Determinants
117 JEE Maths previous year questions on Determinants — options free on every question; 12 include the answer & explanation free, the rest unlock with PYQ Pass.
For \(\alpha ,\beta \in \mathrm{ℝ}\) and a natural number \(n\), let \({A}_{r}=\left|\begin{matrix}r & 1 & \frac{{n}^{2}}{2}+\alpha \\ 2r & 2 & {n}^{2}-\beta \\ 3r-2 & 3 & \frac{n(3n-1)}{2}\end{matrix}\right|\). Then \(2{A}_{10}-{A}_{8}\) is
[JEE Main 2024, 6 Apr (Shift 1)]
\(4\alpha +2\beta\)
\(A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\)
\(2 A_{10}-A_8=\left|\begin{array}{ccc}20 & 1 & \frac{n^2}{2}+a \\ 40 & 2 & n^2-\beta \\ 56 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|-\left|\begin{array}{ccc}8 & 1 & \frac{n^2}{2}+a \\ 16 & 2 & n^2-\beta \\ 22 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\)
\(=\left|\begin{array}{ccc}12 & 1 & \frac{n^2}{2}+a \\ 24 & 2 & n^2-\beta \\ 34 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\)
\(=\left|\begin{array}{ccc}0 & 1 & \frac{n^2}{2}+a \\ 0 & 2 & n^2-\beta \\ -2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\)
\(=-2\left(\left(n^2-\beta\right)-\left(n^2+2 \alpha\right)\right)\)
\(=-2(-\beta-2 \alpha)\)
\(=4 \alpha+2 \beta\)
Let \(A=\left[\begin{matrix}\alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5\end{matrix}\right]\) and \(B=\left[\begin{matrix}1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{matrix}\right]+adj\left(A\right)\). If \(\det (B)=66\), then \(\det (adj(A))\) equals:
[JEE Main 2026, 8 Apr (Shift 2)]
\(441\)
\(A=\left[\begin{array}{lll}\alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5\end{array}\right]\)
\(\Rightarrow|A|=15 \alpha-10+16=15 \alpha+6\)
Now \(\operatorname{adj}(\mathrm{A})=\left[\begin{array}{ccc}15 & 3 & -6 \\ -10 & 5 \alpha & 4 \\ 8 & -4 \alpha & 3 \alpha-2\end{array}\right]\)
\(\therefore \mathrm{B}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -5 \alpha & 0 \\ 0 & 4 \alpha & -2 \alpha\end{array}\right]+\left[\begin{array}{ccc}15 & 3 & -6 \\ -10 & 5 \alpha & 4 \\ 8 & -4 \alpha & 3 \alpha-2\end{array}\right]\)
\(\Rightarrow B=\left[\begin{matrix}16 & 3 & -6 \\ -10 & 0 & 4 \\ 8 & 0 & \alpha -2\end{matrix}\right]\)
\(|\mathrm{B}|=30 \alpha+36=66\)
\(\Rightarrow \alpha=1\)
\(\Rightarrow|\mathrm{A}|=21\)
\(\Rightarrow|\operatorname{adj}(A)|=|A|^2=(21)^2=441\)
Let \(a\in R\) and A be a matrix of order \(3\times 3\) such that \(\det (A)=-4\) and \(A+I=\left[\begin{matrix}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{matrix}\right]\), where \(I\) is the identity matrix of order \(3\times 3\).
If \(\det ((\mathrm{a}+1)adj((\mathrm{a})\mathrm{A}))\) is \({2}^{\mathrm{m}}{3}^{\mathrm{n}},\mathrm{m},\mathrm{n}\in\) \({0,1,2,\ldots ..20}\), then \(\mathrm{m}+\mathrm{n}\) is equal to :
[JEE Main 2025, 2 Apr (Shift 1)]
\(16\)
Since \(A+I=[\begin{matrix}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{matrix}]\),
we subtract the identity matrix \(I\) to get \(A\):
\(A=[\begin{matrix}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{matrix}]−[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}]=[\begin{matrix}0 & a & 1 \\ 2 & 0 & 0 \\ a & 1 & 1\end{matrix}]\)
We compute the determinant of \(A\):
\(\det (A)=∣\begin{matrix}0 & a & 1 \\ 2 & 0 & 0 \\ a & 1 & 1\end{matrix}∣\)
\(−2a+2=−4\text{ }⟹\text{ }−2a=−6\text{ }⟹\text{ }a=3\)
\(\det ((a+1)\text{adj}(aI)A)={[(a+1){a}^{2}]}^{3}\det (A)\)
Substitute \(a=3\) and \(\det (A)=−4\):
\((a+1){a}^{2}=(3+1)⋅{3}^{2}=4⋅9=36\) \({[36]}^{3}⋅(−4)={36}^{3}⋅(−4)\)
\({36}^{3}⋅(−4)={2}^{6}⋅{3}^{6}⋅(−{2}^{2})=−{2}^{8}⋅{3}^{6}\)
\(\mathrm{∣}\det ((a+1)\text{adj}(aI)A)\mathrm{∣}={2}^{8}⋅{3}^{6}\)
Hence, \(m=8\) and \(n=6\).
