Step 1: Understanding the Concept:
Geometric (cis-trans) isomerism occurs when each carbon atom of the double bond is attached to two different groups.
Step 2: Key Formula or Approach:
Check the groups on each vinylic carbon (\(C=C\)).
Step 3: Detailed Explanation:
(A) But-1-ene (\(\text{CH}_2=\text{CH-CH}_2\text{CH}_3\)): The terminal carbon is bonded to two identical hydrogen atoms. Therefore, it cannot show cis-trans isomerism.
(B) But-2-ene (\(\text{CH}_3\text{-CH=CH-CH}_3\)): Each carbon is bonded to H and methyl. It shows isomers.
(C) 3,4-Dimethylhex-3-ene: Each carbon is bonded to methyl and ethyl. It shows isomers.
(D) Pent-2-ene (\(\text{CH}_3\text{-CH=CH-CH}_2\text{CH}_3\)): One carbon is bonded to H, methyl; the other to H, ethyl. It shows isomers.
Step 4: Final Answer:
But-1-ene does not exhibit cis-trans isomerism.