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 |[1,p,p^2-qr],[1,q,q^2-rp],[1,r,r^2-pq]|= 0

A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}] and B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}] Where p_(i), q_(i),r_(i) are the co-factors of the elements l_(i), m_(i), n_(i) for i=1,2,3 . If (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines then (p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2)) and (p_(3),q_(),r_(3)) are

If the lengths of the perpendiculars from the vertices of a triangle ABC on the opposite sides are p_(1), p_(2), p_(3) then prove that (1)/(p_(1)) + (1)/(p_(2)) + (1)/(p_(3)) = (1)/(r) = (1)/(r_(1)) + (1)/(r_(2)) + (1)/(r_(3)) .

Using the properties of determinant, show that : |[1,p+q,p^2+q^2],[1,q+r,q^2+r^2],[1,r+p,r^2+p^2]| = (p-q)(q-r)(r-p)

Let A={p , q , r , s}\ a n d\ B={1,2,3}dot Which of the following relations from A to B is not function? (i) R_(1) = {(p, 1), (q, 2), (r, 1), (s, 2)} (ii) R_(2) = {(p, 1), (q, 2), (r, 1), (s, 1)} (iii) R_(3) = {(p, 1), (q, 2), (r, 2), (r, 2)} (iv) R_(4) = {(p, 2), (q, 3), (r, 2), (s, 2)}

Show that: (9p – 5q)^2 + 180pq = (9p + 5q)^2

Angle Q and R of a DeltaPQR are 25^(@) and 65^(@) . Write which of the following is true (i) PQ^(3) + QR^(2)= RP^(2) (ii) PQ^(2) + RP^(2)= QR^(2) (iii) RP^(2) + QR^(2)= PQ^(2)

5pq(p^2 - q^2) -: 2p(p+q)