Statement 1: `"Delta"=|m y+n z m q+n r m b+n c k z-m x k r-m p k c-m a-n x-k y-n p-k q-n a-k b|` is equal to 0. Statement 2: The value of skew symmetric matrix of order 3 is zero.

Statement1: if n in Na n dn is not a multiple of 3 and (1+x+x^2)^n=sum_(r=0)^(2n)a_r x^r , then the value of sum_(r=0)^n(-1)^r a r^n C_r is zero Statement 2: The coefficient of x^n in the expansion of (1-x^3)^n is zero, if n=3k+1orn=3k+2.

On Z, define ox by (m ox n) = m^(n) + n^(m) : AA m, n in Z . Is ox binary on Z?

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|

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If P= {m, n} and Q = {n, m}, then P xx Q= {(m, n),(n, m)}. (ii) If A and B are non-empty sets, then A xx B is a non-empty set of ordered pairs (x, y) such that x in A and y in B. (iii) If A={1,2}, B={3,4} then A xx (B cap phi)=phi

For 3xx3 matrices Ma n dN , which of the following statement (s) is (are) NOT correct ? N^T M N is symmetricor skew-symmetric, according as m is symmetric or skew-symmetric. M N-N M is skew-symmetric for all symmetric matrices Ma n dNdot M N is symmetric for all symmetric matrices M a n dN (a d jM)(a d jN)=a d j(M N) for all invertible matrices Ma n dNdot

If A = {p, q, r, s}, B = {m, n, q, s, t} and C = {m, n, p, q, s}, then verify the associative property of union of sets.

Assume X, Y, Z, W and P are matrices of order 2xx n, 3xxk, 2xxp, n xx 3 and p xx k , respectively. If n = p, then the order of the matrix 7X-5Z is:

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

Two sets A and B are such that n(A) = m and (B) = k. If the number of subsets A and B is! 144, find the values of m and k.