How many questions from Matrices and Determinants appear in JEE?

Matrices and Determinants has appeared in 180 of the last 186 JEE Main and JEE Advanced papers — about 97% of them — contributing 342 questions in total across those papers.

Is Matrices and Determinants an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 97% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Matrices and Determinants questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Matrices and Determinants JEE Previous Year Questions with Answers

Q1·MathematicsMultiple correctJEE Advanced 2026

Consider the matrix

M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}

. Let

p, q, r, s, a, b, c

and

d

be integers such that

M^{26} = \begin{bmatrix} p & q \\ r & s \end{bmatrix}

and

\sum_{k=1}^{26} M^k = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

. Then which of the following statements is (are) TRUE ?

(A)There exists a $2 \times 2$ invertible matrix $N$ with real entries such that $MN = N \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$
(B)The value of $a$ is 378
(C)For any two given integers $m$ and $n$ , there exist unique integers $x$ and $y$ such that $px + qy = m$ and $rx + sy = n$
(D)For each positive real number $t$ , the system of linear equations $(a + t)x + by = 1$ and $cx + (d + t)y = -1$ has a unique solution

Q2·MathematicsMultiple correctJEE Advanced 2026

Let

\mathbb{R}

denote the set of all real numbers and let

i = \sqrt{-1}

. Consider the matrices

S = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

and

T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}

. Let

a, b, c, d

be real numbers such that

ST = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

. Let

H = \{x + iy : x, y \in \mathbb{R}

and

y > 0\}

. Then which of the following statements is (are) TRUE ?

(A) $\dfrac{b + ia}{d + ic} = i$
(B)If $\omega = \dfrac{-1 + i\sqrt{3}}{2}$ , then $\dfrac{a\omega + b}{c\omega + d} = \omega$
(C)If $m$ is an integer greater than 2 such that $(ST)^{2} = (ST)^{m}$ , then $m$ is an integer multiple of 8
(D)If $z \in H$ , then $\dfrac{az + b}{cz + d} \in H$

Q3·MathematicsSingle correctJEE Advanced 2026

Which one of the following matrices can be obtained by performing elementary row transformations on the

3 \times 3

identity matrix ?

(A) $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{bmatrix}$
(C) $\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 2 & 5 & 8 \end{bmatrix}$
(D) $\begin{bmatrix} 1 & 1 & 1 \\ -1 & 1 & 2 \\ 0 & 2 & 3 \end{bmatrix}$

Q4·MathematicsSingle correctJEE Main 2026

Let

A = \begin{bmatrix} \alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{bmatrix}

and

B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \mathrm{adj}(A)

. If

\det(B) = 66

, then

\det(\mathrm{adj}(A))

equals:

(A)289
(B)361
(C)441
(D)529

Q5·MathematicsSingle correctJEE Main 2026

If the system of linear equations:

x + y + z = 6

,

x + 2y + 5z = 10

,

2x + 3y + \lambda z = \mu

has infinitely many solutions, then the value of

\lambda + \mu

equals:

(A)12
(B)16
(C)22
(D)28

Q6·MathematicsSingle correctJEE Main 2026

The sum of all possible values of θ ∈ [0, 2π], for which the system of equations :

x\cos 3\theta - 8y - 12z = 0

x\cos 2\theta + 3y + 3z = 0

x + y + 3z = 0

has a non-trivial solution, is equal to :

(A)π
(B)2π
(C)3π
(D)4π

Q7·MathematicsNumericalJEE Main 2026

Let A =

\begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}

satisfy A

^2

+ α(adj(adj(A))) + β(adj(A)(adj(adj(A)))) =

\begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}

for some α, β ∈ ℝ. Then (α − β)

^2

is equal to ______

Q8·MathematicsSingle correctJEE Main 2026

Let

A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}

and

B = [b_{ij}]

,

1 \leq i, j \leq 3

. If

B = A^{99} - I

, then the value of

\frac{b_{31} - b_{21}}{b_{32}}

is :

(A)99
(B)199
(C)149
(D)159

Q9·MathematicsSingle correctJEE Main 2026

Let

A

be a

3 \times 3

matrix such that

A^T \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}5\\2\\2\end{bmatrix}

,

A^T \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}3\\1\\1\end{bmatrix}

,

A \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}3\\4\\4\end{bmatrix}

and

A \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}1\\3\\1\end{bmatrix}

. If

\det(A) = 1

, then

\det(\mathrm{adj}(A^2 + A))

is equal to:

(A)16
(B)25
(C)49
(D)64

Q10·MathematicsSingle correctJEE Main 2026

Consider the system of linear equations in

x

,

y

,

z

:

x + 2y + tz = 0

,

6x + y + 5tz = 0

,

3x + t^2 y + f(t)z = 0

, where

f : \mathbb{R} \to \mathbb{R}

is a differentiable function. If this system has infinitely many solutions for all

t \in \mathbb{R}

, then

f

(A)is a constant function
(B)is strictly increasing on $\mathbb{R}$
(C)is strictly decreasing on $\mathbb{R}$
(D)has two critical points

Q11·MathematicsSingle correctJEE Main 2026

Let M be a

3 \times 3

matrix such that

M \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}

,

M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}

and

M \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}

. If

M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}

, then

x + y + z

equals :

(A)4
(B)5
(C)7
(D)11

Q12·MathematicsSingle correctJEE Main 2026

If

f : \mathbf{N} \to \mathbf{Z}

is defined by

f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}

, k ∈ N, and

\sum_{n=1}^{k} f(n) = 98

, then k is equal to :

(A)3
(B)4
(C)5
(D)6

Q13·MathematicsSingle correctJEE Main 2026

Let

A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}

. Then the sum of all elements of the matrix

\mathrm{adj}(\mathrm{adj}(2(\mathrm{adj}A)^{-1}))

is equal to:

(A) $3$
(B) $4$
(C) $-4$
(D) $-3$

Q14·MathematicsSingle correctJEE Main 2026

Let

S = \left\{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0 \right\}

be a set of

2 \times 2

matrices. Then the number of matrices in

S

, for which the sum of the diagonal elements is equal to 4, is:

(A) $20$
(B) $17$
(C) $21$
(D) $19$

Q15·MathematicsSingle correctJEE Main 2026

If the system of equations:

x + y + z = 5

x + 2y + 3z = 9

x + 3y + \lambda z = \mu

has infinitely many solutions, then the value of

\lambda + \mu

is:

(A)16
(B)18
(C)19
(D)21

Q16·MathematicsSingle correctJEE Main 2026

Let

A=\begin{bmatrix}1 & 2\\ 1 & \alpha\end{bmatrix}

and

B=\begin{bmatrix}3 & 3\\ \beta & 2\end{bmatrix}

. If

A^{2}-4A+I=O

and

B^{2}-5B-6I=O

, then among the two statements : (S1):

[(B-A)(B+A)]^{T}=\begin{bmatrix}13 & 15\\ 7 & 10\end{bmatrix}

and (S2):

\det(\operatorname{adj}(A+B))=-5

,

(A)only (S1) is correct
(B)only (S2) is correct
(C)both (S1) and (S2) are correct
(D)both (S1) and (S2) are wrong

Q17·MathematicsNumericalJEE Main 2026

Consider the matrices

A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}

and

B = \begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}

. If matrices P and Q are such that

PA = B

and

AQ = B

, then the absolute value of the sum of the diagonal elements of

2(P + Q)

is ________.

