Matrices & Determinants · JEE Main previous year questions

2024 20 questions

1. Q172024 · 9 Apr 2024 · Shift 1

   Let $\lambda ,\mu \in \mathbf{R}$. If the system of equations $3x+5y+\lambda z=3$ $7x+11y-9z=2$ $97x+155y-189z=\mu$ has infinitely many solutions, then $\mu +2\lambda$ is equal to :

   1. A.24
   2. B.25
   3. C.27
   4. D.22
   Show answer

   Answer: B

2. Q202024 · 9 Apr 2024 · Shift 1

   Let $\overrightarrow{OA}=2\vec{a}$, $\overrightarrow{OB}=6\vec{a}+5\vec{b}$ and $\overrightarrow{OC}=3\vec{b}$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$ is $15\text{ sq. units}$, then the area (in sq. units) of the quadrilateral $OABC$ is equal to:

   1. A.32
   2. B.38
   3. C.35
   4. D.40
   Show answer

   Answer: C

3. Q292024 · 9 Apr 2024 · Shift 1· Numerical answer

   Let $A$ be a non-singular matrix of order 3. If $\det (3\mathrm{adj}(2\mathrm{adj}((\det A)A)))=3^{-13}\cdot 2^{-10}$ and $\det (3\mathrm{adj}(2A))=2^m\cdot 3^n$, then $|3m+2n|$ is equal to ______.

   Show answer

   Answer: 14

4. Q142024 · 9 Apr 2024 · Shift 2

   Let $B=\left[\begin{array}{cc}1& 3\\ 1& 5\end{array}\right]$ and $A$ be a $2\times 2$ matrix such that $A{B}^{-1}=A^{-1}$. If $BC{B}^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta -\alpha$ is equal to

   1. A.16
   2. B.10
   3. C.8
   4. D.2
   Show answer

   Answer: B

5. Q232024 · 9 Apr 2024 · Shift 2· Numerical answer

   Consider the matrices : $A=\left[\begin{array}{cc}2& -5\\ 3& m\end{array}\right],B=\left[\begin{array}{c}20\\ m\end{array}\right]$ and $X=\left[\begin{array}{c}x\\ y\end{array}\right]$. Let the set of all $m$, for which the system of equations $AX=B$ has a negative solution (i.e., $x<0$ and $y<0$), be the interval $(a,b)$. Then $8\underset{a}{\overset{b}{\int }}|A|dm$ is equal to ____.

   Show answer

   Answer: 450

6. Q12024 · 8 Apr 2024 · Shift 1

   Let $A=\left[\begin{array}{ccc}2& a& 0\\ 1& 3& 1\\ 0& 5& b\end{array}\right]$. If $A^3=4A^2-A-21I$, where $I$ is the identity matrix of order $3\times 3$, then $2a+3b$ is equal to

   1. A.$-10$
   2. B.$-12$
   3. C.$-13$
   4. D.$-9$
   Show answer

   Answer: C

7. Q62024 · 6 Apr 2024 · Shift 2

   If $A$ is a square matrix of order 3 such that $\det (A)=3$ and $\det \left(\mathrm{adj}\left(-4\mathrm{adj}\left(-3\mathrm{adj}\left(3\mathrm{adj}\left((2A{)}^{-1}\right)\right)\right)\right)\right)=2^m{3}^n$, then $m+2n$ is equal to :

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

   Answer: B

8. Q282024 · 6 Apr 2024 · Shift 2· Numerical answer

   If the system of equations $2x+7y+\lambda z=3$ $3x+2y+5z=4$ $x+\mu y+32z=-1$ has infinitely many solutions, then $(\lambda -\mu )$ is equal to ____ :

   Show answer

   Answer: 38

9. Q152024 · 5 Apr 2024 · Shift 2

   Let $\alpha \beta \ne 0$ and $A=\left[\begin{array}{ccc}\beta & \alpha & 3\\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{array}\right]$. If $B=\left[\begin{array}{ccc}3\alpha & -9& 3\alpha \\ -\alpha & 7& -2\alpha \\ -2\alpha & 5& -2\beta \end{array}\right]$ is the matrix of cofactors of the elements of $A$, then $\det (AB)$ is equal to :

