X and Y are partners. y wants to admit his son K into business. Can K become the partner of the firm without the consent of X?

No in the absence of a written partnership Deed. The consent of all the partners is required to admit a new partner. Thus K can not be admitted.

But if there is a partnership deed with a written agreement Y can solely admit any new pather without the consent of other partners. Then K can become the partner of the firm.