Matrices & Determinants · JEE Main previous year questions

2023 31 questions

1. Q42023 · 13 Apr 2023 · Shift 1

   For the system of linear equations $$2x+4y+2az=b$$ $$x+2y+3z=4$$ $$2x-5y+2z=8$$ which of the following is NOT correct?

   1. A.It has infinitely many solutions if $a=3,b=8$
   2. B.It has infinitely many solutions if $a=3,b=6$
   3. C.It has unique solution if $a=b=8$
   4. D.It has unique solution if $a=b=6$
   Show answer

   Answer: B

2. Q102023 · 13 Apr 2023 · Shift 1

   Let $B=\left[\begin{array}{ccc}1& 3& \alpha \\ 1& 2& 3\\ \alpha & \alpha & 4\end{array}\right]$, $\alpha >2$ be the adjoint of a matrix $A$ and $|A|=2$. Then $[\begin{array}{ccc}\alpha & -2\alpha & \alpha \end{array}]B\left[\begin{array}{c}\alpha \\ -2\alpha \\ \alpha \end{array}\right]$ is equal to :

   1. A.$32$
   2. B.$-16$
   3. C.$0$
   4. D.$16$
   Show answer

   Answer: B

3. Q152023 · 13 Apr 2023 · Shift 1

   The number of symmetric matrices of order 3, with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$ is :

   1. A.$10^9$
   2. B.$9^{10}$
   3. C.$10^6$
   4. D.$6^{10}$
   Show answer

   Answer: C

4. Q172023 · 13 Apr 2023 · Shift 2

   Let for $A=\left[\begin{array}{ccc}1& 2& 3\\ \alpha & 3& 1\\ 1& 1& 2\end{array}\right]$, $|A|=2$. If $|2\mathrm{adj}(2\mathrm{adj}(2\mathrm{A}))|=32^n$, then $3n+\alpha$ is equal to

   1. A.$11$
   2. B.$9$
   3. C.$12$
   4. D.$10$
   Show answer

   Answer: A

5. Q22023 · 12 Apr 2023 · Shift 1

   Let $A=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right]$. If $B=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]A\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$, then the sum of all the elements of the matrix ${\sum }_{n=1}^{50}{B}^n$ is equal to

   1. A.$50$
   2. B.$75$
   3. C.$100$
   4. D.$125$
   Show answer

   Answer: C

6. Q22023 · 11 Apr 2023 · Shift 1

   Let $S=\{M=[a_{ij}],a_{ij}\in \{0,1,2\},1\le i,j\le 2\}$ be a sample space and $A=\{M\in S:M\text{ is invertible}\}$ be an event. Then $P(A)$ is equal to :

   1. A.$\frac{47}{81}$
   2. B.$\frac{49}{81}$
   3. C.$\frac{50}{81}$
   4. D.$\frac{16}{27}$
   Show answer

   Answer: C

7. Q92023 · 11 Apr 2023 · Shift 1

   Let A be a $2\times 2$ matrix with real entries such that $A^{\prime }=\alpha A+\text{I}$, where $\alpha \in \mathbb{R}-\{-1,1\}$. If $\det (A^2-A)=4$, then the sum of all possible values of $\alpha$ is equal to :

   1. A.2
   2. B.$\frac{3}{2}$
   3. C.0
   4. D.$\frac{5}{2}$
   Show answer

   Answer: D

8. Q32023 · 11 Apr 2023 · Shift 2

   If the system of linear equations $$7x+11y+\alpha z=13$$ $$5x+4y+7z=\beta$$ $$175x+194y+57z=361$$ has infinitely many solutions, then $\alpha +\beta +2$ is equal to :

   1. A.$6$
   2. B.$4$
   3. C.$5$
   4. D.$3$
   Show answer

   Answer: B

9. Q202023 · 10 Apr 2023 · Shift 1

   If A is a $3\times 3$ matrix and $|A|=2$, then $|3\mathrm{adj}(|3A|A^2)|$ is equal to :

   1. A.$3^{12}\cdot 6^{10}$
   2. B.$3^{11}\cdot 6^{10}$
   3. C.$3^{12}\cdot 6^{11}$
   4. D.$3^{10}\cdot 6^{11}$
   Show answer

   Answer: B

10. Q202023 · 10 Apr 2023 · Shift 2

    If $A=\frac{1}{5!6!7!}\left[\begin{array}{ccc}5!& 6!& 7!\\ 6!& 7!& 8!\\ 7!& 8!& 9!\end{array}\right]$, then $|\mathrm{adj}(\mathrm{adj}(2A))|$ is equal to :

    1. A.$2^{12}$
    2. B.$2^{20}$
    3. C.$2^8$
    4. D.$2^{16}$
    Show answer

    Answer: D

11. Q262023 · 10 Apr 2023 · Shift 2· Numerical answer

    Let the equations of two adjacent sides of a parallelogram ABCD be $2x-3y=-23$ and $5x+4y=23$. If the equation of its one diagonal AC is $3x+7y=23$ and the distance of A from the other diagonal is d, then $50d^2$ is equal to _________.

