`("lim")_(xvec1)((1-x)(1-x^2)(1-x^(2n)))/({(1-x)(1-x^2)(1-x^n)}^2),n in N ,e q u a l s` `^2n P_n` (b) `^2n C_n` (c) `(2n)!` (d) none of these

If f(x+1)+f(x-1)=2f(x)a n df(0),=0, then f(n),n in N , is nf(1) (b) {f(1)}^n (c) 0 (d) none of these

If y=tan^(-1)((2^x)/(1+2^(2x+1))),t h e n(dy)/(dx)a tx=0 is (a)1 (b) 2 (c) 1n 2 (d) none of these

If ("lim")_(xvec1)(asin(x-1)+bcos(x-1)+4)/(x^2-1)=-2,t h e n|a+b| is_______

The value of lim_(x->oo)((2^(x^n))^(1/e^x)-(3^(x^n))^(1/e^x))/(x^n) (where n in N) is (a) logn(2/3) (b) 0 (c) nlogn(2/3) (d) none of defined

If f^(prime)(x)=sqrt(2x^2-1)a n dy=f(x^2),t h e n(dy)/(dx)a tx=1 is 2 (b) 1 (c) -2 (d) none of these

If |x^n x^(n+2)x^(2n)1x^a a x^(n+5)x^(a+6)x^(2n+5)|=0,AAx in R ,w h e r en in N , then value of a is n b. n-1 c. n+1 d. none of these

If |[x^n, x^(n+2),x^(2n)],[1,x^a, a],[x^(n+5),x^(a+6),x^(2n+5)]|=0,AAx in R ,where n in N , then value of a is (a) n (b) n-1 (c) n+1 (d) none of these

The value of (lim_(n->oo)[(2n)/(2n^2-1)cos(n+1)/(2n-1)-n/(1-2n)dot(n(-1)^n)/(n^2+1)]i s 1 (b) -1 (c) 0 (d) none of these

The coefficient of x^r[0lt=rlt=(n-1)] in lthe expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2++(x+2)^(n-1) is ^n C_r(3^r-2^n) b. ^n C_r(3^(n-r)-2^(n-r)) c. ^n C_r(3^r+2^(n-r)) d. none of these

If |x|<1,t h e n1+n((2x)/(1+x))+(n(n+1))/(2!)((2x)/(1+x))^2+...... is equal to