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`

If f(x)=Pi_(k=1)^(999)(x^(2)-47x+k) . then product of all real roots of f(x)=0 is

If f(x) = alphax^(n) prove that alpha = (f'(1))/n

If f(x) = sum_(k=2)^(n) (x-(1)/(k-1))(x-(1)/(k)) , then the product of root of f(x) = 0 as n rarr oo , is

If int_(0)^(npi) f(cos^(2)x)dx=k int_(0)^(pi) f(cos^(2)x)dx , then the value of k, 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)=3x^(2)+k|sin x| is differentiable at x=0 then the value of k is-

The value of parameter alpha , for which the function f(x) = 1+alpha x, alpha!=0 is the inverse of itself

The value of parameter alpha , for which the function f(x) = 1+alpha x, alpha!=0 is the inverse of itself

Let f(x)=|(4x+1,-cosx,-sinx),(6,8sinalpha,0),(12sinalpha, 16sin^(2)alpha,1+4sinalpha)| and f(0)=0 . If the sum of all possible values of alpha is kpi for alpha in [0, 2pi] , then the value of k is equal to

f(x)=(x-k)/(5) and g(x)=5x+7 . If f(g(x))=g(f(x)) , what is the value of k?