Matrices & Determinants · JEE Main previous year questions

2021 24 questions

1. Q32021 · 1 Sept 2021 · Shift 2

   Consider the system of linear equations $-x+y+2z=0$ $3x-ay+5z=1$ $2x-2y-az=7$ Let $S_1$ be the set of all $a\in \mathbb{R}$ for which the system is inconsistent and $S_2$ be the set of all $a\in \mathbb{R}$ for which the system has infinitely many solutions. If $n(S_1)$ and $n(S_2)$ denote the number of elements in $S_1$ and $S_2$ respectively, then

   1. A.$n(S_1)=2$, $n(S_2)=2$
   2. B.$n(S_1)=1$, $n(S_2)=0$
   3. C.$n(S_1)=2$, $n(S_2)=0$
   4. D.$n(S_1)=0$, $n(S_2)=2$
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   Answer: C

2. Q72021 · 1 Sept 2021 · Shift 2

   Let $J_{n,m}={\int }_0^{\frac{1}{2}}\frac{x^n}{x^m-1}dx$, $\forall n>m$ and $n,m\in \mathbb{N}$. Consider a matrix $A=[a_{ij}{]}_{3\times 3}$ where $a_{ij}=\{\begin{array}{ll}{J}_{6+i,3}-J_{i+3,3},& i\le j\\ 0,& i>j\end{array}$. Then $|\mathrm{adj}{A}^{-1}|$ is :

   1. A.$(15{)}^2\times 2^{42}$
   2. B.$(15{)}^2\times 2^{34}$
   3. C.$(105{)}^2\times 2^{38}$
   4. D.$(105{)}^2\times 2^{36}$
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   Answer: C

3. Q302021 · 26 Aug 2021 · Shift 2· Numerical answer

   Let A be a $3\times 3$ real matrix. If $\det (2\mathrm{Adj}(2\mathrm{Adj}(\mathrm{Adj}(2\mathrm{A}))))=2^{41}$, then the value of $\det ({\mathrm{A}}^2)$ is equal to ________.

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   Answer: 4

4. Q202021 · 27 Jul 2021 · Shift 2

   Let A and B be two $3\times 3$ real matrices such that $(A^2-B^2)$ 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 :

   1. A.2
   2. B.4
   3. C.1
   4. D.0
   Show answer

   Answer: D

5. Q272021 · 27 Jul 2021 · Shift 2· Numerical answer

   If $A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$ and $M=A+A^2+A^3+\dots +A^{20}$, then the sum of all the elements of the matrix M is equal to ____.

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   Answer: 2020

6. Q112021 · 25 Jul 2021 · Shift 1

   The values of a and b, for which the system of equations 2x + 3y + 6z = 8 x + 2y + az = 5 3x + 5y + 9z = b has no solution, are :

   1. A.a = 3, $b\ne 13$
   2. B.$a\ne 3,b\ne 13$
   3. C.$a\ne 3$, b = 3
   4. D.a = 3, b = 13
   Show answer

   Answer: A

7. Q42021 · 22 Jul 2021 · Shift 2

   Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times 2$ matrices. The probability that such formed matrix have all different entries and are non-singular, is :

   1. A.$\frac{45}{162}$
   2. B.$\frac{21}{81}$
   3. C.$\frac{22}{81}$
   4. D.$\frac{43}{162}$
   Show answer

   Answer: D

8. Q82021 · 22 Jul 2021 · Shift 2

   Let $A=[a_{ij}]$ be a real matrix of order $3\times 3$, such that $a_{i1}+a_{i2}+a_{i3}=1$, for $i=1,2,3$. Then, the sum of all the entries of the matrix $A^3$ is equal to :

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

   Answer: C

9. Q232021 · 22 Jul 2021 · Shift 2· Numerical answer

   Let $A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$. Then the number of $3\times 3$ matrices B with entries from the set $\{1,2,3,4,5\}$ and satisfying $\text{AB}=\text{BA}$ is ________.

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   Answer: 3125

10. Q52021 · 20 Jul 2021 · Shift 1

    Let $A=\left[\begin{array}{cc}2& 3\\ a& 0\end{array}\right],a\in \mathrm{R}$ be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If $\det (Q)=9$, then the modulus of the sum of all possible values of determinant of P is equal to :

    1. A.36
    2. B.24
    3. C.45
    4. D.18
    Show answer

    Answer: A

11. Q92021 · 20 Jul 2021 · Shift 1

    Let $A=[a_{ij}]$ be a $3\times 3$ matrix, where $a_{ij}=\{\begin{array}{llc}1& ,& ifi=j\\ -x& ,& if|i-j|=1\\ 2x+1& ,& otherwise.\end{array}$ Let a function $f:\mathrm{R}\to \mathrm{R}$ be defined as $f(x)=\det (A)$. Then the sum of maximum and minimum values of f on R is equal to:

    1. A.$-\frac{20}{27}$
    2. B.$\frac{88}{27}$
    3. C.$\frac{20}{27}$
    4. D.$-\frac{88}{27}$
    Show answer

    Answer: D

12. Q222021 · 20 Jul 2021 · Shift 1· Numerical answer

    Let $A=\left(\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\right)$ and $B=7A^{20}-20A^7+2I$, where I is an identity matrix of order $3\times 3$. If $B=[b_{ij}]$, then $b_{13}$ is equal to ________.