Let \(P=\left[{p}_{ij}\right]\)and \(Q=\left[{q}_{ij}\right]\) be two square matrices of order \(3\) such that \({q}_{ij}={2}^{\left(i+j−1\right)}{p}_{ij}\) and \(det\left(Q\right)={2}^{10}\). Then the value of \(det\left(adj\left(adjP\right)\right)\) is:
[JEE Main 2026, 24 Jan (Shift 2)]
16
\(\left|Q\right|=\left|\begin{matrix}2{\text{p}}_{11} & {2}^{2}{\text{p}}_{12} & {2}^{3}{\text{p}}_{13} \\ {2}^{2}{\text{p}}_{21} & {2}^{3}{\text{p}}_{22} & {2}^{4}{\text{p}}_{23} \\ {2}^{3}{\text{p}}_{31} & {2}^{4}{\text{p}}_{32} & {2}^{5}{\text{p}}_{33}\end{matrix}\right|={2}^{10}\)
\(\Rightarrow {2}^{2}⋅2⋅{2}^{3}\left|\begin{matrix}{\text{p}}_{11} & {\text{p}}_{12} & {\text{p}}_{13} \\ 2{\text{p}}_{21} & 2{\text{p}}_{22} & 2{\text{p}}_{23} \\ {2}^{2}{\text{p}}_{31} & {2}^{2}{\text{p}}_{32} & {2}^{2}{\text{p}}_{33}\end{matrix}\right|={2}^{10}\)
\(\Rightarrow {2}^{9}\left|\begin{matrix}{\text{p}}_{11} & {\text{p}}_{12} & {\text{p}}_{13} \\ {\text{p}}_{21} & {\text{p}}_{22} & {\text{p}}_{23} \\ {\text{p}}_{31} & {\text{p}}_{32} & {\text{p}}_{33}\end{matrix}\right|={2}^{10}\)
\(\Rightarrow |\text{P}|=2\)
\(|\operatorname{adj}(\operatorname{adj}(P))|=|P|^{(n-1)^2}\)
\(=|P{|}^{4}={2}^{4}=16\)
For some a, b, let \(f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \text { a } & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}\right|, x \neq 0, \lim _{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}\). Then \((\lambda+\mu+\nu)^2\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
16
\(f\left(x\right)=\left|\begin{matrix}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{b} \\ \mathrm{a} & 1 & \mathrm{b}+\frac{\sin x}{x}\end{matrix}\right|\\ \lim _{x\to 0}f\left(x\right)=\left|\begin{matrix}a+1 & 1 & b \\ a & 1+1 & b \\ a & 1 & b+1\end{matrix}\right|\\ =\left(a+1\right)\left(2\left(b+1\right)-b\right)+1\left(ab-a\left(b+1\right)\right)-ba\\ =\left(a+1\right)\left(b+2\right)-a-ab\\ =b+a+2=\lambda +\mu a+vb\\ \text{On comparing, we get}\\ \lambda =2,\mu =1,v=1\\ \Rightarrow {\left(\lambda +\mu +v\right)}^{2}=16\)
If \(A\) is a square matrix of order 3 , then \(\left|\operatorname{Adj}\left(\operatorname{Adj} A^2\right)\right|\) is:
\(|A|^8\)
\(\text{ As }|adj(adj\mathrm{A})|=|\mathrm{A}{|}^{{(\mathrm{n}-1)}^{2}}\text{,}\\ \text{where }\mathrm{n}\text{ is the order of the determinant. }\\ \text{ So, }\left|adj\left(adj{\mathrm{A}}^{2}\right)\right|={\left|{\mathrm{A}}^{2}\right|}^{(3-1{)}^{2}}\\ ={\left|{\mathrm{A}}^{2}\right|}^{4}=|\mathrm{A}{|}^{8}\)
Let \(A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]\) and \(P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]\). The sum of the prime factors of \(\left|P^{-1} A P-2 I\right|\) is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
\(26\)
\(\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{I}\right|=\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{P}^{-1} \mathrm{P}\right|\)
\(=\left|\mathrm{P}^{-1}(\mathrm{~A}-2 \mathrm{I}) \mathrm{P}\right|\)
\(=\left|\mathrm{P}^{-1}\right||\mathrm{A}-2 \mathrm{I}||\mathrm{P}|\)
\(=|\mathrm{A}-2 \mathrm{I}|\)
\(=\left|\begin{array}{ccc}0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0\end{array}\right|\)
\(=69\)
So, Prime factor of 69 is 3 and 23
So, sum \(=26\)
If \(f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\) for all \(x \in R\), then \(2 f(0)+f^{\prime}(0)\) is equal to
[JEE Main 2024, 31 Jan (Shift 2)]
\(42\)
\(f(0)=\left|\begin{array}{ccc}0 & 1 & 1 \\ 2 & 0 & 6 \\ 0 & 4 & -2\end{array}\right|\)
\(=-2(-2-4)=12\)
Now
\(f^{\prime}(x)=\left|\begin{array}{ccc}3 x^2 & 4 x & 3 \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\)\(+\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 3 x+1 \\ 6 x & 2 & 3 x^2 \\ x^3-x & 4 & x^2-2\end{array}\right|\)\(+\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 3 x+1 \\ 3 x^2+2 & 2 x & x^3+6 \\ 3 x^2-1 & 0 & 2 x\end{array}\right|\)
\({f}^{'}\left(0\right)=\left|\begin{matrix}0 & 0 & 3 \\ 2 & 0 & 6 \\ 0 & 4 & -2\end{matrix}\right|+\left|\begin{matrix}0 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 4 & -2\end{matrix}\right|+\left|\begin{matrix}0 & 1 & 1 \\ 2 & 0 & 6 \\ -1 & 0 & 0\end{matrix}\right|\)
\(=-2(0-12)+0+0-1(0+6)+1(0-0)\)
\(=24-6=18\)
Now
\(2 f(0)+f^{\prime}(0)= 2.12+18\)
\(= 24+18=42\)
If \(\alpha \neq a , \beta \neq b , \gamma \neq c\) and \(\left|\begin{array}{lll}\alpha & b & c \\ a & \beta & c \\ a & b & \gamma\end{array}\right|=0\), then \(\frac{ a }{\alpha- a }+\frac{ b }{\beta- b }+\frac{\gamma}{\gamma- c }\) is equal to:
[JEE Main 2024, 08 Apr (Shift 2)]
\(0\)
Solution of determinant by using operation of determinant
\({R}_{1}\to {R}_{1}-{R}_{2}\\ {R}_{2}\to {R}_{2}-{R}_{3}\)
\(\left|\begin{matrix}\alpha -a & b-\beta & 0 \\ 0 & \beta -b & c-\gamma \\ a & b & \gamma \end{matrix}\right|=0\)
\(\left(\alpha -a\right)\left(\beta -b\right)\left(\gamma -c\right)\left|\begin{matrix}1 & -1 & 0 \\ 0 & 1 & -1 \\ \frac{a}{\alpha -a} & \frac{b}{\beta -b} & \frac{\gamma }{\gamma -c}\end{matrix}\right|=0\)
\(\frac{a}{\alpha -a}\left(1-0\right)-\frac{b}{\beta -b}\left(-1-0\right)+\frac{\gamma }{\gamma -c}\left(1-0\right)=0\)
\(\frac{a}{\alpha -a}+\frac{b}{\beta -b}+\frac{\gamma }{\gamma -c}=0\)
If \(\alpha \neq a , \beta \neq b , \gamma \neq c\) and \(\left|\begin{array}{lll}\alpha & b & c \\ a & \beta & c \\ a & b & \gamma\end{array}\right|=0\), then \(\frac{ a }{\alpha- a }+\frac{ b }{\beta- b }+\frac{\gamma}{\gamma- c }\) is equal to:
[JEE Main 2024, 08 Apr (Shift 2)]
\(0\)
Solution of determinant by using operation of determinant
\({R}_{1}\to {R}_{1}-{R}_{2}\\ {R}_{2}\to {R}_{2}-{R}_{3}\)
\(\left|\begin{matrix}\alpha -a & b-\beta & 0 \\ 0 & \beta -b & c-\gamma \\ a & b & \gamma \end{matrix}\right|=0\)
\(\left(\alpha -a\right)\left(\beta -b\right)\left(\gamma -c\right)\left|\begin{matrix}1 & -1 & 0 \\ 0 & 1 & -1 \\ \frac{a}{\alpha -a} & \frac{b}{\beta -b} & \frac{\gamma }{\gamma -c}\end{matrix}\right|=0\)
\(\frac{a}{\alpha -a}\left(1-0\right)-\frac{b}{\beta -b}\left(-1-0\right)+\frac{\gamma }{\gamma -c}\left(1-0\right)=0\)
\(\frac{a}{\alpha -a}+\frac{b}{\beta -b}+\frac{\gamma }{\gamma -c}=0\)
