Matrices & Determinants · JEE Main previous year questions

2025 16 questions

1. Q132025 · 8 Apr 2025 · Shift 2

   Let $A=\left[\begin{array}{ccc}2& 2+p& 2+p+q\\ 4& 6+2p& 8+3p+2q\\ 6& 12+3p& 20+6p+3q\end{array}\right]$. If $\det (\mathrm{adj}(\mathrm{adj}(3A)))=2^m\cdot 3^n,m,n\in \mathbb{N}$, then $m+n$ is equal to

   1. A.$22$
   2. B.$20$
   3. C.$24$
   4. D.$26$
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   Answer: C

2. Q232025 · 4 Apr 2025 · Shift 1· Numerical answer

   Let $A=\left[\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right]$. If for some $\theta \in (0,\pi )$, $A^2=A^T$, then the sum of the diagonal elements of the matrix $(A+I{)}^3+(A-I{)}^3-6A$ is equal to _________ .

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   Answer: 6

3. Q62025 · 4 Apr 2025 · Shift 2

   Let the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $n⩾3$. Then the sum of all the elements of ${\mathrm{A}}^{50}$ is :

   1. A.44
   2. B.39
   3. C.52
   4. D.53
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   Answer: D

4. Q202025 · 3 Apr 2025 · Shift 1

   Let $A$ be a matrix of order $3\times 3$ and $|A|=5$. If $|2\mathrm{adj}(3A\mathrm{adj}(2A))|=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma }$, $\alpha ,\beta ,\gamma \in N$, then $\alpha +\beta +\gamma$ is equal to

   1. A.26
   2. B.27
   3. C.25
   4. D.28
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   Answer: B

5. Q252025 · 3 Apr 2025 · Shift 2· Numerical answer

   Let $I$ be the identity matrix of order $3\times 3$ and for the matrix $A=\left[\begin{array}{ccc}\lambda & 2& 3\\ 4& 5& 6\\ 7& -1& 2\end{array}\right]$, $|A|=-1$. Let $B$ be the inverse of the matrix $\mathrm{adj}\left(A\mathrm{adj}\left(A^2\right)\right)$. Then $|(\lambda B+I)|$ is equal to_____

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   Answer: 38 or -38

6. Q12025 · 2 Apr 2025 · Shift 1

   Let $A=\left[\begin{array}{cc}\alpha & -1\\ 6& \beta \end{array}\right],\alpha >0$, such that $\det (A)=0$ and $\alpha +\beta =1$. If $I$ denotes $2\times 2$ identity matrix, then the matrix $(I+A{)}^8$ is :

   1. A.$\left[\begin{array}{cc}257& -64\\ 514& -127\end{array}\right]$
   2. B.$\left[\begin{array}{cc}766& -255\\ 1530& -509\end{array}\right]$
   3. C.$\left[\begin{array}{cc}1025& -511\\ 2024& -1024\end{array}\right]$
   4. D.$\left[\begin{array}{cc}4& -1\\ 6& -1\end{array}\right]$
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   Answer: B

7. Q32025 · 2 Apr 2025 · Shift 1

   Let $a\in R$ and $A$ be a matrix of order $3\times 3$ such that $\det (A)=-4$ and $A+I=\left[\begin{array}{ccc}1& a& 1\\ 2& 1& 0\\ a& 1& 2\end{array}\right]$, where $I$ is the identity matrix of order $3\times 3$. If $\det ((a+1)\mathrm{adj}((a-1)A))$ is $2^m{3}^n,\mathrm{m},\mathrm{n}\in \{0,1,2,\dots ,20\}$, then $\mathrm{m}+\mathrm{n}$ is equal to :

   1. A.14
   2. B.17
   3. C.15
   4. D.16
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   Answer: D

8. Q62025 · 2 Apr 2025 · Shift 2

   Let $A$ be a $3\times 3$ real matrix such that $A^2(A-2I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha ,\beta$, and $\gamma$ are real constants, then $\alpha +\beta +\gamma$ is equal to :

   1. A.76
   2. B.12
   3. C.4
   4. D.20
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   Answer: B

9. Q22025 · 29 Jan 2025 · Shift 2

   Let $A=[a_{ij}]$ be a $2\times 2$ matrix such that $a_{ij}\in \{0,1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:

