1. We identify each structure in List-I and match it with the correct IUPAC name in List-II.

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Step 1: Identify structure (B)

Given:

$$(\mathrm{CH_3})_2\mathrm{C}(\mathrm{C_3H_7})_2$$

This means one central carbon is attached to:

• two methyl groups,
• two propyl groups.

So the skeleton is:

$$\mathrm{CH_3-CH_2-CH_2-C(CH_3)_2-CH_2-CH_2-CH_3}$$

The longest chain has $7$ carbons $\Rightarrow$ heptane. At carbon $4$, there are two methyl substituents.

Hence the name is:

$$\boxed{4,4\text{-Dimethylheptane}}$$

So,

$$(B) \to (III)$$

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Step 2: Use options to narrow remaining matches

Only options A and B have:

$$(B)\to(III)$$

So the answer must be either A or B.

Difference between A and B is in matching of (C) and (D):

• Option A: $(C)\to(IV)$ and $(D)\to(I)$
• Option B: $(C)\to(I)$ and $(D)\to(IV)$

So we now identify (C) and (D).

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Step 3: Identify the alkene/diene structures

From the given list, the two remaining names are:

• (I) $4$-Methylpent-$1$-ene
• (IV) $2$-Methyl-$1,3$-pentadiene

Name (I): $4$-Methylpent-$1$-ene

Main chain: pent-$1$-ene

$$\mathrm{CH_2=CH-CH_2-CH(CH_3)-CH_3}$$

This has one double bond.

Name (IV): $2$-Methyl-$1,3$-pentadiene

Main chain: penta-$1,3$-diene

$$\mathrm{CH_2=C(CH_3)-CH=CH-CH_3}$$

This has two double bonds.

Thus:

• the structure with two double bonds must match (IV),
• the structure with one double bond must match (I).

From the given figure, structure (C) is the diene and structure (D) is the mono-alkene. Therefore,

$$(C)\to(IV), \qquad (D)\to(I)$$

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Step 4: Identify structure (A)

Now only name (II) remains for (A).

Name (II) is:

$$3\text{-Ethyl-}5\text{-methylheptane}$$

So,

$$(A)\to(II)$$

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Step 5: Final matching

Thus the correct matching is:

$$(A)\to(II), \quad (B)\to(III), \quad (C)\to(IV), \quad (D)\to(I)$$

This corresponds to Option A.

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Step 6: Compare with stored correct answer

Stored correct answer: A

Our derived answer: A

So they agree.