If `_1=|1 1 1x^2y^2z^2x y z|a n d_2=|1 1 1y z z xx y x y z|,` without expanding prove that `_1=_2`

If D1=|1 1 1 x^2 y^2 z^2 x y z| and D2=|1 1 1y z z xx y x y z| , without expanding prove that D1=D2 .

If D_1=|[1, 1, 1],[x^2,y^2,z^2],[x, y, z]| and D_2=|[1, 1, 1],[yz, zx,xy], [x, y, z]| without expanding prove that D_1=D_2

Let delta=|A x x^2 1 B y y^2 1 C z z^2 1|a n d_1=|A B C x y z y z z xx y|, then show that delta = delta1

Without expanding prove that det[[x+y,z,1y+z,x,1z+x,y,1]]=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 .

Without expanding prove that abs[[x+y,y+z,z+x],[z,x,y],[1,1,1]]=0

If a x1 2+b y1 2+c z1 2=a x2 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d ,a x2 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_1y_1z_1x_2y_2z_2x_3y_3z_3|=(d-f){((d+2f))/(a b c)}^(1//2)

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_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_1y_1z_1x_2y_2z_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_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_1y_1z_1x_2y_2z_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)

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