Q18·MathematicsSingle correctJEE Main 2026

Let α, β ∈ ℝ be such that the system of linear equations

x+2y+z=5

2x+y+\alpha z=5

8x+4y+\beta z=18

has no solution. Then

\frac{\beta}{\alpha}

is equal to :

(A)−4
(B)4
(C)8
(D)−8

Q19·MathematicsSingle correctJEE Main 2026

If the system of equations

x + 5y + 6z = 4

,

2x + 3y + 4z = 7

,

x + 6y + az = b

has infinitely many solutions, then the point

(a, b)

lies on the line

(A) $y - x = 3$
(B) $x - y = 3$
(C) $x + y = 11$
(D) $x + y = 12$

Q20·MathematicsSingle correctJEE Main 2026

Let A, B and C be three

2\times2

matrices with real entries such that

B = (I + A)^{-1}

and

A + C = I

. If

BC = \begin{bmatrix} 1 & -5 \\ -1 & 2 \end{bmatrix}

and

CB\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 12 \\ -6 \end{bmatrix}

, then

x_1 + x_2

is

(A)2
(B)0
(C)–2
(D)4

Q21·MathematicsNumericalJEE Main 2026

Let A =

\begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}

and B be two matrices such that

A^{100} = 100B + I

. Then the sum of all the elements of

B^{100}

is ______.

Q22·MathematicsSingle correctJEE Main 2026

Let

P = [p_{ij}]

and

Q = [q_{ij}]

be two square matrices of order 3 such that

q_{ij} = 2^{(i + j - 1)} p_{ij}

and

\det(Q) = 2^{10}

. Then the value of det(adj(adj P)) is :

(A)32
(B)16
(C)81
(D)124

Q23·MathematicsSingle correctJEE Main 2026

Let

f(x) = \int \frac{7x^{10} + 9x^{8}}{(1 + x^{2} + 2x^{9})^{2}}dx

,

x > 0

,

\lim_{x \to 0} f(x) = 0

and

f(1) = \frac{1}{4}

. If

A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^{2} & 4 & 1 \end{bmatrix}

and B = adj(adj A) be such that

|B| = 81

, then

\alpha^{2}

is equal to

(A)2
(B)3
(C)1
(D)4

Q24·MathematicsNumericalJEE Main 2026

The number of 3 × 2 matrices A, which can be formed using the elements of the set

\{-2,\ -1,\ 0,\ 1,\ 2\}

such that the sum of all the diagonal elements of

\mathrm{A}^{\mathrm{T}}\mathrm{A}

is 5, is ______

Q25·MathematicsNumericalJEE Main 2026

Let

|A| = 6

, where A is a

3 \times 3

matrix. If

|\mathrm{adj}(3\mathrm{adj}(A^{2} \cdot \mathrm{adj}(2A)))| = 2^{m} \cdot 3^{n}

,

m, n \in \mathbf{N}

, then

m + n

is equal to __________.

Q26·MathematicsNumericalJEE Main 2026

Let

A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix}

and B be a matrix such that

B(I - A) = I + A

. Then the sum of the diagonal elements of

B^{T}B

is equal to______.

Q27·MathematicsSingle correctJEE Main 2026

Among the statements : I : If

\begin{vmatrix} 1 & \cos\alpha & \cos\beta \\ \cos\alpha & 1 & \cos\gamma \\ \cos\beta & \cos\gamma & 1 \end{vmatrix} = \begin{vmatrix} 0 & \cos\alpha & \cos\beta \\ \cos\alpha & 0 & \cos\gamma \\ \cos\beta & \cos\gamma & 0 \end{vmatrix}

, then

\cos^2\alpha + \cos^2\beta + \cos^2\gamma = \frac{3}{2}

, and II : If

\begin{vmatrix} x^2 + x & x + 1 & x - 2 \\ 2x^2 + 3x - 1 & 3x & 3x - 3 \\ x^2 + 2x + 3 & 2x - 1 & 2x - 1 \end{vmatrix} = px + q

, then

p^2 = 196q^2

,

(A)both are false
(B)only II is true
(C)both are true
(D)only I is true

Q28·MathematicsSingle correctJEE Main 2026

The system of linear equations

x + y + z = 6

2x + 5y + az = 36

x + 2y + 3z = b

has

(A)unique solution for $a = 8$ and $b = 16$
(B)infinitely many solutions for $a = 8$ and $b = 14$
(C)infinitely many solutions for $a = 8$ and $b = 16$
(D)unique solution for $a = 8$ and $b = 14$

Q29·MathematicsSingle correctJEE Main 2026

If

X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}

is a solution of the system of equations

AX = B

, where adj

A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}

and

B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}

, then

|x + y + z|

is equal to :

(A)3
(B) $\frac{3}{2}$
(C)1
(D)2

Q30·MathematicsSingle correctJEE Main 2026

Let n be the number obtained on rolling a fair die. If the probability that the system

x - ny + z = 6

x + (n-2)y + (n+1)z = 8

(n-1)y + z = 1

Has a unique solution is

\frac{k}{6}

, then the sum of k and all possible values of n is :

(A)21
(B)24
(C)20
(D)22

Q31·MathematicsNumericalJEE Main 2026

Let A be a

3 \times 3

matrix such that

A + A^{T} = O

. If

A\begin{bmatrix}1\\-1\\0\end{bmatrix} = \begin{bmatrix}3\\3\\2\end{bmatrix}

,

A^{2}\begin{bmatrix}1\\-1\\0\end{bmatrix} = \begin{bmatrix}-3\\19\\-24\end{bmatrix}

and

\det(\text{adj}(2\text{adj}(A + I))) = (2)^{\alpha}.(3)^{\beta}.(11)^{\gamma}

,

\alpha

,

\beta

,

\gamma

are non-negative integers, then

\alpha + \beta + \gamma

is equal to ______

Q32·MathematicsSingle correctJEE Main 2026

If

A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}

, then the determinant of the matrix

(A^{2025} - 3A^{2024} + A^{2023})

is

(A)28
(B)12
(C)24
(D)16

Q33·MathematicsNumericalJEE Main 2026

For some α, β ∈ R, let

A=\begin{bmatrix}\alpha & 2\\ 1 & 2\end{bmatrix}

and

B=\begin{bmatrix}1 & 1\\ 1 & \beta\end{bmatrix}

be such that

A^{2}-4A+2I=B^{2}-3B+I=O

. Then

(\det(\operatorname{adj}(A^{3}-B^{3})))^{2}

is equal to ………

Q34·MathematicsSingle correctJEE Main 2026

If the system of equations

3x + y + 4z = 3

2x + \alpha y - z = -3

X + 2y + z = 4

has no solution, then the value of

\alpha

is equal to:

(A)19
(B)4
(C)13
(D)23

Q35·MathematicsSingle correctJEE Main 2026

For the matrices

A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}

and

B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix}

, if

(A^{15} + B)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}

, then among the following which one is true?

