If x,y,z are dinstinct and ` |[x,x^3,x^4-1],[y,y^3,y^4-1],[z,z^3,z^4-1]|=0 ` then prove that `(xyz)(xy+yz+zx)=(x+y+z) `

If x, y, z are different and Delta=|[x, x^2, 1+x^3],[y, y^2, 1+y^3],[z, z^2, 1+z^3]|=0 then show that 1+xyz=0

If x, y, z are different and Delta=|[x, x^2, 1+x^3],[y, y^2, 1+y^3],[z, z^2, 1+z^3]|=0 then show that 1+xyz=0

show tha |[1,x,x^2-yz],[1,y,y^2-zx],[1,z,z^2-xy]| =0

If x!=y!=z and |x x^2 1+x^3 y y^2 1+y^3 z z^2 1+z^3|=0 , then prove that x y z=-1 .

Prove that : Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]]=xyz(x-y)(y-z)(z-x)

For any scalar p prove that =|[x,x^2, 1+p x^3],[y, y^2, 1+p y^3],[z, z^2 ,1+p z^3]|=(1+p x y z)(x-y)(y-z)(z-x) .

If xneynez" and " |{:(x,x^(2),1+x^(3)),(y,y^(2),1+y^(3)),(z,z^(2),1+z^(3)):}|=0, then xyz =

prove that: |(y+z,z,y),(z,z+x,x),(y,x,x+y)|=4xyz

If cos^-1 x+cos^-1 y+cos^-1 z=pi and x+y+z= 3/2, then prove that x=y=z

If x,y,z in R and |{:(x,y^2,z^3),(x^4,y^5,z^6),(x^7,y^8,z^9):}|=2 then find the value of |{:(y^5z^6(z^3-y^3),x^4z^6(x^3-z^3),x^4y^5(y^3-x^3)),(y^2z^3(y^6-z^6),xz^3(z^6-x^6), xy^2(x^6-y^6)),(y^2z^3(z^3-y^3),xz^3(x^3-z^3),xy^2(y^3-x^3)):}|