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

Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0) . Let z=px+qy , where p, q gt 0 . Condition on p and q so that the minimum of z occurs at (3, 0) and (1, 1) is

Show that the points P(- 2 , 3, 5) , Q (1, 2, 3) and R(7, 0, -1) are collinear.

The locus of the orthocenter of the triangle formed by the lines (1+p)x-py+p(1+p)=0,(1+q)x-qy+q(1+q)=0 and y=0 , where p ne q , is

A random variable x has the following point distribution : {:(x:,0,1,2,3,4,5,6,7,),(p(x):,0,p,2p,2p,3p,p^(2),2p^(2),7p^(2),):} +p The value of p is :

It is given that events A and B are such that : P(A) = 1/4 ,P (A//B) =1/2 and P (B//A) =2/3 . Then P(B) is :

T_p,T_q,T_r are the pth , qth and rth terms of an A.P., then |(T_p,T_q,T_r),(p,q,r),(1,1,1)| equals :

Find the unit vector in the direction of vector overline(PQ) , where P and Q are the points P=(1,2,3).Q=(4,5,6) respectively.

find the direction ratios of the vector, joining the points P(2,3,0) and Q(-1,-2,-3) direction from P to Q.

If p_(1),p_(2),p_(3) are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then p_(1)p_(2)p_(3) equals