Question

why an element having z>82 emits radiations. if due to instability then why it is unstable ???

Answer 1

Elements which are unstable have a nucleus which is unstable. It will eventually spit out a charged particle, or perhaps a neutron.

And resultant a daughter atom is formed.

Answer 2

A nucleus can be unstable if there are lots of protons in it, remember protons are positively charged so they repel each other in a chemical sense. So how does a nucleus become stable if it contains protons? the short ranged Nuclear Forces hold nucleons together.
now the nucleons inside the nucleons are in constant motion and sometimes if their have enough energy for a short time to move some nucleons out of the nucleus it will be repelled and radiate out