Let A be a matrix of order \(3\times 3\) and \(|A|=5\). If \(|2adj(3\mathrm{A}adj(2\mathrm{A}))|={2}^{\alpha }\cdot {3}^{\beta }\cdot {5}^{\gamma }\alpha ,\beta ,\gamma \in \mathrm{N}\) then \(\alpha +\beta +\gamma\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
\(27\)
First, use property (3) for \(\text{adj}(2A)\): Since \(n=3\), \(\text{adj}(2A)={2}^{3−1}\text{adj}(A)=4\text{ adj}(A)\)
Now substitute back:
\(3A⋅\text{adj}(2A)=3A⋅4\text{ adj}(A)=12⋅(A⋅\text{adj}(A))\)
Using property (4): \(A⋅\text{adj}(A)=\mathrm{∣}A\mathrm{∣}I=5I\)
\(\text{ }⟹\text{ }3A⋅\text{adj}(2A)=12\times 5I=60I\)
For scalar identity matrix \(kI\), \(\text{adj}(kI)={k}^{n−1}I\) (for \(n=3\), this is \({k}^{2}I\)):
\(\text{adj}(60I)={60}^{2}I\)
First, \(2⋅\text{adj}(60I)=2\times {60}^{2}I\) Using property (1), determinant of \(mI\) (3x3) is \({m}^{3}\):
\(∣2\times {60}^{2}I∣={(2\times {60}^{2})}^{3}\)
Write \(60={2}^{2}\times 3\times 5\), so:
\({(2\times ({2}^{2}\times 3\times 5{)}^{2})}^{3}={2}^{3}\times ({2}^{2}\times 3\times 5{)}^{6}\) \(={2}^{3}\times {2}^{12}\times {3}^{6}\times {5}^{6}=\)
Comparing with \({2}^{\alpha }⋅{3}^{\beta }⋅{5}^{\gamma }\):
\(\alpha =15\), \(\beta =6\), \(\gamma =6\)
\(\alpha +\beta +\gamma =15+6+6=27\)
\(\text { If } \mathrm{A} \text { is } 3 \times 3 \text { matrix such that } \operatorname{det}(\mathrm{A})=2 \text {. Then } \operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))\) (24 Jan, Shift I, Memory Based)
\(2^{32}\)
Given:- \(A\) is order 3 square matrix \(|A|=2\)
To find: \(|(\operatorname{adj}\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A)))|\)
we know \(\mid\) adj \(A\left|=|A|^{n-1}\right.\)
\(\begin{aligned}&|A|=2 \quad and \quad n=3 \\& \mid \operatorname{adj}(\operatorname{adj}(\operatorname{adi}(\operatorname{adj} A) \mid=|A|^{(n-1)^4} \\& =|A|^{2^4} \\&=2^{2^4} \\&=2^{16}\end{aligned}\)
\(\text { Let } A \text { and } B \text { are non-singular commutative matrices. Then } A\left[\left(\operatorname{adj} A^{-1}\right)\left(a d j\left(B^{-1}\right)\right)\right]^{-1} B \text { is equal to }\)
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Among the statements:
I: If \(\left|\begin{matrix}1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{matrix}\right|=\left|\begin{matrix}0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{matrix}\right|\), then \({\cos }^{2}\alpha +{\cos }^{2}\beta +{\cos }^{2}\gamma =\frac{3}{2}\), and
II: If \(\left|\begin{matrix}{x}^{2}+x & x+1 & x-2 \\ 2{x}^{2}+3x-1 & 3x & 3x-3 \\ {x}^{2}+2x+3 & 2x-1 & 2x-1\end{matrix}\right|=px+q\), then \({p}^{2}=196{q}^{2}\).
[JEE Main 2026, 23 Jan (Shift 1)]
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Let \(A=\left[\begin{matrix}1 & 2 \\ 0 & 1\end{matrix}\right]\) and \(B=I+adj(A)+(adjA{)}^{2}+\ldots +(adjA{)}^{10}\).Then, the sum of all the elements of the matrix \(B\) is:EndFragment
[JEE Main 2024, 04 Apr (Shift 2)]
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Let \(A=\left[\begin{matrix}1 & 2 \\ 0 & 1\end{matrix}\right]\) and \(B=I+adj(A)+(adjA{)}^{2}+\ldots +(adjA{)}^{10}\).Then, the sum of all the elements of the matrix \(B\) is:EndFragment
[JEE Main 2024, 04 Apr (Shift 2)]
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If \(f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\) for all \(x \in R\), then \(2 f(0)+f^{\prime}(0)\) is equal to
[JEE Main 2024, 31 Jan (Shift 2)]
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If \(A\) is a square matrix of order 3 such that \(\operatorname{det}(A)=3\) and \(\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A )^{-1}\right)\right)\right)\right)\right)=2^{ m } 3^{ n }\), then \(m +2 n\) is equal to :
[JEE Main 2024, 6 Apr (Shift 2)]
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Let \(A=\left[\begin{matrix}2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q\end{matrix}\right]\).If \(\det (adj(adj(3\mathrm{A})))={2}^{\mathrm{m}}\cdot {3}^{\mathrm{n}},\mathrm{m},\mathrm{n}\in \mathrm{ℕ}\), then \(\mathrm{m}+\mathrm{n}\) is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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\(\text { If } \mathrm{A} \text { is } 3 \times 3 \text { matrix such that } \operatorname{det}(\mathrm{A})=2 \text {. Then } \operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))\) (24 Jan, Shift I, Memory Based)
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If \(X=\left[\begin{matrix}x \\ y \\ z\end{matrix}\right]\) is a solution of the system of equation \(AX=B,\) where \(adjA=\left[\begin{matrix}4 & 2 & 2 \\ −5 & 0 & 5 \\ 1 & −2 & 3\end{matrix}\right]\) and \(B=\left[\begin{matrix}4 \\ 0 \\ 2\end{matrix}\right],\) then \(\left|x+y+z\right|\) is equal to:
[JEE Main 2026, 22 Jan (Shift 2)]
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Let \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{array}\right]\) and \(|2 A|^{3}=2^{21}\) where \(\alpha, \beta \in Z\), Then a value of \(\alpha\) is
[JEE Main 2024, 29 Jan (Shift 1)]
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Let \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{array}\right]\) and \(|2 A|^{3}=2^{21}\) where \(\alpha, \beta \in Z\), Then a value of \(\alpha\) is
[JEE Main 2024, 29 Jan (Shift 1)]
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The values of \(\alpha\), for which \(\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0\), lie in the interval
[JEE Main 2024, 27 Jan (Shift 2)]
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The values of \(\alpha\), for which \(\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0\), lie in the interval
[JEE Main 2024, 27 Jan (Shift 2)]
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If \(f(x)=\left|\begin{array}{ccc}2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x\end{array}\right|\), then \(\frac{1}{5} f^{\prime}(0)\) is equal to :
[JEE Main 2024, 30 Jan (Shift 1)]
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If \(f(x)=\left|\begin{array}{ccc}2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x\end{array}\right|\), then \(\frac{1}{5} f^{\prime}(0)\) is equal to :
[JEE Main 2024, 30 Jan (Shift 1)]
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Consider the matrix \(f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]\).
