the determinant `|{:(a,,b,,aalpha+b),(b,,c,,balpha+c),(aalpha+b,,balpha+c,,0):}|=0` is equal to zero if

The determinant Delta=|{:(a,b,aalpha+c),(b,c,balpha+c),(aalpha+b,balpha+c,0):}| is equal to zero if

The determinant : |(a,b,aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c,0)|=0 if :

The determinant |{:(a,b,aalpha+b),(b,c,aalpha+c),(aalpha+b,balpha+c,0):}|=0 if a,b,c are in

If |{:(a, b, aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c," "0):}|=0 then

If the determinant |(a ,b,2aalpha+3b),(b, c,2balpha+3c),(2aalpha+3b,2balpha+3c,0)|=0 then (a) a , b , c are in H.P. (b) alpha is root of 4a x^2+12 b x+9c=0 or(c) a , b , c are in G.P. (d) a , b , c , are in G.P. only a , b ,c are in A.P.

Show that abs((a,b,aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c,0):} =0 if alpha is not the root of the equation (ax^2+2bx+c)=0 then a,b,c are in G.P.

If the determinant |[a,b,2aalpha+3b],[b,c,2balpha+3c],[2aalpha+3b,2balpha+3c,0]|=0 then (a) a , b , c are in H.P. alpha is root of 4a x^2+12 b x+9c=0 or a , b , c are in G.P. a , b , c , are in G.P. only a , b ,c are in A.P.

If the determinant |[a,b,2aalpha+3b],[b,c,2balpha+3c],[2aalpha+3b,2balpha+3c,0]|=0 then (a)a , b , c are in H.P. alpha is root of 4a x^2+12 b x+9c=0 or a , b , c are in G.P. a , b , c , are in G.P. only a , b ,c are in A.P.

The determination Delta=|{:(,b,c,balpha+c),(,c,d,calpha+d),(,balpha+c,calpha+d,aalpha^(3)-calpha):}| is equal to zero if