    Show answer

    Answer: 529

12. Q32023 · 8 Apr 2023 · Shift 1

    Let $A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 2& -1\\ 0& -1& 2\end{array}\right]$. If $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}2A))|=(16{)}^n$, then $n$ is equal to :

    1. A.$9$
    2. B.$8$
    3. C.$10$
    4. D.$12$
    Show answer

    Answer: C

13. Q22023 · 6 Apr 2023 · Shift 1

    If the system of equations $$x+y+az=b$$ $$2x+5y+2z=6$$ $$x+2y+3z=3$$ has infinitely many solutions, then $2a+3b$ is equal to :

    1. A.28
    2. B.25
    3. C.20
    4. D.23
    Show answer

    Answer: D

14. Q62023 · 6 Apr 2023 · Shift 1

    Let $\text{A}=[{\text{a}}_{\text{ij}}{]}_{2\times 2}$, where ${\text{a}}_{\text{ij}}\ne 0$ for all i, j and ${\text{A}}^2=\text{I}$. Let a be the sum of all diagonal elements of A and $\text{b}=|\text{A}|$. Then $3a^2+4b^2$ is equal to :

    1. A.4
    2. B.3
    3. C.14
    4. D.7
    Show answer

    Answer: A

15. Q52023 · 6 Apr 2023 · Shift 2

    Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $P^{\alpha }+P^{\beta }=\gamma I-29P$ and $P^{\alpha }-P^{\beta }=\delta I-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to :

    1. A.18
    2. B.22
    3. C.24
    4. D.40
    Show answer

    Answer: C

16. Q52023 · 31 Jan 2023 · Shift 1

    For the system of linear equations $x+y+z=6$ $\alpha x+\beta y+7z=3$ $x+2y+3z=14$ which of the following is NOT true ?

    1. A.If $\alpha =\beta =7$, then the system has no solution
    2. B.For every point $(\alpha ,\beta )\ne (7,7)$ on the line $x-2y+7=0$, the system has infinitely many solutions
    3. C.There is a unique point $(\alpha ,\beta )$ on the line $x+2y+18=0$ for which the system has infinitely many solutions
    4. D.If $\alpha =\beta$ and $\alpha \ne 7$, then the system has a unique solution
    Show answer

    Answer: B

17. Q122023 · 31 Jan 2023 · Shift 1

    Let $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right)$. Then the sum of the diagonal elements of the matrix $(A+I{)}^{11}$ is equal to :

    1. A.4094
    2. B.2050
    3. C.6144
    4. D.4097
    Show answer

    Answer: D

18. Q232023 · 31 Jan 2023 · Shift 2· Numerical answer

    Let $\mathrm{A}=[a_{ij}],a_{ij}\in \mathbb{Z}\cap [0,4],1\le i,j\le 2$. The number of matrices A such that the sum of all entries is a prime number $p\in (2,13)$ is ___________.

    Show answer

    Answer: 204

19. Q282023 · 31 Jan 2023 · Shift 2· Numerical answer

    Let A be a $n\times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix $\mathrm{Adj}\left(2\cdot \mathrm{Adj}\left(2{\mathrm{A}}^{-1}\right)\right)$ is $2^{84}$, then n is equal to :

    Show answer

    Answer: 5

20. Q172023 · 30 Jan 2023 · Shift 1

    Let $A=\left(\begin{array}{cc}m& n\\ p& q\end{array}\right)$, $d=|A|\ne 0$ and $|A-d(\mathrm{AdjA})|=0$. Then

    1. A.$1+d^2=m^2+q^2$
    2. B.$1+d^2=(m+q{)}^2$
    3. C.$(1+d{)}^2=m^2+q^2$
    4. D.$(1+d{)}^2=(m+q{)}^2$
    Show answer

    Answer: D

21. Q82023 · 29 Jan 2023 · Shift 1

    Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix A such that $A^2=3A+\alpha I$. If $A^4=21A+\beta I$, then

    1. A.$\alpha =1$
    2. B.$\alpha =4$
    3. C.$\beta =8$
    4. D.$\beta =-8$
    Show answer

    Answer: D

22. Q202023 · 29 Jan 2023 · Shift 2

    The set of all values of $\mathrm{t}\in \mathbb{R}$, for which the matrix $$\left[\begin{array}{ccc}{e}^t& e^{-t}(\sin t-2\cos t)& e^{-t}(-2\sin t-\cos t)\\ e^t& e^{-t}(2\sin t+\cos t)& e^{-t}(\sin t-2\cos t)\\ e^t& e^{-t}\cos t& e^{-t}\sin t\end{array}\right]$$ is invertible, is :

    1. A.$\{k\pi ,k\in \mathbb{Z}\}$
    2. B.$\mathbb{R}$
    3. C.$\left\{(2k+1)\frac{\pi }{2},k\in \mathbb{Z}\right\}$
    4. D.$\left\{k\pi +\frac{\pi }{4},k\in \mathbb{Z}\right\}$
    Show answer

    Answer: B

23. Q262023 · 29 Jan 2023 · Shift 2· Numerical answer

    Let A be a symmetric matrix such that $|\mathrm{A}|=2$ and $\left[\begin{array}{cc}2& 1\\ 3& \frac{3}{2}\end{array}\right]A=\left[\begin{array}{cc}1& 2\\ \alpha & \beta \end{array}\right]$. If the sum of the diagonal elements of A is $s$, then $\frac{\beta s}{{\alpha }^2}$ is equal to ____.

    Show answer

    Answer: 5

24. Q182023 · 25 Jan 2023 · Shift 1

    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 $|\mathrm{adj}(\mathrm{adj}{A}^2)|$ is equal to

    1. A.$6^4$
    2. B.$2^8$
    3. C.$4^8$
    4. D.$2^4$
    Show answer

    Answer: B

25. Q62023 · 25 Jan 2023 · Shift 2

    Let A, B, C be $3\times 3$ matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements (S1) ${\mathrm{A}}^{13}{\mathrm{B}}^{26}-{\mathrm{B}}^{26}{\mathrm{A}}^{13}$ is symmetric (S2) ${\mathrm{A}}^{26}{\mathrm{C}}^{13}-{\mathrm{C}}^{13}{\mathrm{A}}^{26}$ is symmetric Then,

    1. A.Only S2 is true
    2. B.Only S1 is true
    3. C.Both S1 and S2 are false
    4. D.Both S1 and S2 are true
    Show answer

    Answer: A