In the absence of peroxide,what is the first step in the addition of $HBr$ to propene?

The reaction of propene with $HOCl$ proceeds via the addition of

The product $(P)$ in the given reaction is:

Addition of $HBr$ to propene yields $2-$bromopropane,while in the presence of benzoyl peroxide,the same reaction yields $1-$bromopropane. Explain and give the mechanism.

Upon heating with acidic $KMnO_4$,an organic compound produces hexan$-1,6-$dioic acid as the major product. The starting compound is

Name the alkenes which will yield $1-$chloro$-1-$methylcyclohexane by their reaction with $HCl$. Write the reactions involved.