(A) $x = 5$ , $y = 7$
(B) $x = 18$ , $y = 11$
(C) $x = 11$ , $y = 2$
(D) $x = 16$ , $y = 3$

Q36·MathematicsSingle correctJEE Advanced 2025

Consider the matrix

P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}

. Let the transpose of a matrix X be denote by

X^{T}

. Then the number of

3 \times 3

invertible matrices Q with integer entries, such that

Q^{-1} = Q^{T}

and

PQ = QP

, is

(A)32
(B)8
(C)16
(D)24

Q37·MathematicsMultiple correctJEE Advanced 2025

Let

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

and

P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}

. Let

Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix}

for some non-zero real numbers x, y and z for which there is a 2 × 2 matrix R with all entries being non-zero real numbers, such that QR = RP. Then which of the following statements is (are) TRUE?

(A)The determinant of Q − 2I is zero
(B)The determinant of Q − 6I is 12
(C)The determinant of Q − 3I is 15
(D)yz = 2

Q38·MathematicsSingle correctJEE Main 2025

Let

\alpha

be a solution of

x^2+x+1=0

, and for some

a

and

b

in

\mathbb{R}

,

\begin{bmatrix}4 & a & b\end{bmatrix}\begin{bmatrix}1 & 16 & 13\\ -1 & -1 & 2\\ -2 & -14 & -8\end{bmatrix}=\begin{bmatrix}0 & 0 & 0\end{bmatrix}

. If

\dfrac{4}{\alpha^4}+\dfrac{m}{\alpha^a}+\dfrac{n}{\alpha^b}=3

, then

m+n

is equal to ___

(A) $3$
(B) $11$
(C) $7$
(D) $8$

Q39·MathematicsSingle correctJEE Main 2025

Let

A=\begin{bmatrix}2 & 2+p & 2+p+q\\ 4 & 6+2p & 8+3p+2q\\ 6 & 12+3p & 20+6p+3q\end{bmatrix}

. If

\det\big(\mathrm{adj}(\mathrm{adj}(3A))\big)=2^m\cdot3^n

,

m,n\in\mathbb{N}

, then

m+n

is equal to:

(A) $22$
(B) $24$
(C) $26$
(D) $20$

Q40·MathematicsIntegerJEE Main 2025

The number of singular matrices of order 2, whose elements are from the set

\{2,3,6,9\}

is ______.

Q41·MathematicsSingle correctJEE Main 2025

Let the system of equations

x+5y-z=1

,

4x+3y-3z=7

,

24x+y+\lambda z=\mu

;

\lambda,\mu\in\mathbb{R}

, have infinitely many solutions. Then the number of the solutions of this system, if

x,y,z

are integers and satisfy

7\le x+y+z\le 77

, is

(A)3
(B)6
(C)5
(D)4

Q42·MathematicsSingle correctJEE Main 2025

Let

A

be a

3\times3

matrix such that

|\text{adj}(\text{adj}(\text{adj}\,A))|=81

. If

S=\left\{n\in\mathbb{Z}:\left(|\text{adj}(\text{adj}\,A)|\right)^{(n-1)^2/2}=|A|^{(3n^2-5n-4)}\right\}

, then

\displaystyle\sum_{n\in S}\left|A^{(n^2+n)}\right|

is equal to

(A)866
(B)750
(C)820
(D)732

Q43·MathematicsSingle correctJEE Main 2025

Let the system of equations

2x+3y+5z=9

,

7x+3y-2z=8

,

12x+3y-(4+\lambda)z=16-\mu

, have infinitely many solutions. Then the radius of the circle centred at

(\lambda,\mu)

and touching the line

4x=3y

is

(A) $\dfrac{17}{5}$
(B) $\dfrac{7}{5}$
(C)7
(D) $\dfrac{21}{5}$

Q44·MathematicsSingle correctJEE Main 2025

Let the matrix

A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}

satisfy

A^n=A^{n-2}+A^2-I

for

n\ge3

. Then the sum of all the elements of

A^{50}

is:

(A)53
(B)52
(C)39
(D)44

Q45·MathematicsIntegerJEE Main 2025

Let

A=\begin{bmatrix}\cos\theta & 0 & -\sin\theta\\0 & 1 & 0\\\sin\theta & 0 & \cos\theta\end{bmatrix}

. 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 ______.

Q46·MathematicsSingle correctJEE Main 2025

Let

A

be a matrix of order

3\times 3

and

|A|=5

. If

\left|2\,\mathrm{adj}\big(3A\,\mathrm{adj}(2A)\big)\right|=2^{\alpha}\cdot 3^{\beta}\cdot 5^{\gamma}

,

\alpha,\beta,\gamma\in\mathbb{N}

, then

\alpha+\beta+\gamma

is equal to:

(A)25
(B)26
(C)27
(D)28

Q47·MathematicsSingle correctJEE Main 2025

If

y(x)=\begin{vmatrix}\sin x & \cos x & \sin x+\cos x+1\\27 & 28 & 27\\1 & 1 & 1\end{vmatrix}

,

x\in\mathbb{R}

, then

\dfrac{d^2y}{dx^2}+y

is equal to:

(A) $-1$
(B)28
(C)27
(D)1

Q48·MathematicsIntegerJEE Main 2025

Let

I

be the identity matrix of order

3\times 3

and for the matrix

A=\begin{bmatrix}\lambda & 2 & 3\\4 & 5 & 6\\7 & -1 & 2\end{bmatrix}

,

|A|=-1

. Let

B

be the inverse of the matrix

\mathrm{adj}\big(A\,\mathrm{adj}(A^2)\big)

. Then

|(\lambda B+I)|

is equal to __________.