   1. A.64
   2. B.343
   3. C.125
   4. D.216
   Show answer

   Answer: D

10. Q32024 · 4 Apr 2024 · Shift 1

    Let $\alpha \in (0,\infty )$ and $A=\left[\begin{array}{ccc}1& 2& \alpha \\ 1& 0& 1\\ 0& 1& 2\end{array}\right]$. If $\det \left(\mathrm{adj}\left(2A-A^T\right)\cdot \mathrm{adj}\left(A-2A^T\right)\right)=2^8$, then $(\det (A){)}^2$ is equal to:

    1. A.16
    2. B.36
    3. C.49
    4. D.1
    Show answer

    Answer: A

11. Q252024 · 4 Apr 2024 · Shift 1· Numerical answer

    Let $A$ be a $3\times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=3\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$. Then the maximum value of $\det (\mathrm{A})$ is ____.

    Show answer

    Answer: 27

12. Q192024 · 4 Apr 2024 · Shift 2

    Let $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ and $B=I+\mathrm{adj}(A)+(\mathrm{adj}A{)}^2+\dots +(\mathrm{adj}A{)}^{10}$. Then, the sum of all the elements of the matrix $B$ is:

    1. A.$-110$
    2. B.22
    3. C.$-124$
    4. D.$-88$
    Show answer

    Answer: D

13. Q232024 · 4 Apr 2024 · Shift 2· Numerical answer

    Let $A$ be a $2\times 2$ symmetric matrix such that $A\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 7\end{array}\right]$ and the determinant of $A$ be 1. If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2\times 2$, then $\alpha +\beta$ equals _________.

    Show answer

    Answer: 5

14. Q32024 · 31 Jan 2024 · Shift 2

    Let $A$ be a $3\times 3$ real matrix such that $A\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)=2\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right),A\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right)=4\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right),A\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=2\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$. Then, the system $(A-3I)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$ has :

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

    Answer: C

15. Q292024 · 31 Jan 2024 · Shift 2· Numerical answer

    Let A be a $3\times 3$ matrix and $\det (A)=2$. If $n=\det (\underset{2024-\text{ times}}{\underbrace{\mathrm{adj}(\mathrm{adj}(\dots ..(\mathrm{adj}A)))}})$, then the remainder when $n$ is divided by 9 is equal to ____.

    Show answer

    Answer: 7

16. Q142024 · 29 Jan 2024 · Shift 1

    Let A be a square matrix such that ${\text{AA}}^{\text{T}}=\text{I}$. Then $\frac{1}{2}\text{A}\left[(\text{A}+{\text{A}}^{\text{T}}{)}^2+(\text{A}-{\text{A}}^{\text{T}}{)}^2\right]$ is equal to

    1. A.${\text{A}}^2+{\text{A}}^{\text{T}}$
    2. B.${\text{A}}^3+\text{I}$
    3. C.${\text{A}}^3+{\text{A}}^{\text{T}}$
    4. D.${\text{A}}^2+\text{I}$
    Show answer

    Answer: C

17. Q142024 · 29 Jan 2024 · Shift 2

    Let $\overrightarrow{OA}=\vec{a},\overrightarrow{OB}=12\vec{a}+4\vec{b}$ and $\overrightarrow{OC}=\vec{b}$, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $\frac{\text{area of the quadrilateral OA BC}}{\text{area of S}}$ is equal to _________.

    1. A.7
    2. B.6
    3. C.8
    4. D.10
    Show answer

    Answer: C

18. Q112024 · 27 Jan 2024 · Shift 1

    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 :

    1. A.Statement I is false but Statement II is true
    2. B.Both Statement I and Statement II are false
    3. C.Both Statement I and Statement II are true
    4. D.Statement I is true but Statement II is false
    Show answer

    Answer: C

19. Q272024 · 27 Jan 2024 · Shift 1· Numerical answer

    Let $A=\left[\begin{array}{ccc}2& 0& 1\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]$, $B=[B_1,B_2,B_3]$, where $B_1,B_2,B_3$ are column matrices, and $${\text{AB}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],{\text{AB}}_2=\left[\begin{array}{c}2\\ 3\\ 0\end{array}\right],{\text{AB}}_3=\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]$$ If $\alpha =|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then ${\alpha }^3+{\beta }^3$ is equal to ________.

    Show answer

    Answer: 28

20. Q272024 · 27 Jan 2024 · Shift 2· Numerical answer

    Let $A$ be a $2\times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $|A-xI|=0$ be $-1$ and $3$, then the sum of the diagonal elements of the matrix $A^2$ is _______.

    Show answer

    Answer: 10