JEE Advance - Mathematics (2010 - Paper 2 Offline - No. 12)
Let $k$ be a positive real number and let
$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{array}{ccc} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{array}\right] . \end{aligned} $$
If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$
is equal to _________.
[ Note : adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$ ].
Explanation
$$ |A|=(2 k+1)^3,|B|=0 $$
( $\therefore$ B is a skew - symmetric matrix of order 3 )
$$ \begin{aligned} \text { Let }(\operatorname{adj} \mathrm{A}) & =|\mathrm{A}|^{n-1} \\\\ \left((2 k+1)^3\right)^2 & =10^6 \\\\ (2 k+1)^6 & =10^6 \Rightarrow 2 k+1=10 \\\\ 2 k=9 \Rightarrow[k] & =4 \end{aligned} $$
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