Matrices & Determinants
Adjoint of Adjoint — Finding α²
nta_pyq_2026_jan
Grade 12
Question:
Let $f(x)=\displaystyle\int\frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}dx$, $x>0$, $\lim_{x\to0}f(x)=0$ and $f(1)=\dfrac{1}{4}$. If $A=\begin{bmatrix}0&0&1\\\frac{1}{4}&f'(1)&1\\\alpha^2&4&1\end{bmatrix}$ and $B=\text{adj}(\text{adj}\,A)$ be such that $|B|=81$, then $\alpha^2$ is equal to
Step-by-Step Solution
Key Concept: For $f$: divide by $x^{18}$, let $t=x^{-9}+x^{-7}+2$; then $f(x)=\tfrac{x^9}{1+x^2+2x^9}+C$. $f'(1)=1$. $|A|=\begin{vmatrix}0&0&1\\1/4&1&1\\\alpha^2&4&1\end{vmatrix}=1-\alpha^2$.
$\alpha^2=4$.
Correct Answer: 4