Given below are two statements :
Statement I : \(f(-x)\) is the inverse of the matrix \(f(x)\).
Statement II : \(f(x) f(y)=f(x+y)\).
In the light of the above statements, choose the correct answer from the options given below
[JEE Main 2024, 27 Jan (Shift 1)]
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Consider the matrix \(f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]\).
Given below are two statements :
Statement I : \(f(-x)\) is the inverse of the matrix \(f(x)\).
Statement II : \(f(x) f(y)=f(x+y)\).
In the light of the above statements, choose the correct answer from the options given below
[JEE Main 2024, 27 Jan (Shift 1)]
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If A, B and \(\left(adj\left({\mathrm{A}}^{-1}\right)+adj\left({\mathrm{B}}^{-1}\right)\right)\) are non-singular matrices of same order, then the inverse of \(\mathrm{A}{\left(adj\left({\mathrm{A}}^{-1}\right)+adj\left({\mathrm{B}}^{-1}\right)\right)}^{-1}\mathrm{B},\) is equal to
[JEE Main 2025, 23 Jan (Shift 1)]
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Let A be a \(3\times 3\) matrix such that \(|adj(adj(adj\mathrm{A}))|=81\). If
\(S=\left\{n \in \mathbb{Z}:(|\operatorname{adj}(\operatorname{adj} A)|)^{\frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}\right\}\) then \(\sum _{\mathrm{n}\in \mathrm{S}}\left|{\mathrm{A}}^{\left({\mathrm{n}}^{2}+\mathrm{n}\right)}\right|\) is equal to
[JEE Main 2025, 7 Apr (Shift 1)]
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For a \(3 \times 3\) matrix M , let trace \((\mathrm{M})\) denote the sum of all the diagonal elements of M . Let A be a \(3 \times 3\) matrix such that \(|A|=\frac{1}{2}\) and trace \((A)=3\). If \(B=\operatorname{adj}(\operatorname{adj}(2 A))\), then the value of \(|B|+\operatorname{trace}(B)\) equals:
[JEE Main 2025, 22 Jan (Shift 2)]
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Let \(A=\left[\begin{matrix}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{matrix}\right]\) and \(\det (\mathrm{A}-\alpha \mathrm{I})=0\), where \(\alpha\) is a real number. If the largest possible value of \(\alpha\) is \(p\), then the circle \((x-p{)}^{2}+(y-2p{)}^{2}=320\), intersects the co-ordinate axes at
[JEE Main 2026, 4 Apr (Shift 2)]
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If \(A\) is a square matrix of order 3 , then \(\left|\operatorname{Adj}\left(\operatorname{Adj} A^2\right)\right|\) is:
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Let \(\alpha \in (0,\infty )\) and\(A=\left[\begin{matrix}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{matrix}\right]\). If \(\det \left(adj\left(2A-{A}^{T}\right)\cdot adj\left(A-2{A}^{T}\right)\right)={2}^{8}\), then \((\det (A){)}^{2}\) is equal to :
[JEE Main 2024, 4 Apr (Shift 1)]
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Let\(\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]=\left[\begin{matrix}{\log }_{5}128 & {\log }_{4}5 \\ {\log }_{5}8 & {\log }_{4}25\end{matrix}\right].\) If \({\mathrm{A}}_{\mathrm{ij}}\) is the cofactor of \({\mathrm{a}}_{\mathrm{ij}},{\mathrm{C}}_{\mathrm{ij}}=\sum _{\mathrm{k}=1}^{2}{\mathrm{a}}_{\mathrm{ik}}{\mathrm{A}}_{\mathrm{jk}},1\leq \mathrm{i},\mathrm{j}\leq 2,\) and \(\mathrm{C}=\left[{\mathrm{C}}_{\mathrm{ij}}\right]\), then \(8|\mathrm{C}|\) is equal to :
[JEE Main 2025, 29 Jan (Shift 1)]
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If \(X=\left[\begin{matrix}x \\ y \\ z\end{matrix}\right]\) is a solution of the system of equation \(AX=B,\) where \(adjA=\left[\begin{matrix}4 & 2 & 2 \\ −5 & 0 & 5 \\ 1 & −2 & 3\end{matrix}\right]\) and \(B=\left[\begin{matrix}4 \\ 0 \\ 2\end{matrix}\right],\) then \(\left|x+y+z\right|\) is equal to:
[JEE Main 2026, 22 Jan (Shift 2)]
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Let \(A=\left[\begin{matrix}1 & 2 \\ 1 & \alpha \end{matrix}\right]\) and \(B=\left[\begin{matrix}3 & 3 \\ \beta & 2\end{matrix}\right]\). If \({A}^{2}-4A+I=O\) and \({B}^{2}-5B-6I=O\), then among the two statements:
\(\left({S}_{1}\right):{\left[\left(B-A\right)\left(B+A\right)\right]}^{T}=\left[\begin{matrix}13 & 15 \\ 7 & 10\end{matrix}\right]\) and
\(({S}_{2}):\det (adj(A+B))=-5\),
[JEE Main 2026, 2 Apr (Shift 1)]
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Among the statements:
I: If \(\left|\begin{matrix}1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{matrix}\right|=\left|\begin{matrix}0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{matrix}\right|\), then \({\cos }^{2}\alpha +{\cos }^{2}\beta +{\cos }^{2}\gamma =\frac{3}{2}\), and
II: If \(\left|\begin{matrix}{x}^{2}+x & x+1 & x-2 \\ 2{x}^{2}+3x-1 & 3x & 3x-3 \\ {x}^{2}+2x+3 & 2x-1 & 2x-1\end{matrix}\right|=px+q\), then \({p}^{2}=196{q}^{2}\).
