If `f(x), g(x)` and h(x) are second degree polynomials. Prove that `sum_(r=1)^n Delta_r` is independent of x where `Delta_r=|[2r+1, 6n(n+2),f(x) ],[2^(r-1), 3*2^(n+1)-6, g(x)],[ r(n-r+1), n(n+1)(n+2),h(x)]|`

Let Delta_r=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3n-1))/2]|dot Show that sum_(r=1)^n Delta_r=0

Let Delta_r=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3n-1))/2]|dot Show that sum_(r=1)^n Delta_r=0

If f(x)=(a^x)/(a^x+sqrt(a ,)),(a >0), then find the value of sum_(r=1)^(2n-1)2f(r/(2n))

If Delta_(r)=|(r-1,n,6),((r-1)^(2),2n^(2),4n-2),((r-1)^(3),3n^(3),3n^(2)-3n)| then sum_(r=1)^(n)Delta_(r)=

If f_r(x),g_r(x),h_r(x),r=1,2,3 are polynomials such that f_r(a)=g_r(a)=h_r(a),r=1,2,3a n d F(x)=|[f_1(x),f_2(x),f_3(x)],[g_1(x),g_2(x),g_3(x)],[h_1(x),h_2(x),h_3(x)]| then F^(prime)(x)a tx=a is____________________

If f_r(x),g_r(x),h_r(x),r=1,2,3 are polynomials such that f_r(a)=g_r(a)=h_r(a),r=1,2,3a n d F(x)=|[f_1(x),f_2(x),f_3(x)],[g_1(x),g_2(x),g_3(x)],[h_1(x),h_2(x),h_3(x)]| then F^(prime)(x)a tx=a is____________________

If f_r(x),g_r(x),h_r(x),r=1,2,3 are polynomials such that f_r(a)=g_r(a)=h_r(a),r=1,2,3a n d F(x)=|[f_1(x),f_2(x),f_3(x)],[g_1(x),g_2(x),g_3(x)],[h_1(x),h_2(x),h_3(x)]| then F^(prime)(x)a tx=a is____________________

If S_r=|{:(2r,x,n(n+1)),(6r^2-1,y,n^2(2n+3)),(4r^3-2nr,z,n^3(n+1)):}| , then the value of sum_(r=1)^(n)S_r is independent of