Prove that ` |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,b,,c),(x,,y,,z),(yz,,xz,,xy):}|`

Factorise : |{:(x,y,z),(x^2,y^2,z^2),(yz,zx,xy):}|

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 u=ax+by+cz , v=ay+bz+cx , w=az+bx+cy , then the value of |{:(a,b,c),(b,c,a),(c,a,b):}|xx|{:(x,y,z),(y,z,x),(z,x,y):}| is

prove that, |{:(x,b,c),(a,y,c),(a,b,z):}|=(x-a)(y-b)(z-c)((x)/(x-a)+(y)/(y-b)+(z)/(z-c)-2)

By using properties of determinants , show that : {:|( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) |:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

|{:(1+x,1,1),(1,1+y,1),(1,1,1+z):}|=xy+yz+zx+xyz

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 : a = y : b = z : c , then prove that (a^(2) + b^(2) + c^(2))(x^(2) + y^(2) + z^(2)) = (ax + by + cz)^(2)

Factorise : x^(2)+y^(2)+z^(2)-xy-yz-zx

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.