If x+y+z=0,show that `|[1,1,1],[x,y,z],[x^3,y^3,z^3]|`=0

Show that : |[x, y, z ],[x^2,y^2,z^2],[x^3,y^3,z^3]|=x y z(x-y)(y-z)(z-x)dot

proof |[x,y,z],[x^(2),y^(2),z^(2)],[yz,zx,xy]| = |[1,1,1],[x^(2),y^(2),z^(2)],[x^(3),y^(3),z^(3)]|

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

If x+y+z=0 show that x^3+y^3+z^3=3x y z .

If x!=0,y!=0,z!=0 and |[1+x,1,1],[1+y,1+2y,1],[1+z,1+z,1+3z]|=0 , then x^(-1)+y^(-1)+z^(-1) is equal to a.1 b.-1 c.-3 d. none of these

If 2^x = 3^y =12^z show that 1/z= 1/y+2/x

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)

Without expanding prove that: |[x+y, y+z, z+x],[z ,x, y],[1, 1 ,1]|=0 .

If x,y,z are different and Delta=|[x,x^2,x^3-1],[y,y^2,y^3-1],[z,z^2,z^3-1]|=0 , then using properties of determinants, show that xyz=1

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)