Prove that: `|[1+x_(1),x_(2),x_(3)],[x_(1),1+x_(2),x_(3)],[x_(1),x_(2),1+x_(3)]|=1+x_(1)+x_(2)+x_(3)`

A triangle has vertices A_(i) (x_(i),y_(i)) for i= 1,2,3,. If the orthocenter of triangle is (0,0) then prove that |{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|=0

Value of |{:(1+x_(1),1+x_(1)x,1+x_(1)x^(2)),(1+x_(2),1+x_(2)x,1+x_(2)x^(2)),(1+x_(3),1+x_(3)x,1+x_(3)x^(2)):}| depends upon

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

Consider the quantities such that x_(1),x_(2),….x_(10),-1 lex_(1),x_(2)….,x_(10)le 1 and x_(1)^(3)+x_(2)^(3)+…+x_(10)^(3)=0 , then the maximum value of x_(1)+x_(2)+….+x_(10) is

Value of |{:(1+x_(1),,1+x_(1)x,,1+x_(1)x^(2)),(1+x_(2),,1+x_(2)x,,1+x_(2)x^(2)),(1+x_(3),,1+x_(3)x,,1+x_(3)x^(2)):}| depends upon

The value of ,2x_(1)y_(1),x_(1)y_(2)+x_(2)y_(1),x_(1)y_(3)+x_(3)y_(1)x_(1)y_(2)+x_(2)y_(1),2x_(2)y_(2),x_(2)y_(3)+x_(3)y_(2)x_(1)y_(3)+x_(3)y_(1),x_(2)y_(3)+x_(3)y_(2),2x_(3)y_(3)]| is

If the coordinates of the vertices of an equilateral triangle with sides of length a are (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3)) then |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^2=(3a^(4))/4

prove that |(1,x,x+1),(2x,x(x-1),x(x+1)),(3x(1-x),x(x-1)(x-2),x(x+1)(x-1))|=6x^(2)(1-x^(2))