Q49·MathematicsSingle correctJEE Main 2025

Let

A=\begin{bmatrix}\alpha & -1\\ 6 & \beta\end{bmatrix}

,

\alpha>0

, such that

\det(A)=0

and

\alpha+\beta=1

. If

I

denotes

2\times2

identity matrix, then the matrix

(1+A)^8

is:

(A) $\begin{bmatrix}4 & -1\\ 6 & -1\end{bmatrix}$
(B) $\begin{bmatrix}257 & -64\\ 514 & -127\end{bmatrix}$
(C) $\begin{bmatrix}1025 & -511\\ 2024 & -1024\end{bmatrix}$
(D) $\begin{bmatrix}766 & -255\\ 1530 & -509\end{bmatrix}$

Q50·MathematicsSingle correctJEE Main 2025

Let

a\in R

and

A

be a matrix of order

3\times3

such that

\det(A)=-4

and

A+I=\begin{bmatrix}1 & a & 1\\ 2 & 1 & 0\\ a & 1 & 2\end{bmatrix}

, where

I

is the identity matrix of order

3\times3

. If

\det((a+1)\,\mathrm{adj}((a-1)A))

is

2^m\,3^n

,

m,n\in\{0,1,2,\ldots,20\}

, then

m+n

is equal to:

(A)14
(B)17
(C)15
(D)16

Q51·MathematicsSingle correctJEE Main 2025

If the system of linear equations

3x+y+\beta z=3

;

2x+\alpha y-z=-3

;

x+2y+z=4

has infinitely many solutions, then the value of

22\beta-9\alpha

is:

(A)49
(B)31
(C)43
(D)37

Q52·MathematicsSingle correctJEE Main 2025

Let

A

be a

3\times3

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^4=\alpha A^2+\beta A+\gamma I

, where

\alpha,\beta,\gamma

are real constants, then

\alpha+\beta+\gamma

is equal to:

(A)12
(B)20
(C)76
(D)4

Q53·MathematicsSingle correctJEE Main 2025

If the system of equations

2x+\lambda y+3z=5

,

3x+2y-z=7

,

4x+5y+\mu z=9

has infinitely many solutions, then

(\lambda^2+\mu^2)

is equal to:

(A)22
(B)18
(C)26
(D)30

Q54·MathematicsSingle correctJEE Main 2025

Let

\alpha,\beta

(\alpha\ne\beta)

be the values of m, for which the equations

x+y+z=1

;

x+2y+4z=m

and

x+4y+10z=m^2

have infinitely many solutions. Then the value of

\displaystyle\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)

is equal to:

(A)440
(B)3080
(C)3410
(D)560

Q55·MathematicsSingle correctJEE Main 2025

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 Z

, then

\alpha+\beta

is equal to

(A)280
(B)168
(C)210
(D)224

Q56·MathematicsSingle correctJEE Main 2025

Let

A=[a_{ij}]=\begin{bmatrix}\log_5 128 & \log_4 5\\ \log_5 8 & \log_4 25\end{bmatrix}

. If

A_{ij}

is the cofactor of

a_{ij}

,

C_{ij}=\displaystyle\sum_{k=1}^{2}a_{ik}A_{jk}

,

1\le i, j\le 2

, and

C=[C_{ij}]

, then

8|C|

is equal to:

(A)262
(B)288
(C)242
(D)222

Q57·MathematicsIntegerJEE Main 2025

Let

S=\{m\in Z: A^{m^2}+A^{m}=3I-A^{-6}\}

, where

A=\begin{bmatrix}2 & -1\\ 1 & 0\end{bmatrix}

. Then

n(S)

is equal to ______.

Q58·MathematicsSingle correctJEE Main 2025

Let M and m respectively be the maximum and the minimum values of

f(x)=\begin{vmatrix}1+\sin^2 x & \cos^2 x & 4\sin 4x\\ \sin^2 x & 1+\cos^2 x & 4\sin 4x\\ \sin^2 x & \cos^2 x & 1+4\sin 4x\end{vmatrix}

,

x\in R

. Then

M^4-m^4

is equal to:

(A)1280
(B)1295
(C)1040
(D)1215

Q59·MathematicsIntegerJEE Main 2025

Let M denote the set of all real matrices of order

3\times3

and let

S=\{-3, -2, -1, 1, 2\}

. Let

S_1=\{A=[a_{ij}]\in M: A=A^T

and

a_{ij}\in S, \forall i, j\}

,

S_2=\{A=[a_{ij}]\in M: A=-A^T

and

a_{ij}\in S, \forall i, j\}

,

S_3=\{A=[a_{ij}]\in M: a_{11}+a_{22}+a_{33}=0

and

a_{ij}\in S, \forall i, j\}

. If

n(S_1\cup S_2\cup S_3)=125\alpha

, then

\alpha

equals.

Q60·MathematicsSingle correctJEE Main 2025

Let

A=\begin{bmatrix}\frac{1}{\sqrt{2}}&-2\\0&1\end{bmatrix}

and

P=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}

,

\theta>0

. If

B=PAP^T

,

C=P^TB^{10}P

and the sum of the diagonal elements of C is

\frac{m}{n}

, where

\gcd(m,n)=1

, then

m+n

is:

(A)65
(B)127
(C)258
(D)2049

Q61·MathematicsSingle correctJEE Main 2025

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:

(A)13
(B)10
(C)11
(D)12

Q62·MathematicsSingle correctJEE Main 2025

For some

a,b

, let

f(x)=\begin{vmatrix} a+\dfrac{\sin x}{x} & 1 & b \\ a & 1+\dfrac{\sin x}{x} & b \\ a & 1 & b+\dfrac{\sin x}{x}\end{vmatrix}

,

x\neq 0

,

\displaystyle\lim_{x\to 0}f(x)=\lambda+\mu a+\nu b

. Then

(\lambda+\mu+\nu)^2

is equal to:

(A)25
(B)9
(C)36
(D)16

Q63·MathematicsIntegerJEE Main 2025

Let A be a

3\times3

matrix such that

X^T AX=O

for all nonzero

3\times1

matrices

X=\begin{bmatrix}x\\y\\z\end{bmatrix}

. If

A\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}1\\4\\-5\end{bmatrix}

,

A\begin{bmatrix}1\\2\\1\end{bmatrix}=\begin{bmatrix}0\\4\\-8\end{bmatrix}

, and

\det(\operatorname{adj}(2(A+I)))=2^{\alpha}3^{\beta}5^{\gamma}

,

\alpha,\beta,\gamma\in\mathbb{N}

, then

\alpha^2+\beta^2+\gamma^2

is _______

Q64·MathematicsSingle correctJEE Main 2025

If the system of equations

2x-y+z=4

,

5x+\lambda y+3z=12

,

100x-47y+\mu z=212

, has infinitely many solutions, then

\mu-2\lambda

is equal to

(A) $56$
(B) $59$
(C) $55$
(D) $57$

Q65·MathematicsSingle correctJEE Main 2025

If the system of equations

(\lambda-1)x+(\lambda-4)y+\lambda z=5

\lambda x+(\lambda-1)y+(\lambda-4)z=7

(\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9

has infinitely many solutions, then

\lambda^2+\lambda

is equal to

(A) $10$
(B) $12$
(C) $6$
(D) $20$

Q66·MathematicsSingle correctJEE Main 2025

If

A

,

B

and

(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1}))

are non-singular matrices of same order, then the inverse of

A\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)^{-1}B

, is equal to

(A) $AB^{-1}+A^{-1}B$
(B) $\operatorname{adj}(B^{-1})+\operatorname{adj}(A^{-1})$
(C) $\dfrac{1}{|AB|}\big(\operatorname{adj}(B)+\operatorname{adj}(A)\big)$
(D) $\dfrac{AB^{-1}}{|A|}+\dfrac{BA^{-1}}{|B|}$

