The reaction that gives (E)-2-methylhex-3-ene as the major product is

Cocept:

Option (1) 

This is Peterson olefination: In which hydroxy silanes under go anti-elimination upon reaction with acids and sy elimination upon reaction with bases. Leaving groups should be antiperiplanar to each other.

Option (2) 

Proceeds via acid hydrolysis followed by dehydration which gives an alkene with no stereochemistry.

Option (3)

This is the Wittig reaction in which t-BuOK forms phosphorus ylide which attacks on carbonyl carbon to form a four-membered ring structure followed by elimination to give a Z-alkene.

Option (4) 

This reaction proceeds via Julia olefination in which an aldehyde reacts with phenyl sulphones. It involves a four-membered ring intermediate which after elimination gives E-alkene.

Explanation:-

The mechanisms of these reactions are given below.

Conclusion:-

Hence option (4) is the correct one.