1
JEE Main 2023 (Online) 12th April Morning Shift
Numerical
+4
-1

Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\ n & n^{2}+n+2 & n^{2} \\ n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If $$\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96$$, then $$n$$ is equal to _____________.

2
JEE Main 2023 (Online) 11th April Morning Shift
Numerical
+4
-1

Let $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$$, where $$a, c \in \mathbb{R}$$. If $$A^{3}=A$$ and the positive value of $$a$$ belongs to the interval $$(n-1, n]$$, where $$n \in \mathbb{N}$$, then $$n$$ is equal to ___________.

3
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1

Let $$\mathrm{S}$$ be the set of values of $$\lambda$$, for which the system of equations $$6 \lambda x-3 y+3 z=4 \lambda^{2}$$, $$2 x+6 \lambda y+4 z=1$$, $$3 x+2 y+3 \lambda z=\lambda$$ has no solution. Then $$12 \sum_\limits{i \in S}|\lambda|$$ is equal to ___________.

4
JEE Main 2023 (Online) 31st January Evening Shift
Numerical
+4
-1
Let A be a $n \times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right) \cdot$ is $2^{84}$, then $\mathrm{n}$ is equal to :
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