If p, q, r are not in G.P. and `|[1,q/p,alpha+q/p],[1,r/p,alpha+r/q],[palpha+q,qalpha+r,0]|=0` Show that `palpha^2+2qalpha+r=0`

If p,q,r are not in geometric progression and |[1,q/p,alpha+q/p],[1,r/q,alpha+r/q],[palpha+q,qalpha+r,0]|=0 then prove that palpha^2+2q alpha^2+r=0

If p, q. r in G.P and 1/(p-a),1/(r-p),1/(q-r) are in AP.show that p+4q+r=0

If p, q, r are in A.P., q, r, s are in G.P. and r, s, t are in H.P., prove that p, r, t are in G.P.

If p, q, r are in G.P and the equation px^2+2qx+r=0 and dx^2+2ex+f=0 have a common root, then show that d/p, e/q, f/r are in AP.

If (p^2+q^2), (pq+qr), (q^2+r^2) are in GP then Prove that p, q, r are in G.P.

Let alpha and beta be the roots of equation px^2+qx+r=0,pne0 . If p, q, r are in A.P. and 1/alpha+1/beta =4, then the value of abs(alpha-beta) is :

If p+q+r=0=a+b+c , then the value of the determinant |[p a, q b, r c],[ q c ,r a, p b],[ r b, p c, q a]|i s (a) 0 (b) p a+q b+r c (c) 1 (d) none of these

Let alpha and beta be the roots of equation px^(2)+qx+r=0" " pne0."If " p , q , r "are in A.P" and (1)/(alpha)+(1)/(beta)=4 , "then the value of " |alpha-beta| is -

If p sin alpha = q sin (120^(@) + alpha) = r sin (240^(@) + alpha),"prove that", pq + qr +rp = 0

In /_\PQR , angleP = 60^@ , show that q + r = (q-p)/2