Which one of the following compounds can exist as cis-trans isomers?

The correct option is 1, 2-Dimethylcyclohexane.

To determine which compound can exist as cis-trans (geometric) isomers, we need to examine the structures of the given molecules:
1. 2-Methylhex-2-ene: The double-bonded carbon-2 is bonded to two identical methyl groups (-CH₃). For cis-trans isomerism to occur in an alkene, each carbon of the double bond must be attached to two different groups. Since carbon-2 is bonded to two identical groups, it cannot show cis-trans isomerism.
2. 1, 1-Dimethylcyclopropane: In cyclic compounds, cis-trans isomerism requires substituents on different ring carbons. In this molecule, both methyl groups are attached to the same carbon (carbon-1), which means there is no possibility of cis-trans isomerism.
3. 1, 2-Dimethylcyclohexane: This cyclic compound has two methyl groups on adjacent carbons (carbon-1 and carbon-2). Because the ring prevents free rotation, these methyl groups can point either to the same side of the ring plane (the cis isomer) or to opposite sides of the ring plane (the trans isomer). Thus, it can exist as cis-trans isomers.
4. Pent-1-ene: The double-bonded carbon-1 is bonded to two identical hydrogen atoms. Since one of the alkene carbons has two identical groups attached to it, this molecule cannot exhibit cis-trans isomerism.

Therefore, 1, 2-Dimethylcyclohexane is the only compound among the options that can exist as cis-trans isomers.