Q67·MathematicsSingle correctJEE Main 2025

The system of equations

x+y+z=6,\ x+2y+5z=9,\ x+5y+\lambda z=\mu,

has no solution if :

(A) $\lambda=17,\ \mu\neq 18$
(B) $\lambda\neq 17,\ \mu=18$
(C) $\lambda=15,\ \mu\neq 18$
(D) $\lambda=17,\ \mu=18$

Q68·MathematicsSingle correctJEE Main 2025

Let

A=[a_{ij}]

be a

3\times 3

matrix such that

A\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix},\ A\begin{bmatrix}4\\1\\3\end{bmatrix}=\begin{bmatrix}1\\1\\0\end{bmatrix}

and

A\begin{bmatrix}2\\1\\2\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}

, then

a_{23}

equals :

(A) $-1$
(B) $0$
(C) $2$
(D) $1$

Q69·MathematicsIntegerJEE Main 2025

Let

A

be a

3\times3

matrix with

\det A=-2

. If

\det\big(3\,\mathrm{adj}(-6\,\mathrm{adj}(3A))\big)=2^{m}\cdot 3^{n}

with

m>n

, find

4m+2n

(per the paper).

Q70·MathematicsSingle correctJEE Main 2025

For a

3\times3

matrix

M

, let trace

(M)

denote the sum of all the diagonal elements of

M

. Let

A

be a

3\times3

matrix such that

|A|=\dfrac{1}{2}

and trace

(A)=3

. If

B=\operatorname{adj}(\operatorname{adj}(2A))

, then the value of

|B|+\operatorname{trace}(B)

equals:

(A)56
(B)132
(C)174
(D)280

Q71·MathematicsSingle correctJEE Main 2025

If the system of linear equations:

x+y+2z=6

,

2x+3y+az=a+1

,

-x-3y+bz=2b

, where

a,b\in\mathbb{R}

, has infinitely many solutions, then

7a+3b

is equal to:

(A)9
(B)12
(C)16
(D)22

Q72·MathematicsIntegerJEE Advanced 2024

Let

S = \left\{A\begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\}\right\}

, where

|A|

denotes the determinant of

A

. Then the number of elements in

S

is ______ .

Q73·MathematicsSingle correctJEE Advanced 2024

Let

\alpha

and

\beta

be the distinct roots of the equation

x^{2} + x - 1 = 0

. Consider the set

T = \{1, \alpha, \beta\}

. For a

3 \times 3

matrix

M = (a_{ij})_{3\times3}

, define

R_{i} = a_{i1} + a_{i2} + a_{i3}

and

C_{j} = a_{1j} + a_{2j} + a_{3j}

for

i = 1, 2, 3

and

j = 1, 2, 3

. Match each entry in List-I to the correct entries in List-II. The correct option is:

List-I	List-II
P.The number of matrices $M = (a_{ij})_{3\times3}$ with all entries in $T$ such that $R_{i} = C_{j} = 0$ for all $i, j$ , is	1.1
Q.The number of symmetric matrices $M = (a_{ij})_{3\times3}$ with all entries in $T$ such that $C_{j} = 0$ for all $j$ , is	2.12
R.Let $M = (a_{ij})_{3\times3}$ be a skew symmetric matrix such that $a_{ij} \in T$ for $i > j$ . Then the number of elements in the set $\left\{\begin{pmatrix} x \\ y \\ z \end{pmatrix} : x, y, z \in R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix}\right\}$ is	3.infinite
S.Let $M = (a_{ij})_{3\times3}$ be a matrix with all entries in $T$ such that $R_{i} = 0$ for all $i$ . Then the absolute value of determinant of $M$ is	4.6
	5.0

(A)(P) $\to$ (4) (Q) $\to$ (2) (R) $\to$ (5) (S) $\to$ (1)
(B)(P) $\to$ (2) (Q) $\to$ (4) (R) $\to$ (1) (S) $\to$ (5)
(C)(P) $\to$ (2) (Q) $\to$ (4) (R) $\to$ (3) (S) $\to$ (5)
(D)(P) $\to$ (1) (Q) $\to$ (5) (R) $\to$ (3) (S) $\to$ (4)

Q74·MathematicsNumericalJEE Main 2024

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 ________.

Q75·MathematicsSingle correctJEE Main 2024

Let

B=\begin{bmatrix}1&3\\1&5\end{bmatrix}

and A be a

2\times 2

matrix such that

AB^{-1}=A^{-1}

. If

BCB^{-1}=A

and

C^{4}+\alpha C^{2}+\beta I=O

, then

2\beta-\alpha

is equal to:

(A) $16$
(B) $2$
(C) $8$
(D) $10$

Q76·MathematicsSingle correctJEE Main 2024

Let

\lambda, \mu \in \mathbb{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:

(A)25
(B)24
(C)27
(D)22

Q77·MathematicsNumericalJEE Main 2024

Consider the matrices

A=\begin{bmatrix}2&-5\\3&m\end{bmatrix}

,

B=\begin{bmatrix}20\\m\end{bmatrix}

and

X=\begin{bmatrix}x\\y\end{bmatrix}

. 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\displaystyle\int_{a}^{b}|A|\,dm

is equal to _______.

Q78·MathematicsSingle correctJEE Main 2024

Let

A=\begin{bmatrix}2 & a & 0\\ 1 & 3 & 1\\ 0 & 5 & b\end{bmatrix}

. If

A^3=4A^2-A-21I

, where

I

is the identity matrix of order

3\times3

, then

2a+3b

is equal to:

(A) $-10$
(B) $-13$
(C) $-9$
(D) $-12$

Q79·MathematicsNumericalJEE Main 2024

Let

A=\begin{bmatrix}2 & -1\\ 1 & 1\end{bmatrix}

. If the sum of the diagonal elements of

A^{13}

is

3^n

, then

n

is equal to ___

Q80·MathematicsSingle correctJEE Main 2024

If

\alpha\ne a

,

\beta\ne b

,

\gamma\ne c

and

\begin{vmatrix}\alpha&b&c\\a&\beta&c\\a&b&\gamma\end{vmatrix}=0

, then

\dfrac{a}{\alpha-a}+\dfrac{b}{\beta-b}+\dfrac{\gamma}{\gamma-c}

is equal to

(A) $2$
(B) $3$
(C) $0$
(D) $1$

Q81·MathematicsSingle correctJEE Main 2024

If the system of equations

x+4y-z=\lambda

,

7x+9y+\mu z=-3

,

5x+y+2z=-1

has infinitely many solutions, then

(2\mu+3\lambda)

is equal to

(A) $2$
(B) $-3$
(C) $3$
(D) $-2$

Q82·MathematicsNumericalJEE Main 2024

Let

\alpha\beta\gamma=45

;

\alpha,\beta,\gamma\in\mathbb{R}

. If

x(\alpha,1,2)+y(1,\beta,2)+z(2,3,\gamma)=(0,0,0)

for some

x,y,z\in\mathbb{R}

,

xyz\neq 0

, then

6\alpha+4\beta+\gamma

is equal to _______.

