Let $J_{n, m}=\int_0^{\frac{1} { 2}} \frac{x^n}{x^m-1} d x, \forall nm$ and $n, m \in N$. Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$, where $a_{i j}=\left\{\begin{array}{cc}J_{6+i, 3}-J_{i+3,3}, & i \leq j \ 0, & ij\end{array}\right.$ . Then $\left|\operatorname{adj} A^{-1}\right|$ is: [JEE Main 2021, 1 Sep (Shift 2)]

Question

Let $J_{n, m}=\int_0^{\frac{1} { 2}} \frac{x^n}{x^m-1} d x, \forall n>m$ and $n, m \in N$. Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$, where $a_{i j}=\left\{\begin{array}{cc}J_{6+i, 3}-J_{i+3,3}, & i \leq j \ 0, & i>j\end{array}\right.$ . Then $\left|\operatorname{adj} A^{-1}\right|$ is: [JEE Main 2021, 1 Sep (Shift 2)]

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Let \(J_{n, m}=\int_0^{\frac{1} { 2}} \frac{x^n}{x^m-1} d x, \forall n>m\) and \(n, m \in N\). Consider a matrix \(A=…