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

Match the following lists (where [x] represents the greatest integer function) and then choose the correct code. Codes : {:(,"a b c d"),((1),"s r q p"),((2),"q p s p"),((3), "s r p q"),((4),"p p q r"):}

Prove that |A| = | ((q+r)^2 ,p^2,p^2),(q^2 ,(r + p)^2, q^2),(r^2 , r^2 , (p+q)^2)| = 2pqr(p+q+r)^3

Find the values of p,q,r and s if [{:(p^2-1, 0, -31-q^3),(7, r+1, 9), ( -2 , 8 , s-1):}] =[{:(1,0,-4),(7, 3/2 , 9),(-2, 8 , pi):}]

The vertices of a triangle are (p q ,1/(p q)),(q r ,1/(q r)), and (r q ,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 (1,2) (b) 2,-1) (1,-1) (d) 2,3)

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 x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

If [{:(0,p,3),(2,q^2 , -1),(r, 1,0):}] is skew-symmetric , find the values of p , q and r