Matrices & Determinants · JEE Main previous year questions

2022 22 questions

1. Q32022 · 29 Jul 2022 · Shift 1

   Let $\mathrm{A}$ and $\mathrm{B}$ be two $3\times 3$ non-zero real matrices such that $\mathrm{AB}$ is a zero matrix. Then

   1. A.the system of linear equations $AX=0$ has a unique solution
   2. B.the system of linear equations $AX=0$ has infinitely many solutions
   3. C.B is an invertible matrix
   4. D.$\mathrm{adj}(\mathrm{A})$ is an invertible matrix
   Show answer

   Answer: B

2. Q242022 · 29 Jul 2022 · Shift 1· Numerical answer

   The number of matrices of order $3\times 3$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is ____.

   Show answer

   Answer: 282

3. Q22022 · 29 Jul 2022 · Shift 2

   Which of the following matrices can NOT be obtained from the matrix $\left[\begin{array}{cc}-1& 2\\ 1& -1\end{array}\right]$ by a single elementary row operation ?

   1. A.$\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right]$
   2. B.$\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$
   3. C.$\left[\begin{array}{cc}-1& 2\\ -2& 7\end{array}\right]$
   4. D.$\left[\begin{array}{cc}-1& 2\\ -1& 3\end{array}\right]$
   Show answer

   Answer: C

4. Q182022 · 28 Jul 2022 · Shift 1

   Let the matrix $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_n=\mathrm{Adj}(B_{n-1})$ for all $n\ge 1$, then $\det (B_4)$ is equal to :

   1. A.$3^{28}$
   2. B.$3^{30}$
   3. C.$3^{32}$
   4. D.$3^{36}$
   Show answer

   Answer: C

5. Q32022 · 28 Jul 2022 · Shift 2

   Let $A$ and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

   1. A.$A^4-B^4$ is a symmetric matrix
   2. B.$AB-BA$ is a symmetric matrix
   3. C.$B^5-A^5$ is a skew-symmetric matrix
   4. D.$AB+BA$ is a skew-symmetric matrix
   Show answer

   Answer: C

6. Q262022 · 27 Jul 2022 · Shift 1· Numerical answer

   Let $S$ be the set containing all $3\times 3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is 6 is _________.

   Show answer

   Answer: 5376

7. Q132022 · 27 Jul 2022 · Shift 2

   Let $A=\left(\begin{array}{cc}4& -2\\ \alpha & \beta \end{array}\right)$. If ${\mathrm{A}}^2+\gamma \mathrm{A}+18\mathrm{I}=\mathrm{O}$, then $\det (\mathrm{A})$ is equal to ____.

   1. A.$-18$
   2. B.$18$
   3. C.$-50$
   4. D.$50$
   Show answer

   Answer: B

8. Q282022 · 27 Jul 2022 · Shift 2· Numerical answer

   Consider a matrix $A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If $\frac{\det (\mathrm{adj}(\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))))}{(\alpha -\beta {)}^{16}(\beta -\gamma {)}^{16}(\gamma -\alpha {)}^{16}}=2^{32}\times 3^{16}$, then the number of such 3 - tuples $(\alpha ,\beta ,\gamma )$ is ____.

   Show answer

   Answer: 42

9. Q152022 · 26 Jul 2022 · Shift 1

   Let A be a $2\times 2$ matrix with $\det (\mathrm{A})=-1$ and $\det ((\mathrm{A}+\mathrm{I})(\mathrm{Adj}(\mathrm{A})+\mathrm{I}))=4$. Then the sum of the diagonal elements of A can be :

   1. A.$-1$
   2. B.2
   3. C.1
   4. D.$-\sqrt{2}$
   Show answer

   Answer: B

10. Q42022 · 30 Jun 2022 · Shift 1

    Let $A=\left[\begin{array}{ccc}1& -2& \alpha \\ \alpha & 2& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& \alpha \\ -1& 2\\ 4& -5\end{array}\right]$, $\alpha \in C$. Then the absolute value of the sum of all values of $\alpha$ for which $\det (AB)=0$ is :

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

    Answer: A

11. Q122022 · 30 Jun 2022 · Shift 1

    Let A and B be two square matrices of order 2. If $\det (A)=2$, $\det (B)=3$ and $\det \left(\left(\det (5(\det A)B)\right)A^2\right)=2^a{3}^b{5}^c$ for some a, b, c, $\in \mathrm{N}$, then $a+b+c$ is equal to :

    1. A.10
    2. B.12
    3. C.13
    4. D.14
    Show answer

    Answer: B

12. Q262022 · 28 Jun 2022 · Shift 2· Numerical answer

    If the system of linear equations $2x-3y=\gamma +5$, $\alpha x+5y=\beta +1$, where $\alpha ,\beta ,\gamma \in \mathbb{R}$ has infinitely many solutions then the value of $|9\alpha +3\beta +5\gamma |$ is equal to ________.

