If `x in R,a_(i),b_(i),c_(i) in R` for `i=1,2,3` and `|{:(a_(1)+b_(1)x,a_(1)x+b_(1),c_(1)),(a_(2)+b_(2)x,a_(2)x+b_(2),c_(2)),(a_(3)+b_(3)x,a_(3)x+b_(3),c_(3)):}|=0`, then which of the following may be true ?

if quad /_=[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

if Delta=det[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

consider the fourth -degree polynomial equation |{:(a_(1)+b_(1)x,,a_(1)x^(2)+b_(1),,c_(1)),(a_(2)+b_(2)x^(2),,a_(2)x^(2)+b_(2),,c_(2)),(a_(3)+b_(3)x^(2),,a_(3)x^(2)+b_(3),,c_(3)):}|=0 Without expanding the determinant find all the roots of the equation.

The determinant |(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|

Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let Delta=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then prove that