If x + y + z = 0 , prove that `|(ax,by,cz),(cy,az,bx),(bz,cx,ay)|=xyz|(a,b,c),(c,a,b),(b,c,a)|`

Prove that |(a,b+c,a^(2)),(b,c+a,b^(2)),(c,a+b,c^(2))|=-(a+b+c)xx(a-b)(b-c)(c-a).

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 a) u^(2) + v^(2) + w^(2) - 2 uvw b) u^(3)+ v^(3) + w^(3) - 3 uvw c)0 d)none of these

Prove that |(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|=3abc-a^(3)-b^(3)-c^(3)

prove that abs[[a,b,c],[a+2x,b+2y,c+2z],[x,y,z]]=0

If x,y, and z are in A.P ax , by and cz are in G.P. and a,b,c are in HP then prove that (x)/(z) +(z)/(x) = (a)/ (c )+(c )/(a)

|{:(a+x, b, c), (a, b+y, c), (a, b, c+z):}| is equal to :

The determinant Delta=|(a,b,ax+b),(b,c,bx+c),(ax+b,bx+c,c)| is equal to zero, if a)a,b,c are in A.P. b)a,b,c are in G.P. c)a,b,c are in H.P. d) ax^(2) + bx + c = 0

prove |[1, b c, a(b+c)],[ 1, c a, b(c+a)],[ 1, a b, c(a+b)]|=0