If `m` is a positive integer and `D_r=|(2r-1,\ ^m C_r,1),(m^2-1, 2^m ,m+1),(sin^2(m^2),sin^2(m),sin^2(m+1))|` . Prove that `sum_(r=0)^m D_r=0` .

If m is a positive integer and D_r=|2r-1\ ^m C_r1m^2-1 2^m m+1s in^2(m^2)s in^2(m)s in^2(m+1)| . Prove that sum_(r=0)^m D_r=0 .

Let m be a positive integer and Deltar = |[2r-1,.^mC_r,1],[m^2-1,2^m,m+1],[sin^2(m^2),sin^2m,sin(m^2)]| . Then the value of sum_(r=0)^m Deltar

If k a n d n are positive integers and s_k=1^k+2^k+3^k++n^k , then prove that sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+1)dot

If m in N and mgeq2 prove that: |1 1 1\ ^m C_1\ ^(m+1)C_1\ ^(m+2)C_1\ ^m C_2\ ^(m+1)C_2\ ^(m+2)C_2|=1 .

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

The value of the determinant |(1,1,1),(.^(m)C_(1),.^(m +1)C_(1),.^(m+2)C_(1)),(.^(m)C_(2),.^(m +1)C_(2),.^(m+2)C_(2))| is equal to

Prove that sum_(r=1)^(m-1)(2r^2-r(m-2)+1)/((m-r)^m C_r)=m-1/mdot

Prove that : int_(0)^(pi) sin^(2m) x. cos^(2m+1) x dx=0

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n , for epsilonR . Also .^nC_r=C_r On the basis of the above information answer the following questions the value of .^mC_r.^nC_0+^mC_(r-1).^nC_1+^mC_(r-2).^nC_2+….+^mC_1.^nC_(r-1)+^nC_0^nC_r where m,n, r are positive interges and rltm,rltn= (A) .^(mn)C_r (B) .^(m+n)C_r (C) 0 (D) 1

Let T_r a n dS_r be the rth term and sum up to rth term of a series, respectively. If for an odd number n ,S_n=na n dT_n=(T_n-1)/(n^2),t h e nT_m ( m being even)is 2/(1+m^2) b. (2m^2)/(1+m^2) c. ((m+1)^2)/(2+(m+1)^2) d. (2(m+1)^2)/(1+(m+1)^2)