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Let \(M\) and \(m\) respectively be the maximum and the minimum values of \(f\left(x\right)=\left|\begin{matrix}1+{\sin }^{2}x & {\cos }^{2}x & 4\sin 4x \\ {\sin }^{2}x & 1+{\cos }^{2}x & 4\sin 4x \\ {\sin }^{2}x & {\cos }^{2}x & 1+4\sin 4x\end{matrix}\right|,\)\(x\in R\). Then \({M}^{4}-{m}^{4}\) is equal to :
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Let \(\alpha \beta \neq 0\) and \(A=\left[\begin{matrix}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{matrix}\right]\). If \(B=\left[\begin{matrix}3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{matrix}\right]\) is the matrix of cofactors of the elements of \(A\), then \(\det (AB)\) is equal to
[JEE Main 2024, 5 Apr (Shift 2)]
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\(\text { Let } A \text { and } B \text { are non-singular commutative matrices. Then } A\left[\left(\operatorname{adj} A^{-1}\right)\left(a d j\left(B^{-1}\right)\right)\right]^{-1} B \text { is equal to }\)
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Let \(A\) and \(B\) be two square matrices of order 3 such that \(|A|=3\) and \(|B|=2\). Then \(\left|{A}^{T}A(adj(2A){)}^{-1}(adj(4B))(adj(AB){)}^{-1}A{A}^{T}\right|\) is equal to :
[JEE Main 2024, 5 Apr (Shift 1)]
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For a \(3 \times 3\) matrix \(M\) , let trace \((\mathrm{M})\) denote the sum of all the diagonal elements of \(M\) . Let \(A\) be a \(3 \times 3\) matrix such that \(|A|=\frac{1}{2}\) and trace \((A)=3\). If \(B=\operatorname{adj}(\operatorname{adj}(2 A))\), then the value of \(|B|+\operatorname{trace}(B)\) equals:
[JEE Main 2025, 22 Jan (Shift 2)]
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Let \(P=\left[{p}_{ij}\right]\)and \(Q=\left[{q}_{ij}\right]\) be two square matrices of order \(3\) such that \({q}_{ij}={2}^{\left(i+j−1\right)}{p}_{ij}\) and \(det\left(Q\right)={2}^{10}\). Then the value of \(det\left(adj\left(adjP\right)\right)\) is:
[JEE Main 2026, 24 Jan (Shift 2)]
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If \(A\) is a square matrix of order 3 such that \(\operatorname{det}(A)=3\) and \(\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A )^{-1}\right)\right)\right)\right)\right)=2^{ m } 3^{ n }\), then \(m +2 n\) is equal to :
[JEE Main 2024, 6 Apr (Shift 2)]
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If \(A\) is a square matrix of order 3 such that \(\operatorname{det}(A)=3\) and \(\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A )^{-1}\right)\right)\right)\right)\right)=2^{ m } 3^{ n }\), then \(m +2 n\) is equal to :
[JEE Main 2024, 6 Apr (Shift 2)]
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Let A be a matrix of order \(3\times 3\) and \(|A|=5\). If \(|2 \operatorname{adj}(3 \operatorname{Aadj}(2 A))|=2^\alpha \cdot 3^\beta \cdot 5^{\gamma} \alpha, \beta, \gamma \in N\) then \(\alpha +\beta +\gamma\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
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If \(A, B\) and \(\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)\) are non-singular matrices of same order, then the inverse of \(A\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)^{-1} B\) is equal to
[JEE Main 2025, 23 Jan (Shift 1)]
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If \(f: \mathrm{N} \rightarrow \mathrm{Z}\) is defined by \(f\left(n\right)=\left|\begin{matrix}n & -1 & -5 \\ -2{n}^{2} & 3(2k+1) & 2k+1 \\ -3{n}^{3} & 3k(2k+1) & 3k(k+2)+1\end{matrix}\right|,k\in N,\text{ and }\)\(\sum _{n=1}^{k}f\left(n\right)=98\) then \(k\) is equal to:
[JEE Main 2026, 5 Apr (Shift 2)]
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Let \(S={\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]:a,b,c,d\in \left\{0,1,2,3,4\right\}\) and \({A}^{2}-4A+3I=0}\) be a set of \(2\times 2\) matrices. Then the number of matrices in \(S\), for which the sum of the diagonal elements is equal to \(4\), is:
[JEE Main 2026, 4 Apr (Shift 1)]
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Let \(A=\left[\begin{matrix}2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q\end{matrix}\right]\).
