2016/050) Let a,b be roots of $x^2-3x + A =0$ and c,d be roots of $x^2-12x+B=0$ if $a,b,c,d$ form an increasing GP then find A and B.

because a,b,c,d are in GP let common ratio be t then we have $b=at,c=at^2,d=at^3$
a,b are roots of $x^2-3x + A =0$ so $a+at= 3$ or $a(1+t) = 3\cdots(1)$
c,d are roots of $x^2-12x+C=0$ so $c+d = 12$ or $at^2+bt^3= 12$ or $at^2(1+t) = 12\cdots(2)$
from (1) and (2) $t^2 = 4$ ot $t=2$ because t has to be positive otherwise $a,b,c,d$ cannot be increasing
from (1) $a = 1$ so $a=1,b=2,c=4,d=8$ and hence $A=ab= 2,C=cd = 32$