Mathematics Question from JEE Main 2026 · 24 January, Shift 2

Let $f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx,\quad x>0,\qquad \lim_{x\to 0}f(x)=0,\qquad f(1)=\frac14.$ If $A= \begin{bmatrix} 0 & 0 & 1\\ \frac14 & f'(1) & 1\\ \alpha^2 & 4 & 1 \end{bmatrix}$ and $B=\operatorname{adj}(\operatorname{adj} A)$ is such that $|B|=81$, then $\alpha^2$ is equal to