If `alpha=e^(i2pi//7)a n df(x)=a_0+sum_(k-0)^(20)a_k x^k ,` then prove that the value of `f(x)+f(alpha, x)+......+f(alpha^6x)` is independent of `alphadot`

Given (1-x^(3))^(n)=sum_(k=0)^(n)a_(k)x^(k)(1-x)^(3n-2k) then the value of 3*a_(k-1)+a_(k) is

If f(x)=3x-2, g(x)=2x+k and if fog=gof, then find the value of k.

Let f(x)=(alpha x)/(x+1), x ne -1. Then, for what value of alpha " is " f[f(x)]=x ?

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then find the value of f(7)dot

Let f(x ) = ( alpha x^2 )/(x+1),x ne -1 the value of alpha for which f(a) =a , (a ne 0) is

If alpha,betaa n dgamma are roots of 2x^3+x^2-7=0 , then find the value of sum(alpha/beta+beta/alpha) .

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

If alpha and beta are the roots of x^(2)+6x-4=0 , find the values of (alpha-beta)^(2) .

If for some differentiable function f(alpha)=0a n df^(prime)(alpha)=0, Statement 1: Then sign of f(x) does not change in the neighbourhood of x=alpha Statement 2: alpha is repeated root of f(x)=0