Prove that
` Delta ={:[( 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)]:}`

If u=ax+by+cz , v=ay+bz+cx , w=ax+bx+cy , then the value of |{:(a,b,c),(b,c,a),(c,a,b):}|xx|{:(x,y,z),(y,z,x),(z,x,y):}| is

Prove that p + q = 0 " if " x^(3) - 3px + 2q " is divisible by" (x + 1)^(2)

Consider int(x^(3)+3x^(2)+2x+1)/(sqrt(x^(2)+x+1))dx =(ax^(2)+bx+c)sqrt(x^(2)+x+1)+lambda int(dx)/(sqrt(x^(2)+x+1)) Now, match the following lists and then choose the correct code. Codes: {:(,a,b,c,d),((1),q,p,s,r),((2),s,p,q,r),((3),r,q,p,s),((4),q,s,p,r):}

Prove that the value of determinants is zero: |a-b b-c c-a x-y y-z z-x p-q q-r r-p|

If A= [(2x,2y,2z),(2a,2b,2c),(2p,2q,2r)] and Delta =[(x,y,z),(a,b,c),(p,q,r)] then det A is

If p, q, r are any real numbers, then (A) max (p, q) lt max (p, q, r) (B) min(p,q) =1/2 (p+q-|p-q|) (C) max (p, q) < min(p, q, r) (D) None of these

If x + y + z = 0, prove that |(ax,by,cz),(cy,az,bx),(bz,cx,ay)|=xyz|(a,b,c),(c,a,b),(b,c,a)|

The equation of a curve passing through (1,0) for which the product of the abscissa of a point P and the intercept made by a normal at P on the x-axis equal twice the square of the radius vector of the point P is (a) ( b ) (c) (d) x^(( e )2( f ))( g )+( h ) y^(( i )2( j ))( k )=( l ) x^(( m )4( n ))( o ) (p) (q) (b) ( r ) (s) (t) x^(( u )2( v ))( w )+( x ) y^(( y )2( z ))( a a )=2( b b ) x^(( c c )4( d d ))( e e ) (ff) (gg) (c) ( d ) (e) (f) x^(( g )2( h ))( i )+( j ) y^(( k )2( l ))( m )=4( n ) x^(( o )4( p ))( q ) (r) (s) (d) None of these

Let a, b, c in R with a gt 0 such that the equation ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0 has non-real roots. If P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b , then

if w is a complex cube root to unity then value of Delta =|{:(a_(1)+b_(1)w,,a_(1)w^(2)+b_(1),,c_(1)+b_(1)bar(w)),(a_(2)+b_(2)w,,a_(2)w^(2)+b_(2),,c_(2)+b_(2)bar(w)),(a_(3)+b_(3)w,,a_(3)w^(2)+b_(3),,c_(3)+b_(3)bar(w)):}| is