If `f(x)=sum_(k=0)^n a_k |x-1|^k`, where `a_i in R`, then

if f(x)=sum_(k=0rarr n)a_(k)|x|^(k), where a_(i)^( prime)s are real constants then f(x) is (A) continuous at x=0 for all a_(i)(B) differentiable at x=0 for all a_(i)(C) differentiable at x=0 for all a_(2k+1) (D) none of these

If sum_(k=0)^(n)f(x+ka)=0, where a>0, then the period of f(x) is

If f:(0,pi)rarr R and is given by f(x)=sum_(k=1)^(n)[1+sin((kx)/(n))] where [] denotes the integral part of x, then the range of f(x) is

Let (1+x^2^2)^2(1+x)^n = sum _(k=0)^(n+4)a_k x^k. If a_1,a_2,a_3 are in rithmetic progression find n.

If a=(e^(2 pi i))/(7) and f(x)=A_(0)+sum_(k=1)^(20)A_(k)x^(k) then the value of sum_(r=0)^(6)f(a^(r)x)=n(A_(0)+A_(n)x^(n)+A_(2n)x^(2n)), then the value of n is

If alpha=e^(2 pi(i)/(11)) and f(x)=5+sum_(k=1)^(60)A_(x)^(k), then the value of sum_(r=0)^(10)f(alpha^(r)x) is

Let f:R rarr R be a function which satisfies f(x+y)=f(x)+f(y) AA x, y in R . If f(1)=2 and g(n)=sum_(k-1)^((n-1))f(k),n in N then the value of n , for which g(n)=20 , is :