If \(\det (adj(adj(3\mathrm{A})))={2}^{\mathrm{m}}\cdot {3}^{\mathrm{n}},\mathrm{m},\mathrm{n}\in \mathrm{ℕ}\), then \(\mathrm{m}+\mathrm{n}\) is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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For some \(a, b\), let \(f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \text { a } & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}\right|, x \neq 0, \lim _{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}\). Then \((\lambda+\mu+\nu)^2\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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Let \(A\) be a \(3 \times 3\) matrix such that \({A}^{T}\left[\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}5 \\ 2 \\ 2\end{matrix}\right],{A}^{T}\left[\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}3 \\ 1 \\ 1\end{matrix}\right],A\left[\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}3 \\ 4 \\ 4\end{matrix}\right]\) and \(A\left[\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}1 \\ 3 \\ 1\end{matrix}\right]\). If \(\operatorname{det}(A)=1\), then \(\operatorname{det}\left(\operatorname{adj}\left(A^2+A\right)\right)\) is equal to:
[JEE Main 2026, 5 Apr (Shift 1)]
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Let\(\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]=\left[\begin{matrix}{\log }_{5}128 & {\log }_{4}5 \\ {\log }_{5}8 & {\log }_{4}25\end{matrix}\right].\) If \({\mathrm{A}}_{\mathrm{ij}}\) is the cofactor of \({\mathrm{a}}_{\mathrm{ij}},{\mathrm{C}}_{\mathrm{ij}}=\sum _{\mathrm{k}=1}^{2}{\mathrm{a}}_{\mathrm{ik}}{\mathrm{A}}_{\mathrm{jk}},1\leq \mathrm{i},\mathrm{j}\leq 2,\), and \(\mathrm{C}=\left[{\mathrm{C}}_{\mathrm{ij}}\right]\), then \(8|\mathrm{C}|\) is equal to :
[JEE Main 2025, 29 Jan (Shift 1)]
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Let \(A=\left[\begin{matrix}1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5\end{matrix}\right]\). Then the sum of all elements of the matrix \(adj\left(adj\left(2(adjA{)}^{-1}\right)\right)\) is equal to:
[JEE Main 2026, 4 Apr (Shift 1)]
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Let \(M\) and \(m\) respectively be the maximum and the minimum values of \(f\left(x\right)=\left|\begin{matrix}1+{\sin }^{2}x & {\cos }^{2}x & 4\sin 4x \\ {\sin }^{2}x & 1+{\cos }^{2}x & 4\sin 4x \\ {\sin }^{2}x & {\cos }^{2}x & 1+4\sin 4x\end{matrix}\right|,\)\(x\in R\). Then \({M}^{4}-{m}^{4}\) is equal to :
[JEE Main 2025, 29 Jan (Shift 1)]
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Let \(a\in \mathrm{R}\) and \(A\) be a matrix of order \(3\times 3\) such that \(\det (A)=-4\) and \(A+I=\left[\begin{matrix}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{matrix}\right]\), where \(I\) is the identity matrix of order \(3\times 3\). If \(\det ((\mathrm{a}+1)adj((\mathrm{a})\mathrm{A}))\) is \({2}^{\mathrm{m}}{3}^{\mathrm{n}},\mathrm{m},\mathrm{n}\in\) \({0,1,2,\ldots ..20}\), then \(\mathrm{m}+\mathrm{n}\) is equal to :
[JEE Main 2025, 2 Apr (Shift 1)]
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Let for \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ a & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|A|=2\). If \(|2 \operatorname{adj}(2 \operatorname{adj}(2 A))|=32^{ n }\), then \(3 n+\alpha\) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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If \(P\) is a \(3 \times 3\) real matrix such that \(P^T=a P+(a-1) I\), where \(a>1\), then
[JEE Main 2023, 30 Jan (Shift 2)]
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If \( A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right) \) and \( \operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0 \), then a possible value of \( \alpha \) is :
[JEE Main 2021, 17 Mar (Shift 1)]
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If \(A\) is a \(3 \times 3\) matrix and \(|A|=2\), then \(\left|3 \operatorname{adj}\left(|3 A| A^2\right)\right|\) is equal to
[JEE Main 2023, 10 Apr (Shift 1)]
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Let \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]\) and \(B=\left[\begin{array}{rr}1 & -i \\ 0 & 1\end{array}\right]\), where \(i=\sqrt{-1}\). If \(M=A^T B A\), then the inverse of the matrix \(A M^{2023} A^T\) is
[JEE Main 2023, 25 Jan (Shift 2)]
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Let \(A=\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right]\). If \(A^{-1}=\alpha I+\beta A, \alpha, \beta \in R . I\) is a \(2 \times 2\) identify matrix, then \(4(\alpha-\beta)\) is equal to:
[JEE Main 2021, 27 Jul (Shift 1)]
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Let \(x, y, z>1\) and \(A=\left[\begin{array}{lll}1 & \log _x y & \log _x z \\\log _y x & 2 & \log _y z \\\log _z x & \log _z y & 3\end{array}\right]\).Then \(\left|\operatorname{adj}\left(\operatorname{adj} A^2\right)\right|\) is equal to
[JEE Main 2023, 25 Jan (Shift 1)]
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Let for \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|A|=2\). If \(|2 \operatorname{adj}(2 \operatorname{adj}(2 A))|=32^{ n }\), then \(3 n+\alpha\) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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If \(A=\left(\begin{array}{ll}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i =\sqrt{-1}\), and \(Q=A^T B A\), then the inverse of the matrix \(A Q^{2021} A^T\) is equal to
[JEE Main 2021, 26 Aug (Shift 1)]
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If \(\Delta =\left|\begin{matrix}x-2 & 2x-3 & 3x-4 \\ 2x-3 & 3x-4 & 4x-5 \\ 3x-5 & 5x-8 & 10x-17\end{matrix}\right|\)
\(=A{x}^{3}+B{x}^{2}+Cx+D\), \(B+C\) is equal to
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If \({a}_{r}=\cos \frac{2r\pi }{9}+i\sin \frac{2r\pi }{9},\)
\(r=1,2,3,...,i=\sqrt{-1},\) then the determinant \(\left|\begin{matrix}{a}_{1} & {a}_{2} & {a}_{3} \\ {a}_{4} & {a}_{5} & {a}_{6} \\ {a}_{7} & {a}_{8} & {a}_{9}\end{matrix}\right|\) is
[JEE Main 2021, 31 Aug (Shift 1)]
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Let \(f(x)=\left|\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{array}\right|\), \(x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right]\). If \(\alpha\) and \(\beta\) respectively are the maximum and the minimum values of \(f\), then
[JEE Main 2023, 1 Feb (Shift 1)]
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If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(k^2\) is:
[JEE Main 2021, 17 Mar (Shift 2)]
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Let \( A=\left[a_{i j}\right] \) and \( B=\left[b_{i j}\right] \) be two \( 3 \times 3 \) real matrices such that \( b_{i j}=(3)^{(i+j-2)} a_{ij} \), where \( i, j=1,2,3 \). If the determinant of \( B \) is \(81\) , then the determinant of \( A \) is:
