Prove that `~(pharrq)=~pharrq=pharr~q`.

If p,q,r are not in G.P. and {:|(1,q/p,alpha+q/p),(1,r/q,alpha+r/q),(palpha+q,qalpha+r,0)|=0 then prove that palpha^2+2qalpha+r=0

ABCD is a parallelogram and a line through A meets DC at P and BC (produced) at Q. Prove that ar(triangleBPC) = ar(triangleDPQ)

In the figure PSDA is a parallelogram. Points Q and R are taken on PS such that PQ = QR = RS and PA||QB||RC. Prove that: ar(trianglePQE) = ar(triangleCFD)

Prove that |{:(p^(2)+1,pq,pr),(pq,q^(2)+1,qr),(pr,qr,r^(2)+1):}|=1+p^(2)+q^(2)+r^(2)

Prove that the area of the parallelogram formed by the lines x cos alpha+y sin alpha=p ,x cos alpha+y sin alpha=q , x cos beta+y sin beta=r and x cos beta+y sin beta=s is ±(p−q)(r−s)cosec(α−β).

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. Prove that a/p(q-r)+b/q(r-p)+c/r(p-q)=0

Sum of first p,q and r terms of an A.P. are a,b,c respectively. Prove that a/p(q-r) +b/q(r-p) + c/r(p-q)=0 .

If pth, qth and rth terms of a HP be respectively a,b and c , has prove that (q-r)bc+(r-p)ca+(p-q)ab=0 .

If the p^(th) , q^(th) and r^(th) terms of a G.P. are a, b and c, respectively. Prove that a^(q-r) b^(r-p)c^(P-q)=1 .

If p, q, r are in A.P. while x, y, z are in G.P., prove that x^(q-r).y^(r-p).z^(p-q) =1 .