Without expanding the determinant , prove that `|{:(ax,by,cz),(x^2,y^2,z^2),(1,1,1):}|=|{:(a,b,c),(x,y,z),(yz,zx,xy):}|`

Without expanding the determinant, prove that (i) |{:(a,a^(2),bc),(b,b^(2),ca),(c,c^(2),ab):}|=|{:(1,a^(2),a^(3)),(1,b^(2),b^(3)),(1,c^(2),c^(3)):}| (ii) |{:(ax,by,cz),(x^(2),y^(2),z^(2)),(1,1,1):}|=|{:(a,b,c),(x,y,z),(yz,zx,xy):}| (iii) |{:(1,bc,b+c),(1,ca,c+a),(1,ab,a+b):}|=|{:(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2)):}|

Without expanding prove that Delta ={:[( x+y,y+z,z+x) ,( z,x,y),( 1,1,1) ]:} =0

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)

Using the property of determinants and without expanding {:[( b+c,q+r,y+z),( c+a,r+p,z+x),( a+b,p+q,x+y) ]:}=2 {:[(a,p,x),( b,q,y),(c,r,z)]:}

Using the property of determinants and without expanding {:[( x,a,x+a),( y,b,y+b),(z,c,z+c)]:} =0

Using Cofactors of elements of third column , evaluate , Delta ={:[( 1,x,yz),(1,y,zx),( 1,z,xy) ]:}

Prove that |{:((1+ax)^(2),(1+ay)^(2),(1+az)^(2)),((1+bx)^(2),(1+by)^(2),(1+bz)^(2)),((1+cx)^(2),(1+cy)^(2),(1+cz)^(2)):}|=2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x) .

The coefficient of x^2 y^3 z^4 in (ax-by+cz)^9 is