[JEE Main 2020, 7 Jan (Shift 2)]
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If A is a 3 × 3 matrix and |A| = 2, then |3adj (|3A| A2)| is equal to:
[JEE Main 2023, 10 Apr (Shift 1)]
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If \( A=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{lll}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right] \), then \( |\operatorname{adj}(\operatorname{adj}(2 A))| \) is equal to
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If \(A=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{lll}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]\) then \(|adj(\operatorname{adj}(2 A))|\) is equal to :
[JEE Main 2023, 10 Apr (Shift 2)]
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Let \(A=\left[{a}_{ij}\right]\) be a \(3\times 3\) matrix, where
\({a}_{ij}=\left\{\begin{matrix}1,\text{ if }i=j \\ -x,\text{ if }|i-j|=1 \\ 2x+1,\text{ otherwise }\end{matrix}\right.\)
Let a function \(f:R\to R\) be defined as \(f(x)=\det (A)\). Then the sum of maximum and minimum values of f on R is equal to :
[JEE Main 2021, 20 Jul (Shift 1)]
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The maximum value of
\(f\left(x\right)=\left|\begin{matrix}{\sin }^{2}x & 1+{\cos }^{2}x & \cos 2x \\ 1+{\sin }^{2}x & {\cos }^{2}x & \cos 2x \\ {\sin }^{2}x & {\cos }^{2}x & \sin 2x\end{matrix}\right|,\)
\(x\in R\) is
[JEE Main 2021, 16 Mar (Shift 2)]
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Let \(A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]\) If \(\mid \operatorname{adj}\left(\operatorname{adj}(\operatorname{adj} 2 A)) \mid=(16)^n\right.\) then \(n\) is equal to
[JEE Main 2023, 8 Apr (Shift 1)]
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Let \(J_{n, m}=\int_0^{\frac{1} { 2}} \frac{x^n}{x^m-1} d x, \forall n>m\) and \(n, m \in N\). Consider a matrix \(A=\left[a_{i j}\right]_{3 \times 3}\), where \(a_{i j}=\left\{\begin{array}{cc}J_{6+i, 3}-J_{i+3,3}, & i \leq j \\ 0, & i>j\end{array}\right.\)
. Then \(\left|\operatorname{adj} A^{-1}\right|\) is:
[JEE Main 2021, 1 Sep (Shift 2)]
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Let the determinant of a square matrix A of order m be m – n, where m and n satisfy 4m + n = 22 and 17m + 4n = 93. If det (n adj(adj(mA))) = 3a5b6c. then a + b + c is equal to:
[JEE Main 2023, 15 Apr (Shift 1)]
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If P is a 3 × 3 real matrix such that PT = aP + (a – 1) I, where a > 1, then
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Let \(\alpha\) be a root of the equation \((a-c) x^2+(b-a) x+\) \((c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix \(\left[\begin{array}{ccc}\alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{array}\right]\) is singular. Then the value of \(\frac{(a-c)^2}{(b-a)(c-b)}+\frac{(b-a)^2}{(a-c)(c-b)}+\frac{(c-b)^2}{(a-c)(b-a)}\) is
[JEE Main 2023, 24 Jan (Shift 1)]
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The set of all values of \(t \in R\), for which the matrix \(\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\ e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\ e^t & e^{-t} \cos t & e^{-t} \sin t\end{array}\right]\) Invertible, is
[JEE Main 2023, 29 Jan (Shift 2)]
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The maximum value of \(f(x)=\left|\begin{array}{ccc}\sin ^2 x & 1+\cos ^2 x & \cos 2 x \\ 1+\sin ^2 x & \cos ^2 x & \cos 2 x \\ \sin ^2 x & \cos ^2 x & \sin 2 x\end{array}\right|, x \in R\) is
[JEE Main 2021, 16 Mar (Shift 2)]
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The number of distinct real roots of \(\left|\begin{matrix}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{matrix}\right|=0\) in the interval \(-\frac{\pi }{4}\leq x\leq \frac{\pi }{4}\) is:
[JEE Main 2021, 25 Jul (Shift 2)]
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Let \( x, y, z>1 \) and \( A=\left[\begin{array}{ccc}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 2 & \log _{y} z \\ \log _{z} x & \log _{z} y & 3\end{array}\right] \).
Then, \( \left|\operatorname{adj}\left(\operatorname{adj} A^{2}\right)\right| \) is equal to
[JEE Main 2023, 25 Jan (Shift 1)]
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Let \(A\) be a \(3 \times 3\) matrix such that \(|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))|=12^4\). Then \(\mid A^{-1}\) adj \(A \mid\) is equal to
[JEE Main 2023, 24 Jan (Shift 2)]
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If \( A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right) \) and \( \operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0 \), then a possible value of \( \alpha \) is:
[JEE Main 2021, 17 Mar (Shift 1)]
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The value of \( \left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right| \) is :
[JEE Main 2021, 26 Feb (Shift 1)]
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Let \(A\) and \(B\) be two \(3 \times 3 \) real matrices such that \(\left(A^2-B^2\right)\) is invertible matrix. If \(A^5=B^5\) and \(A^3 B^2=A^2 B^3\), then the value of the determinant of the matrix \(A^3+B^3\) is equal to :
[JEE Main 2021, 27 Jul (Shift 2)]
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If \(A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], A^{-1}=\alpha A+\beta I\) and \(\alpha+\beta=-2\), then \(4 \alpha^2+\) \(\beta^2+\lambda^2\) is equal to:
[JEE Main 2023, 08 Apr (Shift 2)]
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$$\begin{aligned}&\text { The solutions of the equation }\\&\left|\begin{array}{ccc}1+\sin ^2 x & \sin ^2 x & \sin ^2 x \\\cos ^2 x & 1+\cos ^2 x & \cos ^2 x \\4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0,(0
[JEE Main 2021, 18 Mar (Shift 1)]
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The solutions of the equation \(\left|\begin{array}{ccc}1+\sin ^2 x & \sin ^2 x & \sin ^2 x \\\cos ^2 x & 1+\cos ^2 x & \cos ^2 x \\4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0,(0 [JEE Main 2021, 18 Mar (Shift 1)]
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Let the determinant of a square matrix \(A\) of order \(m\) be \(m-n\), where \(m\) and \(n\) satisfy \(4 m+n=22\) and \(17 m+4 n=\) 93. If \(\operatorname{det}(n \operatorname{adj}(\operatorname{adj}(m A)))=3^a 5^b 6^c\). then \(a+b+c\) is equal to:
[JEE Main 2023, 15 Apr (Shift 1)]
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Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A^{\prime}=\alpha A+I\), where \(\alpha \in R -\{-1,1\}\). If \(\operatorname{det}\left(A^2-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to