Q83·MathematicsSingle correctJEE Main 2024

If

A

is a square matrix of order

3

such that

\det(A)=3

and

\det\left(\operatorname{adj}\left(-4\,\operatorname{adj}\left(-3\,\operatorname{adj}\left(3\,\operatorname{adj}\left((2A)^{-1}\right)\right)\right)\right)\right)=2^{m}3^{n}

, then

m+2n

is equal to:

(A) $3$
(B) $2$
(C) $4$
(D) $6$

Q84·MathematicsNumericalJEE Main 2024

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 ______.

Q85·MathematicsSingle correctJEE Main 2024

For

\alpha,\beta\in\mathbb{R}

and a natural number

n

, let

A_{r}=\begin{vmatrix} r & 1 & \tfrac{n^{2}}{2}+\alpha \\ 2r & 2 & n^{2}-\beta \\ 3r-2 & 3 & \tfrac{n(3n-1)}{2}\end{vmatrix}

. Then

2A_{10}-A_{8}

is

(A) $4\alpha+2\beta$
(B) $2\alpha+4\beta$
(C) $2\beta$
(D) $0$

Q86·MathematicsSingle correctJEE Main 2024

Let A and B be two square matrices of order 3 such that

|A| = 3

and

|B| = 2

. Then

\left|A^T A(\text{adj}(2A))^{-1} (\text{adj}(4B))(\text{adj}(AB))^{-1} A A^T\right|

is equal to:

(A) $64$
(B) $81$
(C) $32$
(D) $108$

Q87·MathematicsSingle correctJEE Main 2024

If the system of equations

11x + y + \lambda z = -5

,

2x + 3y + 5z = 3

,

8x - 19y - 39z = \mu

has infinitely many solutions, then

\lambda^4 - \mu

is equal to:

(A) $49$
(B) $45$
(C) $47$
(D) $51$

Q88·MathematicsSingle correctJEE Main 2024

The values of

m,n

, for which the system of equations

x+y+z=4

,

2x+5y+5z=17

,

x+2y+mz=n

has infinitely many solutions, satisfy the equation:

(A) $m^2+n^2-m-n=46$
(B) $m^2+n^2+m+n=64$
(C) $m^2+n^2+mn=68$
(D) $m^2+n^2-mn=39$

Q89·MathematicsSingle correctJEE Main 2024

Let

\alpha\beta\ne 0

and

A=\begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix}

. If

B=\begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}

is the matrix of cofactors of the elements of

A

, then

\det(AB)

is equal to:

(A)343
(B)125
(C)64
(D)216

Q90·MathematicsSingle correctJEE Main 2024

Let

\alpha\in(0,\infty)

and

A=\begin{bmatrix}1 & 2 & \alpha\\ 1 & 0 & 1\\ 0 & 1 & 2\end{bmatrix}

. If

\det\big(\mathrm{adj}(2A-A^T)\cdot\mathrm{adj}(A-2A^T)\big)=2^8

, then

|A|^2

is equal to:

(A) $1$
(B) $49$
(C) $16$
(D) $4$

Q91·MathematicsSingle correctJEE Main 2024

Let

A=\begin{bmatrix}1&2\\0&1\end{bmatrix}

and

B=I+\operatorname{adj}(A)+(\operatorname{adj}A)^2+\ldots+(\operatorname{adj}A)^{10}

. Then, the sum of all the elements of the matrix B is

(A) $-110$
(B) $22$
(C) $-88$
(D) $-124$

Q92·MathematicsSingle correctJEE Main 2024

Let the system of linear equations

x+(\sqrt2\sin\alpha)y+(\sqrt2\cos\alpha)z=0

,

x+(\cos\alpha)y+(\sin\alpha)z=0

,

x+(\sin\alpha)y-(\cos\alpha)z=0

has a non-trivial solution, then

\alpha\in\left(0,\dfrac\pi2\right)

is equal to:

(A) $\dfrac{3\pi}{4}$
(B) $\dfrac{7\pi}{24}$
(C) $\dfrac{5\pi}{24}$
(D) $\dfrac{11\pi}{24}$

Q93·MathematicsNumericalJEE Main 2024

Let A be a

2\times 2

symmetric matrix such that

A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}3\\7\end{bmatrix}

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

Q94·MathematicsNumericalJEE Main 2024

Let

A

be a

3\times3

matrix of non-negative real elements such that

A\begin{bmatrix}1\\1\\1\end{bmatrix}=3\begin{bmatrix}1\\1\\1\end{bmatrix}

. Then the maximum value of

\det(A)

is ___

Q95·MathematicsNumericalJEE Main 2024

Let

A=I_2-2MM^T

, where

M

is real matrix of order

2\times 1

such that the relation

M^T M=I_1

holds. If

\lambda

is a real number such that the relation

AX=\lambda X

holds for some non-zero real matrix

X

of order

2\times 1

, then the sum of squares of all possible values of

\lambda

is equal to __________.

Q96·MathematicsSingle correctJEE Main 2024

If the system of equations

2x+3y-z=5

,

x+\alpha y+3z=-4

,

3x-y+\beta z=7

has infinitely many solutions, then

13\alpha\beta

is equal to:

(A) $1110$
(B) $1120$
(C) $1210$
(D) $1220$

Q97·MathematicsSingle correctJEE Main 2024

Let the system of equations

x+2y+3z=5

,

2x+3y+z=9

,

4x+3y+\lambda z=\mu

have infinite number of solutions. Then

\lambda+2\mu

is equal to:

(A)28
(B)17
(C)22
(D)15

Q98·MathematicsSingle correctJEE Main 2024

If

A=\begin{bmatrix}\sqrt2 & 1\\ -1 & \sqrt2\end{bmatrix}

,

B=\begin{bmatrix}1 & 0\\ 1 & 1\end{bmatrix}

,

C=ABA^T

and

X=A^T C^2 A

, then

\det X

is equal to:

(A) $243$
(B) $729$
(C) $27$
(D) $891$

Q99·MathematicsSingle correctJEE Main 2024

If the system of linear equations

x-2y+z=-4

;

2x+\alpha y+3z=5

;