    Show answer

    Answer: 58

13. Q22022 · 27 Jun 2022 · Shift 1

    Let the system of linear equations $$\begin{array}{l}x+2y+z=2,\\ \alpha x+3y-z=\alpha ,\\ -\alpha x+y+2z=-\alpha \end{array}$$ be inconsistent. Then $\alpha$ is equal to :

    1. A.$\frac{5}{2}$
    2. B.$-\frac{5}{2}$
    3. C.$\frac{7}{2}$
    4. D.$-\frac{7}{2}$
    Show answer

    Answer: D

14. Q32022 · 27 Jun 2022 · Shift 1

    $x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1,abc\ne 0$, then :

    1. A.$x,y,z$ are in A.P.
    2. B.$x,y,z$ are in G.P.
    3. C.$\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in A.P.
    4. D.$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1-(a+b+c)$
    Show answer

    Answer: C

15. Q272022 · 27 Jun 2022 · Shift 1· Numerical answer

    The positive value of the determinant of the matrix A, whose $$\mathrm{Adj}(\mathrm{Adj}(\mathrm{A}))=\left(\begin{array}{ccc}14& 28& -14\\ -14& 14& 28\\ 28& -14& 14\end{array}\right),$$ is ____.

    Show answer

    Answer: 14

16. Q172022 · 27 Jun 2022 · Shift 2

    Let $A$ and $B$ be two $3\times 3$ matrices such that $AB=I$ and $|A|=\frac{1}{8}$. Then $|adj(Badj(2A))|$ is equal to

    1. A.$16$
    2. B.$32$
    3. C.$64$
    4. D.$128$
    Show answer

    Answer: C

17. Q232022 · 27 Jun 2022 · Shift 2· Numerical answer

    Let A be a matrix of order $2\times 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of A is a prime number $p$, $2<p<8$, then the number of such matrices A is ________.

    Show answer

    Answer: 180

18. Q132022 · 26 Jun 2022 · Shift 1

    Let A be a $3\times 3$ invertible matrix. If $|\mathrm{adj}(24A)|=|\mathrm{adj}(3\mathrm{adj}(2A))|$, then $|A{|}^2$ is equal to :

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

    Answer: C

19. Q32022 · 25 Jun 2022 · Shift 1

    Let A be a $3\times 3$ real matrix such that $A\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$; $A\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right)$ and $A\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 1\\ 2\end{array}\right)$. If $X=(x_1,x_2,x_3{)}^T$ and I is an identity matrix of order 3, then the system $(A-2I)X=\left(\begin{array}{c}4\\ 1\\ 1\end{array}\right)$ has :

    1. A.no solution
    2. B.infinitely many solutions
    3. C.unique solution
    4. D.exactly two solutions
    Show answer

    Answer: B

20. Q82022 · 25 Jun 2022 · Shift 1

    Let $A=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$. If $\mathrm{M}$ and $\mathrm{N}$ are two matrices given by $\mathrm{M}=\sum _{k=1}^{10}{A}^{2k}$ and $\mathrm{N}=\sum _{k=1}^{10}{A}^{2k-1}$ then ${\mathrm{MN}}^2$ is :

    1. A.a non-identity symmetric matrix
    2. B.a skew-symmetric matrix
    3. C.neither symmetric nor skew-symmetric matrix
    4. D.an identity matrix
    Show answer

    Answer: A

21. Q252022 · 25 Jun 2022 · Shift 1· Numerical answer

    Let $\mathrm{A}$ be a $3\times 3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices $\mathrm{A}$ having sum of all the entries equal to $5$, is $\underline{}$.

    Show answer

    Answer: 414

22. Q42022 · 25 Jun 2022 · Shift 2

    The system of equations $-kx+3y-14z=25$ $-15x+4y-kz=3$ $-4x+y+3z=4$ is consistent for all k in the set

    1. A.$\mathbb{R}$
    2. B.$\mathbb{R}-\{-11,13\}$
    3. C.$\mathbb{R}-\{13\}$
    4. D.$\mathbb{R}-\{-11,11\}$
    Show answer

    Answer: D