Given `|(a,a^(2),a^(3)+1),(b,b^(2).b^(3)+1),(c,c^(2)+c^(3)+1)|=0` and `a ne b ne c`.Show that abc=-1

If |{:(a,,a^(2),,1+a^(3)),(b,,b^(2),,1+b^(3)),(c,,c^(2),,1+c^(3)):}|=0 and the vectors vecA =(1, a, a^(2)) , vec(B) = (1, b, b^(2)) , vec(C )(1,c,c^(2)) are non-coplanar then the product abc = ….

Prove that |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|=(a-b)(b-c)(c-a)

If a, b, c are the sides of a scalene triangle such that |[a , a^2 , a^(3)-1 ],[b , b^2 , b^(3)-1],[ c, c^2 , c^(3)-1]|=0 , then the geometric mean of a, b c is

Prove that |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|=(a-b)(b-c)(c-a) .

Sove that |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|^(2)=|(1+a^(2)+a^(4),1+ab+a^(2)b^(2),1+ac+a^(2)c^(2)),(1+ab+a^(2)b^(2),1+b^(2)+b^(4),1+bc+b^(2)c^(2)),(1+ac+a^(2)c^(2),1+ba+b^(2)c^(2),1+c^(2)+c^(4))| and hence show RHS determinent is =(a-b)^(2)(b-c)^(2)(c-a)^(2)

If =|(2a,x_(1),y_(1)),(2b,x_(2),y_(2)),(2c,x_(3),y_(3))|=abc/2 ne 0 then the area area of the triangle whose vertices are (x_(1)/a,y_(1)/a),(x_(2)/b,y_(2)/b),(x_(3)/c,y_(3)/c) is:

If a,b,c are all distinct and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-1]| =0, show that abc(ab+bc+ac) = a+b+c

If a^(2) + b^(2) + c^(3) + ab + bc + ca le 0 for all, a, b, c in R , then the value of the determinant |((a + b +2)^(2),a^(2) + b^(2),1),(1,(b +c + 2)^(2),b^(2) + c^(2)),(c^(2) + a^(2),1,(c +a +2)^(2))| , is equal to

If A_(i),B_(i),C_(i) are the cofactors of a_(i),b_(i),c_(i) respectively,i=1,2,3 in Delta =|(a_(1),b_(2),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| show that |(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))|=Delta^(0)

Without expanding the determinant, prove that {:[( 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)) ]:}