If `|{:(a,b,a alpha+b),(b,c,b alpha+c),(a alpha +b,b alpha+c,0):}|=0`
Prove that a,b,c are in G.P. or `alpha` is a root of `ax^2 + 2bx + c=0`

If |(a,b,aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c,0)|=0, prove that a,b,c, are in G.P. or alpha is a root of ax^(2)+ 2bx+c=0

In the quadratic ax^2+bx+c=0,D=b^2-4ac and alpha+beta,alpha^2+beta^2,alpha^3+beta^3, are in G.P , where alpha,beta are the roots of ax^2+bx+c, then (a) Delta != 0 (b) bDelta = 0 (c) cDelta = 0 (d) Delta = 0

Prove that =|a c c-a +b c b b-c b+c a-b b-c 0 a-c x y z 1+x+y|=0 implies that a ,b ,c are in A.P. or a ,c ,b are in G.P.

In Delta ABC , prove that c cos (A - alpha) + a cos (C + alpha) = b cos alpha

If (a+bx)/(a-bx)=(b-cx)/(b-cx)=(c+dx)/(c-dx)( x ne 0) then show that a, b, c and d are in G.P.

Let a, b, c be real numbers with a = 0 and let alpha,beta be the roots of the equation ax^2 + bx + C = 0 . Express the roots of a^3x^2 + abcx + c^3 = 0 in terms of alpha,beta

If a,b and c are distinct positive real numbers in A.P, then the roots of the equation ax^(2)+2bx+c=0 are

If alpha and beta , alpha and gamma , alpha and delta are the roots of the equations ax^(2)+2bx+c=0 , 2bx^(2)+cx+a=0 and cx^(2)+ax+2b=0 respectively where a , b , c are positive real numbers, then alpha+alpha^(2) is equal to

If alpha is a complex constant such that alpha z^2+z+ baralpha=0 has a real root, then (a) alpha+ bar alpha=1 (b) alpha+ bar alpha=0 (c) alpha+ bar alpha=-1 (d)the absolute value of the real root is 1