$(a)$ How does electropositivity of elements get affected as we move :
$(i)$ down a group
$(ii)$ across a period?
$(b)$ Which atomic property increases both as we move across a period or down a group?

(N/A) $(i)$ Electropositive character increases down a group because the atomic size increases and the effective nuclear charge decreases,making it easier to lose electrons.
$(ii)$ Electropositive character decreases across a period because the atomic size decreases and the effective nuclear charge increases,making it harder to lose electrons.
$(b)$ Atomic number and atomic mass increase both across a period and down a group.