[JEE Main 2023, 11 Apr (Shift 1)]
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If the matrices \( A=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{array}\right], B=\operatorname{adj} A \) and \( C=3 A \), then \( \frac{|\operatorname{adj} B|}{|C|} \) is equal to
[JEE Main 2020, 9 Jan (Shift 1)]
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Let A be a \(3\times 3\) matrix with \(\det (\mathrm{A})=4\). Let \({\mathrm{R}}_{\mathrm{i}}\) denote the \({\mathrm{i}}^{\text{th }}\) row of A. If a matrix B is obtained by performing the operation \({\mathrm{R}}_{2}\to 2{\mathrm{R}}_{2}+5{\mathrm{R}}_{3}\) on 2A, then \(\det (\mathrm{B})\) is equal to:
[JEE Main 2021, 25 Feb (Shift 2)]
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If \(\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2\end{array}\right|=\frac{9}{8}(103 x+81)\), then \(\lambda, \frac{\lambda}{3}\) are the roots of the equation
[JEE Main 2023, 11 Apr (Shift 2)]
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Consider the following system of equations:
\[\begin{aligned}& x+2 y-3 z=a \\& 2 x+6 y-11 z=b \\& x-2 y+7 z=c\end{aligned}\]
where \(a, b\) and \(c\) are real constants. Then the system of equations:
[JEE Main 2021, 26 Feb (Shift 2)]
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If \(A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], A^{-1}=\alpha A+\beta I\) and \(\alpha+\beta=-2\), then \(4 \alpha^2+\) \(\beta^2+\lambda^2\) is equal to:
[JEE Main 2023, 8 Apr (Shift 2)]
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Let \( a-2 b+c=1 \). If \( f(x)=\left|\begin{array}{lll}x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3\end{array}\right| \), then :
[JEE Main 2020, 9 Jan (Shift 2)]
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Let \(A\) be a \(3 \times 3\) matrix with \(\operatorname{det}(A)=4\). Let \(R_i\) denote the \(i^{\text {th }}\) row of \(A\). If a matrix \(B\) is obtained by performing the operation \(R_2 \rightarrow 2 R_2+5 R_3\) on \(2 A\), then \(\operatorname{det}(B)\) is equal to:
[JEE Main 2021, 25 Feb (Shift 2)]
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Let \(B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha>2\) be the adjoint of a matrix \(A\) and \(|A|=2\), then \(\left[\begin{array}{ll}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]\) is equal to:
[JEE Main 2023, 13 Apr (Shift 1)]
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If \( \mathrm{a}+\mathrm{x}=\mathrm{b}+\mathrm{y}=\mathrm{c}+\mathrm{z}+1 \), where \( \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{x}, \mathrm{y}, \mathrm{z} \) are non-zero distinct real numbers, then \( \left|\begin{array}{lll}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{array}\right| \) is equal to
[JEE Main 2020, 5 Sep (Shift 2)]
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Let \(B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha>2\) be the adjoint of a matrix \(A\) and \(|A|=2\), then \(\left[\begin{array}{ll}\alpha-2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]\) is equal to:
[JEE Main 2023, 13 Apr (Shift 1)]
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Let \(A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]\) If \(\mid \operatorname{adj}\left(\operatorname{adj}(\operatorname{adj} 2 A)) \mid=(16)^n\right.\) then \(n\) is equal to
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If the minimum and the maximum values of the function \(f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow R\), defined by \(f(\theta)=\left|\begin{array}{ccc}-\sin ^2 \theta & -1-\sin ^2 \theta & 1 \\-\cos ^2 \theta & -1-\cos ^2 \theta & 1 \\12 & 10 & -2\end{array}\right|\)
are \(m\) and \(M\) respectively, then the ordered pair \((m, M)\) is equal to:
[JEE Main 2020, 5 Sep (Shift 1)]
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The number of distinct real roots of \( \left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0 \) in the interval \( -\frac{\pi}{4} \leq x \leq \frac{\pi}{4} \) is
[JEE Main 2021, 25 Jul (Shift 2)]
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Let \(J_{n, m}=\int_0^{1 / 2} \frac{x^n}{x^m-1} d x, \forall n>m\) and \(n, m \in N\). Consider a matrix \(A=\left[a_{i j}\right]_{3 \times 3}\) where \(a_{i j}=\left\{\begin{array}{cc}J_{6+i, 3}-J_{i+3,3}, & i \leq j \\ 0, & i>j\end{array}\right.\).Then \(\left|\operatorname{adj} A^{-1}\right|\) is:
[JEE Main 2021, 1 Sep (Shift 2)]
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\(\text{ Let }A=\left[\begin{matrix}1 & 2 \\ -1 & 4\end{matrix}\right].\text{ If }{A}^{-1}=\alpha I+\beta A,\alpha ,\beta \in R\text{. I is a }2\times 2\text{ identify matrix, then }4(\alpha -\beta )\text{ is equal to : }\begin{matrix}\end{matrix}\)
[JEE Main 2021, 27 Jul (Shift 1)]
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Let \(A\) and \(B\) be two \(3 \times 3\) real matrices such that \(\left(A^2-B^2\right)\) is invertible matrix. If \(A^5=B^5\) and \(A^3 B^2=A^2 B^3\), then the value of the determinant of the matrix \(A^3+B^3\) is equal to:
[JEE Main 2021, 27 Jul (Shift 2)]
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Let \(A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)\) where \([t]\) denotes the greatest integer less than or equal to \(t\). If \(\operatorname{det}(A)=192\), then the set of values of \(x\) is the interval.
[JEE Main 2021, 27 Aug (Shift 2)]
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If \( a_{r}=\cos \frac{2 r \pi}{9}+i \sin \frac{2 r \pi}{9}, r=1,2,3, \ldots \ldots, i=\sqrt{-1} \),
then determinant \( \left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right| \) is equal to :
[JEE Main 2021, 31 Aug (Shift 1)]
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Let \( A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right] \).
If \( |\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{\mathrm{n}} \), then \( n \) is equal to
[JEE Main 2023, 08 Apr (Shift 1)]
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Let \(A=\left[\begin{array}{cc}m & n \\ p & q\end{array}\right], d=|A| \neq 0,|A-d(\operatorname{Adj} A)|=0\). Then
[JEE Main 2023, 30 Jan (Shift 1)]
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Let \( A=\left[\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right] \), where \( [t] \) denotes
GIF less than or equal to \( t \). If \( |A|=192 \) then \( x \) belongs to:
[JEE Main 2021, 27 Aug (Shift 2)]
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Let \( A \) be a \( 3 \times 3 \) matrix such that
\(∣adj(adj(adjA))|={12}^{4}\). Then \( \left|A^{-1} \operatorname{adj} A\right| \) is equal to
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