If we have 2 GiB RAM and a processor Pentium 4 with 3.2 GHz and 2 MiB Cache , What will you use , Linux 32 bit or 64 bit , and why ?

32bit, the current HW doesn't support/require64Bit.

if your process support64bit archtechure then you can use x64 OS other wise you must be used a32bit OS x86

32bit OS limits the RAM to4GB and processor physically1

well its a two edged sword if u dont intend to have high end applications in future X86 would work for you but if you anticipate to run high end applications X64 works best. NOTE: The32-bit Linux kernel can be recompiled to handle up to64GB of RAM if you're particularly concerned about staying32-bit.

So efficiency wise i would use X86 because at the same RAM size the X64 will be relatively slower.