Show without expanding at any stage that: ` |[(a-1)^2, a^2+1, a],[(b-1)^2, b^2+1,b],[(c-1)^2, c^2+1, c]|=0 `

Without expanding at any stage, prove that |(a-b,1,a),(b-c,1,b),(c-a,1,c)| = |(a,1,b),(b,1,c),(c,1,a)|

Show without expanding at any stage that: |[a+b,b+c,c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|=2|[a,b,c],[b,c,a],[c,a,b]|

Without expanding at any state, prove that |((1)/(a),a,bc),((1)/(b),b,ca),((1)/(c ),c,ab)|=0

Without expanding at any state, prove that |(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b))|=0

Using the property of determinants and without expanding, prove that: |[1,b c, a(b+c)],[1,c a, b(c+a)],[1,a b, c(a+b)]|=0

Without expanding show that |[b^2c^2,b c, b+c],[c^2a^2,c a ,c+a ],[a^2b^2,a b ,a+b]|=0

Without expanding determinant at any stage evaluate |(1,1,1),(a,b,c),(a^(2)-bc,b^(2)-ca,c^(2)-ab)|

Without expanding at any stage, prove that |{:(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2)):}|=|{:(1,a,bc),(1,b,ca),(1,c,ab):}|

Without expanding evaluate the determinant "Delta"=|(1, 1, 1),(a, b, c),( a^2,b^2,c^2)| .

Without expanding at any stage, find the value of : |{:(,a,b,c),(,a+2x, b+2y, c+2z),(,x,y,z):}|