Organic General problems

1)In a book dat i have been reading it is said -"Dipole moment of trans-dichloroethylene is zero while cis-dichloroethylene has a definite dipole moment"...WHY?

This is a very good question.

See about the center, the direction of dipole is not cancelled. The component in the cis conformer remains un-removed. See that image

2) Is there any technique or method to solve the following problems--
(a)How many chain isomers can be obtained from the alkane C6H14 ?
(b) The maximum number of an alkene with molecular formula C4H8 is?
(c) The total number of isomers for the compounds of the formula C4H8O is ?

There is no technique or formula.. you will see some formulas here and there but they are not reliable. This will come directly after a couple of problems that you solve. :)

What is α-hydrogen?

Source: Wikipedia

The alpha carbon in organic chemistry refers to the first carbon that attaches to a functional group (the carbon is attached at the first, or alpha, position).[1] By extension, the second carbon is the beta carbon,[2] and so on. This nomenclature can also be applied to the hydrogen atoms attached to the carbons. A hydrogen attached to an alpha carbon is called an "alpha-hydrogen" (α-hydrogen), a hydrogen on the beta-carbon is a beta-hydrogen, and so on.

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1) For the explanation to the 1st question...how come the dipole moment of trans-2,3-Dichloropentene-2 is not zero?

I did not understand what you mean.. we had to show that it gets cancelled. ANd it does indeed!

3) For the explanation to ques 3.....is the Carbon atom of the methyl group shown in the diagram also a α-Carbon?

Yes that is also an alpha carbon.