3x-y+\beta z=3

has infinitely many solutions, then

12\alpha+13\beta

is equal to

(A) $60$
(B) $64$
(C) $54$
(D) $58$

Q100·MathematicsSingle correctJEE Main 2024

Let A be a

3\times3

real matrix such that

A\begin{pmatrix}1\\0\\1\end{pmatrix}=2\begin{pmatrix}1\\0\\1\end{pmatrix}

,

A\begin{pmatrix}-1\\0\\1\end{pmatrix}=4\begin{pmatrix}-1\\0\\1\end{pmatrix}

,

A\begin{pmatrix}0\\1\\0\end{pmatrix}=2\begin{pmatrix}0\\1\\0\end{pmatrix}

. Then, the system

(A-3I)\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}

has

(A)unique solution
(B)exactly two solutions
(C)no solution
(D)infinitely many solutions

Q101·MathematicsSingle correctJEE Main 2024

If

f(x)=\begin{vmatrix}x^3 & 2x^2+1 & 1+3x\\ 3x^2+2 & 2x & x^3+6\\ x^3-x & 4 & x^2-2\end{vmatrix}

for all

x\in R

, then

2f(0)+f'(0)

is equal to

(A) $48$
(B) $24$
(C) $42$
(D) $18$

Q102·MathematicsNumericalJEE Main 2024

Let A be a

3\times3

matrix and

\det(A)=2

. If

n=\det\left(\text{adj}\left(\text{adj}\left(\ldots\left(\text{adj}A\right)\right)\right)\right)

(adj applied 2024 times). Then the remainder when

n

is divided by 9 is equal to ______.

Q103·MathematicsSingle correctJEE Main 2024

Consider the system of linear equations

x+y+z=5

,

x+2y+\lambda^2 z=9

,

x+3y+\lambda z=\mu

, where

\lambda,\mu\in R

. Then, which of the following statement is NOT correct?

(A)System has infinite number of solutions if $\lambda=1$ and $\mu=13$
(B)System is inconsistent if $\lambda=1$ and $\mu\ne 13$
(C)System is consistent if $\lambda\ne 1$ and $\mu=13$
(D)System has unique solution if $\lambda\ne 1$ and $\mu\ne 13$

Q104·MathematicsSingle correctJEE Main 2024

If

f(x)=\begin{vmatrix}2\cos^2x & 2\sin^2x & 3+\sin^2 2x\\ 3+2\cos^2x & 2\sin^2x & \sin^2 2x\\ 2\cos^2x & 3+2\sin^2x & \sin^2 2x\end{vmatrix}

, then

\dfrac13 f'(0)

is equal to:

(A) $0$
(B) $1$
(C) $2$
(D) $6$

Q105·MathematicsSingle correctJEE Main 2024

Consider the system of linear equations

x+y+z=4\mu

,

x+2y+2\lambda z=10\mu

,

x+3y+4\lambda^2 z=\mu^2+15

, where

\lambda,\mu\in\mathbb{R}

. Which one of the following statements is NOT correct?

(A)The system has unique solution if $\lambda\ne\dfrac12$ and $\mu\ne1,15$
(B)The system is inconsistent if $\lambda=\dfrac12$ and $\mu\ne1$
(C)The system has infinite number of solutions if $\lambda=\dfrac12$ and $\mu=15$
(D)The system is consistent if $\lambda\ne\dfrac12$

Q106·MathematicsSingle correctJEE Main 2024

Let

R=\begin{bmatrix}x&0&0\\0&y&0\\0&0&z\end{bmatrix}

be a non-zero

3\times 3

matrix, where

x\sin\theta=y\sin\left(\theta+\dfrac{2\pi}{3}\right)=z\sin\left(\theta+\dfrac{4\pi}{3}\right)\ne 0

,

\theta\in(0,2\pi)

. For a square matrix

M

, let trace

(M)

denote the sum of all the diagonal entries of

M

. Then, among the statements: (I) Trace

(R)=0

(II) If trace

(\text{adj}(\text{adj}(R)))=0

, then

R

has exactly one non-zero entry.

(A)Both (I) and (II) are true
(B)Neither (I) nor (II) is true
(C)Only (II) is true
(D)Only (I) is true

Q107·MathematicsSingle correctJEE Main 2024

Let

A=\begin{bmatrix}2 & 1 & 2\\ 6 & 2 & 11\\ 3 & 3 & 2\end{bmatrix}

and

P=\begin{bmatrix}1 & 2 & 0\\ 5 & 0 & 2\\ 7 & 1 & 5\end{bmatrix}

. The sum of the prime factors of

|P^{-1}AP-2I|

is equal to:

(A)26
(B)27
(C)66
(D)23

Q108·MathematicsNumericalJEE Main 2024

Let for any three distinct consecutive terms

a,b,c

of an A.P, the lines

ax+by+c=0

be concurrent at the point

P

and

Q\,(\alpha,\beta)

be a point such that the system of equations

x+y+z=6

,

2x+5y+\alpha z=\beta

and

x+2y+3z=4

, has infinitely many solutions. Then

(PQ)^2

is equal to ___.

Q109·MathematicsSingle correctJEE Main 2024

Let

A=\begin{bmatrix}1 & 0 & 0\\ 0 & \alpha & \beta\\ 0 & \beta & \alpha\end{bmatrix}

and

|2A|^3=2^{21}

where

\alpha,\beta\in Z

. Then a value of

\alpha

is

(A) $3$
(B) $5$
(C) $17$
(D) $9$

Q110·MathematicsSingle correctJEE Main 2024

Let A be a square matrix such that

AA^T=I

. Then

\dfrac{1}{2}A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]

is equal to

(A) $A^2+I$
(B) $A^3+I$
(C) $A^2+A^T$
(D) $A^3+A^T$

Q111·MathematicsSingle correctJEE Main 2024

Consider the matrix

f(x)=\begin{bmatrix}\cos x & -\sin x & 0\\ \sin x & \cos x & 0\\ 0 & 0 & 1\end{bmatrix}

. Given below are two statements: Statement I:

f(-x)

is the inverse of the matrix

f(x)

. Statement II:

f(x)\cdot f(y)=f(x+y)

. In the light of the above statements, choose the correct answer from the options given below:

(A)Statement I is false but Statement II is true
(B)Both Statement I and Statement II are false
(C)Statement I is true but Statement II is false
(D)Both Statement I and Statement II are true

Q112·MathematicsNumericalJEE Main 2024

Let

A=\begin{bmatrix}2 & 0 & 1\\ 1 & 1 & 0\\ 1 & 0 & 1\end{bmatrix}

,

B=[B_1,B_2,B_3]

, where

B_1,B_2,B_3

are column matrices, and

AB_1=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix}

,

AB_2=\begin{bmatrix}2\\ 3\\ 0\end{bmatrix}

,

AB_3=\begin{bmatrix}3\\ 2\\ 1\end{bmatrix}

. If

\alpha=|B|

and

\beta

is the sum of all the diagonal elements of

B

, then

\alpha^3+\beta^3

is equal to __________.

