`|{:(1,1,1),(x,y,z),(x^3,y^3,z^3):}|=0`

If x+y+z=0 then show that abs((1,1,1),(x,y,z),(x^3,y^3,z^3)) = 0.

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

If |{:(x,x^2,x^3+1),(y,y^2,y^3+1),(z,z^2,z^3+1):}|=a and x ney nez , prove that, xyz=-1.

The number of positive integral solutions of the equation |(x^3+1,x^2y,x^2z),(xy^2,y^3+1,y^2z),(xz^2,z^2y,z^3+1)|=11 is

Use product {:[( 1,1,2),( 0,2,3),( 3,2,4) ]:} {:[( 2,0,1),(9,2,3),(6,1,2) ]:} to solve the system of equations x-y+2z=1 xy-3z=1 3x-2y+4z =2

Using properties of determinants porve that, |{:(x,x^2,1+px^3),(y,y^2,1+py^3),(z,z^2,1+pz^3):}|=(1+pxyz)(x-y)(y-z)(z-x)

For any scalar c prove that, |{:(x,x^2,1+cx^3),(y,y^2,1+cy^3),(z,z^2,1+cz^3):}|=(1+cxyz)(x-y)(y-z)(z-x)

If x,y,z are all distinct and if abs((x,x^2,1+x^3),(y,y^2,1+y^3),(z,z^2,1+z^3)) =0,show that xyz+1=0

Without expanding prove that Delta ={:|( x+y,y+z,z+x) ,( z,x,y),( 1,1,1) |:} =0

If (x_(1),y_(1),z_(1)) , (x_(2),y_(2),z_(2)) , (x_(3) ,y_(3),z_(3)) and (x_(4) , y_(4) , z_(4)) be the consecutive vertices of a parallelogram, show that x_(1)+x_(3)=x_(2)+x_(4),y_(1)+y_(3)=y_(2)+y_(4) and z_(1)+z_(3)=z_(2)+z_(4) .