Find the value of the `"determinant" |{:(1,x,y+z),(1,y,z+x),(1,z,x+y):}|`

The value of the determinant |{:(1 , x , x+ z) , (1 , y , z + x) , (1 , z , x + y):}| is

The value of the determinant |{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| is equal to

The value of the determinant |{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| is equal to

The value of the determinant |{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| is equal to

Find the value of : |(1+x,y,z),(x,1+y,z),(x,y,1+z)|

Without actual expansion, prove that the following determinants vanish : |{:(x+y,y+z,z+x),(z,x,y),(1,1,1):}|

Determinant (x-y)(y-z)(z-x)(x+y+z)

Without expanding evaluate the folleing determinents (ii) |(x+y,y+z,z+x),(z,x,y),(1,1,1)|

Without expanding the determinants, show that |(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(y-x)(z-y)(z-x)