Using properties of determinants, prove that,
`|{:(1,1+p,1+p+q),(2,3+2p,1+3p+2q),(3,6+3p,1+6p+3q):}|=1`

{:|( 1,1+p,1+p+q),( 2,3+2p,4+3p+2q),( 3,6+3p,10+6p+3q)|:}=1

Show that |[1,1+p,1+p+q],[2,3+2p,1+3p+2q],[3,6+3p,1+6p+3q]| = 1

Let's resolve into factors: (p^2 - 3q^2)^2 - 16 (p^2 - 3q^2) + 63

The vertices of a triangle are (p q ,1/(p q)),(q r ,1/(q r)), and (r p ,1/(r p)), where p ,q and r are the roots of the equation y^3-3y^2+6y+1=0 . The coordinates of its centroid are

In an A.P. if p^(th) term is 1/q and q^(th) term is 1/p , prove that the sum of first pq terms is 1/2(pq+1) , where p ne q .

In the equation x^2-px+q=0 ratio of the roots are 2:3 ,then prove that 6p^2=25q

The base B C of a A B C is bisected at the point (p ,q) & the equation to the side A B&A C are p x+q y=1 & q x+p y=1 . The equation of the median through A is: (p-2q)x+(q-2p)y+1=0 (p+q)(x+y)-2=0 (2p q-1)(p x+q y-1)=(p^2+q^2-1)(q x+p y-1) none of these

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)) .

The points (p+1,1), (2p+1,3) and (2p+2, 2p) are collinear, if

Let .^(n)P_(r) denote the number of permutations of n different things taken r at a time . Then , prove that 1+1.^(1)P_(1)+2.^(2)P_(2)+3.^(3)P_(3)+...+n.^(n)P_(n)=.^(n+1)P_(n+1) .