   1. A.$\frac{5}{8}$
   2. B.$\frac{1}{4}$
   3. C.$\frac{3}{4}$
   4. D.$\frac{3}{8}$
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   Answer: D

10. Q142025 · 29 Jan 2025 · Shift 2

    Let $A=[a_{ij}]$ be a matrix of order $3\times 3$, with $a_{ij}=(\sqrt{2}{)}^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha +\beta \sqrt{2}$, $\alpha ,\beta \in \mathbf{Z}$, then $\alpha +\beta$ is equal to :

    1. A.$210$
    2. B.$280$
    3. C.$224$
    4. D.$168$
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    Answer: C

11. Q212025 · 28 Jan 2025 · Shift 1· Numerical answer

    Let M denote the set of all real matrices of order $3\times 3$ and let $\mathrm{S}=\{-3,-2,-1,1,2\}$. Let ${\mathrm{S}}_1=\{\mathrm{A}=[a_{ij}]\in \mathrm{M}:\mathrm{A}={\mathrm{A}}^{\mathrm{T}}\text{ and }{a}_{ij}\in \mathrm{S},\forall i,j\}$, ${\mathrm{S}}_2=\{\mathrm{A}=[a_{ij}]\in \mathrm{M}:\mathrm{A}=-{\mathrm{A}}^{\mathrm{T}}\text{ and }{a}_{ij}\in \mathrm{S},\forall i,j\}$, ${\mathrm{S}}_3=\{\mathrm{A}=[a_{ij}]\in \mathrm{M}:a_{11}+a_{22}+a_{33}=0\text{ and }{a}_{ij}\in \mathrm{S},\forall i,j\}$. If $n\left({\mathrm{S}}_1\cup {\mathrm{S}}_2\cup {\mathrm{S}}_3\right)=125\alpha$, then $\alpha$ equals ____ .

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    Answer: 1613

12. Q212025 · 24 Jan 2025 · Shift 1· Numerical answer

    Let A be a $3\times 3$ matrix such that ${\mathrm{X}}^{\mathrm{T}}\mathrm{AX}=\mathrm{O}$ for all nonzero $3\times 1$ matrices $\mathrm{X}=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$. If $\mathrm{A}\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 4\\ -5\end{array}\right],\mathrm{A}\left[\begin{array}{l}1\\ 2\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 4\\ -8\end{array}\right]$, and $\det (\mathrm{adj}(2(\mathrm{A}+\mathrm{I})))=2^{\alpha }{3}^{\beta }{5}^{\gamma },\alpha ,\beta ,\gamma \in N$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is

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    Answer: 44

13. Q22025 · 24 Jan 2025 · Shift 2

    Let $\mathrm{A}=[{\mathrm{a}}_{\mathrm{ij}}]$ be a square matrix of order $2$ with entries either $0$ or $1$ . Let $\mathrm{E}$ be the event that $\mathrm{A}$ is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is :

    1. A.$\frac{3}{8}$
    2. B.$\frac{1}{8}$
    3. C.$\frac{3}{16}$
    4. D.$\frac{5}{8}$
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    Answer: A

14. Q82025 · 24 Jan 2025 · Shift 2

    If the system of equations $x+2y-3z=2$ $2x+\lambda y+5z=5$ $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda +\mu$ is equal to :

    1. A.13
    2. B.10
    3. C.12
    4. D.11
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    Answer: C

15. Q132025 · 23 Jan 2025 · Shift 2

    Let $A=[a_{ij}]$ be a $3\times 3$ matrix such that $A\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$, $A\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$ and $A\left[\begin{array}{c}2\\ 1\\ 2\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, then $a_{23}$ equals :

    1. A.2
    2. B.$-1$
    3. C.1
    4. D.0
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    Answer: B

16. Q252025 · 22 Jan 2025 · Shift 1· Numerical answer

    Let $A$ be a square matrix of order 3 such that $\det (A)=-2$ and $\det (3\mathrm{adj}(-6\mathrm{adj}(3A)))=2^{m+n}\cdot 3^{mn},m>n$. Then $4m+2n$ is equal to _________.

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    Answer: 34