Without expanding the determinant prove the following.
`|[b+c,q+r,y+z],[c+a,r+p,z+x],[a+b,p+q,x+y]|=2|[a,p,x],[b,q,y],[c,r,z]|`

Without expanding the determinant prove the following. |[x,a,x+a],[y,b,y+b],[z,c,z+c]|=0

Prove that Delta=|[a+b x, c+d x, p+q x],[ a x+b, c x+d, p x+q],[ u, v, w]|=(1-x^2)|[a , c, p],[ b, d, q],[ u, v, w]| .

By using properties of determinants, prove that |[x+y+2z,x,y],[z,y+z+2x,y],[z,x,z+x+2y]|=2(x+y+z)^3

By using properties of determinants, prove that |[x,x^2,yz],[y,y^2,zx],[z,z^2,xy]|=(x-y)(y-z)(z-x)(xy+yz+zx)

prove that abs[[a,b,c],[a+2x,b+2y,c+2z],[x,y,z]]=0

Without expanding prove that abs[[x+y,y+z,z+x],[z,x,y],[1,1,1]]=0

The corner points of the feasible region determined by the following system of inequalities: 2 x+y le 10, x+3 y le 15, x, y ge 0 are (0,0),(5,0),(3,4) , and (0,5) . Let Z=p x+q y , where p, qgt0 . Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is a) p=q b) p=2q c) p=3q d) q=3p

a,b,c (all positive) are the p th, q th and r th terms of a geometric progression, then |{:(log _(e) a, p, 1), ( log _(e) b ,q, 1), ( log _(e) c, r,1):}| :a)pqr b)0 c) p + q + r d) pq + qr + rp

If p,q,r are in H.P. and p,q,-2r are in G.P. ( p,q,r, gt0) then