If $$x^3 - y^3 = 81$$ and $$x - y = 3$$, then what is the value of $$x^2 + y^2$$ ?

Given : $$x^3 - y^3 = 81$$ ---------------(i)

and $$x - y = 3$$ -----------(ii)

Cubing both sides,

=> $$(x-y)^3=(3)^3$$

=> $$x^3-y^3-3xy(x-y)=27$$

Substituting values from equations (i) and (ii), we get :

=> $$81-3xy(3)=27$$

=> $$9xy=81-27=54$$

=> $$xy=\frac{54}{9}=6$$ ------------(iii)

Also, $$(x-y)^2=x^2+y^2-2xy$$

Substituting values from equations (ii) and (iii), we get :

=> $$(3)^2=(x^2+y^2)-2(6)$$

=> $$9=(x^2+y^2)-12$$

=> $$x^2+y^2=9+12=21$$

=> Ans - (B)