If `x-r` is a factor of the polynomial `f(x)=a_0x^n+a_1x^(n-1)+.....+a_n` repeated `m` times, `(1ltxlt=n),` then `r` is a root of `f^(prime)(x)=0` repeated `m` times.True/False.

If x-c is a factor of order m of the polynomial f(x) of degree n (1 < m < n) , then find the polynomials for which x=c is a root.

Find the coefficient of x^n in the polynomial (x+^n C_0)(x+3^n C_1)xx(x+5^n C_2)[x+(2n+1)^n C_n]dot

If f(x)=m x+ca n df(0)=f^(prime)(0)=1. What is f(2)?

If f(x)=x/(sinx) \ a n d \ g(x)=x/(tanx),w h e r e \ 0ltxlt=1, then in this interval

If f(x) =(p-x^n)^(1/n) , p >0 and n is a positive integer then f[f(x)] is equal to

The equation 3x^(2)-12x+(n-5)=0 has repeated roots. Find the value of of n.

If f(x)=alphax^n , prove that alpha=(f^(prime)(1))/n .

Differentiate the following with respect of x : a_0x^n+a_1x^(n-1)+a_2x^(n-2)++a_(n-1)x+a_ndot

If d/(dx){x^n-a_1x^(n-1)+a_2x^(n-2)++(-1)^n a_n}e^x=x^n e^x , then the value of ar, 0 < r ≤ n, is equal to

Statement 1: sum_(r=0)^n(r+1)^n c_r=(n+2)2^(n-1)dot Statement 2: sum_(r=0)^n(r+1)^n c_r=(1+x)^n+n x(1+x)^(n-1)dot (1) Statement 1 is false, Statement 2 is true. (2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 (3) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1. (4) Statement 1 is true, Statement 2 is false.