If `x_1, x_2` be the roots of the equation `x^2 -3x+ A=0` and `x_3,x_4` be those of the equation `x^2-12x + B = 0` and `x_1 , x_2, x_3, x_4` be an increasing GP. find find A and B.

Explanation

`therefore x_(1),x_(2),x_(3),x_(4) " are in GP. "`
Let `therefore x_(2)=x_(1)r,x_(3)=x_(1)r^(2),x_(4)=x_(1)r^(3)`
`" " [" here, product of " x_(1),x_(2),x_(3),x_(4)" are not given "]`
Given, `x_(1)+x_(2)=3,x_(1)x_(2)=A`
`implies x_(1)(1+r)=3,x_(1)^(2)r=A " " ".....(i)"`
and `x_(3)+x_(4)=12,x_(3)x_(4)=B`
`implies x_(1)r^(2)(1+r)=12,x_(1)^(2)r^(5)=B" " ".....(ii)"`
from Eqs. (i) and (ii),
`r^(2)=4 implies r=2" "[" for increasing GP "]`
From Eqs. (i), `x_(1)=1`
Now, `A=x_(1)^(2)r=1^(2)*2=2 " " [" from Eq. (i)"]`
and `B=x_(1)^(2)r^(5)=1^(2)*2^(5)=32 " " [" from Eq. (ii)"]`.