$O$$_2$$

In the blast-furnace process the chief aim is to reduce the iron oxides present in the ore to metallic iron. Simultaneously we have to remove the acidic impurity $$\text{SiO}_2$$ with the help of a basic flux $$\text{CaO}$$ to form a fusible slag. Let us examine each of the four reactions one by one and verify whether it can actually take place inside the furnace.

We start with reaction (a):

$$\text{CaO}+\text{SiO}_2 \;\longrightarrow\; \text{CaSiO}_3$$

$$\text{CaO}$$ is a basic oxide obtained by the decomposition of limestone. $$\text{SiO}_2$$ is an acidic oxide present as gangue. A rule from elementary inorganic chemistry states that a basic oxide reacts with an acidic oxide to give a salt (slag). Therefore this reaction indeed occurs in the slag zone of the furnace and is essential for removing silica impurity.

Next we look at reaction (b):

$$3\,\text{Fe}_2\text{O}_3 + \text{CO} \;\longrightarrow\; 2\,\text{Fe}_3\text{O}_4 + \text{CO}_2$$

In the upper part of the furnace the temperature is comparatively low. Under such conditions carbon monoxide is not strong enough to reduce $$\text{Fe}_2\text{O}_3$$ completely to Fe, but it can carry out a partial reduction converting $$\text{Fe}_2\text{O}_3$$ to $$\text{Fe}_3\text{O}_4$$. Hence this reaction is also a legitimate step of the overall reduction sequence $$\text{Fe}_2\text{O}_3 \rightarrow \text{Fe}_3\text{O}_4 \rightarrow \text{FeO} \rightarrow \text{Fe}$$ inside the furnace.

Now consider reaction (c):

$$\text{FeO}+\text{SiO}_2 \;\longrightarrow\; \text{FeSiO}_3$$

Although $$\text{FeO}$$ is basic in nature, inside the furnace it is present only transiently and is immediately reduced further to metallic iron by CO according to

$$\text{FeO}+\text{CO}\;\longrightarrow\;\text{Fe}+\text{CO}_2.$$

At the operating temperatures of the furnace the equilibrium is overwhelmingly shifted toward the right in the above reduction step, so $$\text{FeO}$$ is not allowed to persist long enough to react with silica. Moreover the plant operator deliberately adds excess lime ($$\text{CaO}$$) as the preferred basic oxide for combining with $$\text{SiO}_2$$ because the product $$\text{CaSiO}_3$$ is lighter, more fusible and can be tapped off easily. Therefore the formation of $$\text{FeSiO}_3$$ is not observed in practice. Reaction (c) is thus absent from the blast furnace.

Finally we analyse reaction (d):

$$\text{FeO}\;\longrightarrow\;\text{Fe}+\dfrac12\,\text{O}_2$$

This step represents a thermal decomposition (an oxidation of iron) and would require the liberation of free oxygen gas. The interior of a blast furnace is a strongly reducing atmosphere filled with excess carbon monoxide and almost no molecular oxygen. Under such reducing conditions $$\text{FeO}$$ simply cannot supply $$\text{O}_2$$; instead, as already mentioned, it is reduced by CO to yield iron metal. Hence reaction (d) certainly does not occur in the process.

Summarising the discussion:

• Reactions (a) and (b) do take place.
• Reactions (c) and (d) do not take place.

Therefore the reactions that do not occur in the blast furnace are (c) and (d).

Hence, the correct answer is Option C.