Show that :
`|{:(1+x,1,1),(1,1+y,1),(1,1,1+z):}|=xyz(1+(1)/(x)+(1)/(y)+(1)/(z))`

Prove that: {:|(1,x,x^3),(1,y,y^3),(1,z,z^3)| = (x-y)(y-z)(z-x)(x+y+z)

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)| = (x-y)(y-z)(z-x)

Using prperties of determinants, prove that : |(1,1,1+3x),(1+3y,1,1),(1,1+3z,1)| = 9(3xyz + xy + yz + zx) .

Without expanding as far as possible, prove that |{:(1,1,1),(x,y,z),(x^(3),y^(3),z^(3)):}| = (x-y)(y-z)(z-x)(x+y+z) .

Show that |(1,x^2,x^3),(1,y^2,y^3),(1,z^2,z^3)| = (x-y),(y-z)(z-x)(xy+yz+zx)

Show that the lines (x-1)/(2)=(y-3)/(4)=-z and (x-4)/(2)=(y-1)/(-2)=(z-1)/(1) are coplanar Also find the equation of the plane containing the lines.

If xy + yz + zx = 1 , prove that : x/(1-x^2)+y/(1-y^2)+z/(1-z^2)= (4xyz)/((1-x^2)(1-y^2)(1-z^2)) .

If x +y+ z=xyz , prove that : (2x)/(1-x^2)+(2y)/(1-y^2)+(2z)/(1-z^2)= (2x)/(1-x^2). (2y)/ (1-y^2). (2z)/(1-z^2) .