Question

Why Nucleus is responsible for transmission of characters from parents to off Springs?

Answer 1

the nucleus. this contains instructions in the form of genes which are found on chromosomes. your chromosomes make up your DNA.

a mother passes on half her genes to her child.
a father passes on half his genes to his child.

the child therefore gets a mix of his/her mums and dads genes which appear as hereditary characteristics.
As nucleus contains the chromosomes which are contains genes which have hereditary informations we can say that nucleus is responsible for this.