Verified Solution • 2025–26 Session moderate

QJEEM23Maf957: JEE Main 2023 Mathematics - Class 12 JEEM23Maf957 Solution: Let . If ∑ k = 1 n D k = 96 , then is equal to _____________.

Let Dk=∣1amp;2kamp;2k−1namp;n2+n+2amp;n2namp;n2+namp;n2+n+2∣\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\n & n^{2}+n+2 & n^{2} \\n & n^{2}+n & n^{2}+n+2\end{array}\right| . If ∑k=1nDk=96, then nn is equal to _____________.

Step-by-Step Verified Solution

amp;∑k=1nDk=∣∑1amp;2∑kamp;2∑k−∑1namp;n2+n+2amp;n2namp;n2+namp;n2+n+2∣amp;=∣namp;n(n+1)amp;n2namp;n2+n+2amp;n2namp;n2+namp;n2+n+2∣amp;=∣0amp;−2amp;00amp;2amp;−n−2namp;n2+namp;n2+n+2∣amp;=2((−n)(−n−2))=96amp;⇒n2+2n=48amp;⇒n=6,−8amp;⇒n=6\begin{aligned}& \sum_{k=1}^n D_k=\left|\begin{array}{ccc}\sum 1 & 2 \sum k & 2 \sum k-\sum 1 \\n & n^2+n+2 & n^2 \\n & n^2+n & n^2+n+2\end{array}\right| \\\\& =\left|\begin{array}{ccc}n & n(n+1) & n^2 \\n & n^2+n+2 & n^2 \\n & n^2+n & n^2+n+2\end{array}\right| \\\\& =\left|\begin{array}{ccc}0 & -2 & 0 \\0 & 2 & -n-2 \\n & n^2+n & n^2+n+2\end{array}\right| \\\\& =2((-n)(-n-2))=96 \\\\& \Rightarrow n^2+2 n=48 \\\\& \Rightarrow n=6,-8 \\\\& \Rightarrow n=6\end{aligned}

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