JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Derivatives XI

Let $\mathbb{R}$ denote the set of all real numbers. Consider the polynomial function $\mathrm{f}:\mathbb{R}\to \mathbb{R}$ defined by

Here $\frac{{\mathrm{d}}^{10}}{{\mathrm{dx}}^{10}}\left({\left({\mathrm{x}}^2-1\right)}^{10}\right)$ is the $10^{\text{th }}$ order derivative of the function ${\left({\mathrm{x}}^2-1\right)}^{10}$
Then which of the following statements is (are) TRUE ?
(A) The coefficient of ${\mathrm{x}}^8$ in the polynomial $\mathrm{f}(\mathrm{x})$ is $(-10)\left(\frac{18!}{8!}\right)$
(B) The value of $f(1)+f(-1)$ is equal to 10 ! $2^{11}$
(C) The degree of the polynomial $f(\mathrm{x})$ is 10
(D) The constant term of the polynomial $f(\mathrm{x})$ is $-\left(\frac{10!}{5!}\right)$