Prove that `Delta=[a+b x c+dx 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 c w|`

Delta=[[a+bx,c+dx,p+qxax+b,cx+d,px+qu,v,w]]=(1-x^(2))det[[a,c,pb,d,qu,c,w]]

Prove that /_\ |[a+bx, c+dx, p+qx],[-ax+b, cx+d, px+q],[u,v,w]|=(1-x^2) [[a,c,p],[b,d,q],[u,v,w]]

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

If p+q+r=0=a+b+c , then the value of the determnalnt |p a q b r c q c r a p b r b p c q a|i s 0 b. p a+q b+r c c. 1 d. none of these

Using the property of determinants and without expanding, prove that: |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|

Prove that: |{:(a, b, c), (x, y, z), (p, q, r):}|=|{:(y, b, q), (x, a, p), (z, c, r):}|

If vec u , vec v , vec w are noncoplanar vectors and p, q are real numbers, then the equality [3 vec u ,""p vec v , p vec w]-[p vec v ,"" vec w , q vec u]-[2 vec w ,""q vec v , q vec u]=0 holds for (A) exactly one value of (p, q) (B) exactly two values of (p, q) (C) more than two but not all values of (p, q) (D) all values of (p, q)

Prove that [a b c][u v w]=|[a*u, b*u, c*u], [a*v, b*v, c*v], [a*w, b*w, c*w]|

SECTION MULTIPLE CORRECT CHOICE TYPE Example 41 Let a, b, c , p, q be five different non-zero real numbers and x, y, z be three numbers satisfying the system of equations a a-p aq then (a) x + y + z = a + b + c-p-q abc (b) x = pq 0 in (c) y = (a-p)(h-p)(c-p) p(p-q) (a-q)(b-q)(c-q) aq-p) (d) z = b.