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    Answer: 910

13. Q132021 · 18 Mar 2021 · Shift 1

    Let $A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$ and $2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$. If $\mathrm{Tr}(A)$ denotes the sum of all diagonal elements of the matrix A, then $\mathrm{Tr}(A)-\mathrm{Tr}(B)$ has value equal to

    1. A.1
    2. B.2
    3. C.0
    4. D.3
    Show answer

    Answer: B

14. Q112021 · 18 Mar 2021 · Shift 2

    Define a relation R over a class of $n\times n$ real matrices A and B as "ARB iff there exists a non-singular matrix P such that $\mathrm{PA}{\mathrm{P}}^{-1}=\mathrm{B}$". Then which of the following is true?

    1. A.R is reflexive, transitive but not symmetric
    2. B.R is symmetric, transitive but not reflexive.
    3. C.R is reflexive, symmetric but not transitive
    4. D.R is an equivalence relation
    Show answer

    Answer: D

15. Q252021 · 18 Mar 2021 · Shift 2· Numerical answer

    Let I be an identity matrix of order $2\times 2$ and $\mathrm{P}=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$. Then the value of $n\in \mathrm{N}$ for which ${\mathrm{P}}^n=5\mathrm{I}-8\mathrm{P}$ is equal to ____.

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    Answer: 6

16. Q62021 · 16 Mar 2021 · Shift 1

    Let $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right],i=\sqrt{-1}$. Then, the system of linear equations $A^8\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$ has :

    1. A.Exactly two solutions
    2. B.Infinitely many solutions
    3. C.A unique solution
    4. D.No solution
    Show answer

    Answer: D

17. Q272021 · 16 Mar 2021 · Shift 1· Numerical answer

    The total number of $3\times 3$ matrices A having entries from the set $\{0,1,2,3\}$ such that the sum of all the diagonal entries of $A{A}^{\mathrm{T}}$ is 9, is equal to _________.

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    Answer: 766

18. Q302021 · 16 Mar 2021 · Shift 1· Numerical answer

    Let $P=\left[\begin{array}{ccc}-30& 20& 56\\ 90& 140& 112\\ 120& 60& 14\end{array}\right]$ and $A=\left[\begin{array}{ccc}2& 7& {\omega }^2\\ -1& -\omega & 1\\ 0& -\omega & -\omega +1\end{array}\right]$ where $\omega =\frac{-1+i\sqrt{3}}{2}$, and $I_3$ be the identity matrix of order 3. If the determinant of the matrix $(P^{-1}AP-I_3{)}^2$ is $\alpha {\omega }^2$, then the value of $\alpha$ is equal to _________.

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    Answer: 36

19. Q32021 · 26 Feb 2021 · Shift 1

    Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^2$ is 1, then the possible number of such matrices is :

    1. A.6
    2. B.4
    3. C.1
    4. D.12
    Show answer

    Answer: B

20. Q32021 · 26 Feb 2021 · Shift 2

    Consider the following system of equations : $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$, where a, b and c are real constants. Then the system of equations :

    1. A.has no solution for all a, b and c
    2. B.has a unique solution when $5a=2b+c$
    3. C.has infinite number of solutions when $5a=2b+c$
    4. D.has a unique solution for all a, b and c
    Show answer

    Answer: C

21. Q222021 · 26 Feb 2021 · Shift 2· Numerical answer

    If the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 3& 0& -1\end{array}\right]$ satisfies the equation $A^{20}+\alpha A^{19}+\beta A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& 0\\ 0& 0& 1\end{array}\right]$ for some real numbers $\alpha$ and $\beta$, then $\beta -\alpha$ is equal to _________.

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    Answer: 4

22. Q132021 · 25 Feb 2021 · Shift 2

    If for the matrix, $A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right]$, $A{A}^T=I_2$, then the value of ${\alpha }^4+{\beta }^4$ is :

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

    Answer: C

23. Q152021 · 25 Feb 2021 · Shift 2

    The following system of linear equations $2x+3y+2z=9$ $3x+2y+2z=9$ $x-y+4z=8$

    1. A.does not have any solution
    2. B.has a solution $(\alpha ,\beta ,\gamma )$ satisfying $\alpha +{\beta }^2+{\gamma }^3=12$
    3. C.has a unique solution
    4. D.has infinitely many solutions
    Show answer

    Answer: C

24. Q192021 · 25 Feb 2021 · Shift 2

    Let A be a $3\times 3$ matrix with $\det (A)=4$. Let $R_i$ denote the $i^{\text{th}}$ row of A. If a matrix B is obtained by performing the operation $R_2\to 2R_2+5R_3$ on 2A, then $\det (B)$ is equal to :

    1. A.64
    2. B.16
    3. C.128
    4. D.80
    Show answer

    Answer: A