JEE Main 2022 (Online) 24th June Morning ShiftQuestion 1
Let $S=\{\sqrt{n}:1\le n\le50$ and $n$ is odd$\}$.
Let $a\in S$ and $A=\begin{bmatrix}1&0&a\\-1&1&0\\-a&0&1\end{bmatrix}$.
If $\sum_{a\in S}\det(\operatorname{adj}A)=100\lambda$, then $\lambda$ is equal to :
Determinant
Solution
$\det A=a^2+1$, so $\det(\operatorname{adj}A)=(a^2+1)^2=(n+1)^2$ with $a^2=n$ odd, $1\le n\le49$. $\sum(n+1)^2=\sum_{k=1}^{25}(2k)^2=4\cdot\tfrac{25\cdot26\cdot51}{6}=22100$. $100\lambda=22100\Rightarrow\lambda=221$. Option (B).