Q113·MathematicsNumericalJEE Main 2024

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 __________.

Q114·MathematicsSingle correctJEE Main 2024

The values of

\alpha

, for which

\begin{vmatrix} 1 & \dfrac32 & \alpha+\dfrac32 \\ 1 & \dfrac13 & \alpha+\dfrac13 \\ 2\alpha+3 & 3\alpha+1 & 0\end{vmatrix}=0

, lie in the interval

(A) $(-2,1)$
(B) $(-3,0)$
(C) $\left(-\dfrac32,\dfrac32\right)$
(D) $(0,3)$

Q115·MathematicsMultiple correctJEE Advanced 2023

Let M =

(a_{ij})

,

i, j \in \{1, 2, 3\}

, be the

3 \times 3

matrix such that

a_{ij} = 1

if

j + 1

is divisible by

i

, otherwise

a_{ij} = 0

. Then which of the following statements is(are) true?

(A) $M$ is invertible
(B)There exists a nonzero column matrix $\begin{pmatrix} a_{1} \\ a_{2} \\ a_{3} \end{pmatrix}$ and such that $M \begin{pmatrix} a_{1} \\ a_{2} \\ a_{3} \end{pmatrix} = \begin{pmatrix} -a_{1} \\ -a_{2} \\ -a_{3} \end{pmatrix}$
(C)The set $\left\{ X \in \mathbb{R}^{3} : MX = \mathbf{0} \right\} \neq \{\mathbf{0}\}$ , where $\mathbf{0} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$
(D)The matrix $(M - 2I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix

Q116·MathematicsIntegerJEE Advanced 2023

Let

R = \left\{ \begin{pmatrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{pmatrix} : a, b, c, d \in \{0, 3, 5, 7, 11, 13, 17, 19\} \right\}

. Then the number of invertible matrices in R is

Q117·MathematicsSingle correctJEE Advanced 2023

Let

\alpha

,

\beta

and

\gamma

be real numbers. Consider the following system of linear equations

x + 2y + z = 7

x + \alpha z = 11

2x - 3y + \beta z = \gamma

Match each entry in List-I to the correct entries in List-II. The correct option is:

List – I	List – II
P.If $\beta = \frac{1}{2}(7\alpha - 3)$ and $\gamma = 28$ , then the system has	1.A unique solution
Q.If $\beta = \frac{1}{2}(7\alpha - 3)$ and $\gamma \neq 28$ , then the system has	2.No solution
R.If $\beta \neq \frac{1}{2}(7\alpha - 3)$ where $\alpha = 1$ and $\gamma \neq 28$ , then the system has	3.Infinitely many solution
S.If $\beta \neq \frac{1}{2}(7\alpha - 3)$ where $\alpha = 1$ and $\gamma = 28$ , then the system has	4. $x = 11$ , $y = -2$ and $z = 0$ as a solution
	5. $x = -15$ , $y = 4$ and $z = 0$ as a solution

(A)(P) → (3) (Q) → (2) (R) → (1) (S) → (4)
(B)(P) → (3) (Q) → (2) (R) → (5) (S) → (4)
(C)(P) → (2) (Q) → (1) (R) → (4) (S) → (5)
(D)(P) → (2) (Q) → (1) (R) → (1) (S) → (3)

Q118·MathematicsSingle correctJEE Main 2023

Let the system of linear equations

-x+2y-9z=7

,

-x+3y+7z=9

,

-2x+y+5z=8

,

-3x+y+13z=\lambda

has a unique solution

x=\alpha, y=\beta, z=\gamma

. Then the distance of the point

(\alpha,\beta,\gamma)

from the plane

2x-2y+z=\lambda

is

(A)9
(B)11
(C)13
(D)7

Q119·MathematicsSingle correctJEE Main 2023

Let the determinant of a square matrix A of order m be

m-n

, where m and n satisfy

4m+n=22

and

17m+4n=93

. If

\det(n\,\mathrm{adj}(\mathrm{adj}(mA)))=3^a 5^b 6^c

, then

a+b+c

is equal to:

(A)96
(B)101
(C)109
(D)84

Q120·MathematicsSingle correctJEE Main 2023

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?

(A)It has infinitely many solutions if $a=3,\,b=6$
(B)It has unique solution if $a=b=6$
(C)It has unique solution if $a=b=8$
(D)It has infinitely many solution if $a=3,\,b=8$

Q121·MathematicsSingle correctJEE Main 2023

If the system of equations

2x + y - z = 5

,

2x - 5y + \lambda z = \mu

,

x + 2y - 5z = 7

has infinitely many solutions, then

(\lambda + \mu)^2 + (\lambda - \mu)^2

is equal to

(A)916
(B)912
(C)920
(D)904

Q122·MathematicsSingle correctJEE Main 2023

Let for

A = \begin{bmatrix} 1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}

,

|A| = 2

. If

\left|2\,\text{adj}\,(2\,\text{adj}\,(2A))\right| = 32^n

, then

3n + \alpha

is equal to

(A)10
(B)9
(C)12
(D)11

Q123·MathematicsSingle correctJEE Main 2023

Let

B=\begin{bmatrix} 1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4 \end{bmatrix},\,\alpha>2

be the adjoint of a matrix

A

and

|A|=2

, then

[\alpha\,-2\alpha\,\alpha]\begin{bmatrix} \alpha \\ -2\alpha \\ \alpha \end{bmatrix}

is equal to:

(A) $16$
(B) $32$
(C) $-16$
(D) $0$

Q124·MathematicsSingle correctJEE Main 2023

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:

(A) $6^{10}$
(B) $9^{10}$
(C) $10^{3}$
(D) $10^{6}$

Q125·MathematicsNumericalJEE Main 2023

Let

D_k=\begin{vmatrix}1&2k&2k-1\\n&n^2+n+2&n^2\\n&n^2+n&n^2+n+2\end{vmatrix}

. If

\sum_{k=1}^{n}D_k=96

, then n is equal to

Q126·MathematicsSingle correctJEE Main 2023

Let

A=\begin{bmatrix}1&\frac{1}{51}\\0&1\end{bmatrix}

. If

B=\begin{bmatrix}1&2\\-1&-1\end{bmatrix}A\begin{bmatrix}-1&-2\\1&1\end{bmatrix}

, then the sum of all the elements of the matrix

\sum_{n=1}^{50}B^n

is equal to

(A)100
(B)50
(C)75
(D)125

Q127·MathematicsSingle correctJEE Main 2023

If

\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix}=\dfrac{9}{8}(103x+81)

, then

\lambda,\dfrac{\lambda}{3}

are the roots of the equation

(A) $4\lambda^2+24\lambda+27=0$
(B) $4\lambda^2-24\lambda+27=0$
(C) $4\lambda^2+24\lambda-27=0$
(D) $4\lambda^2-24\lambda-27=0$

Q128·MathematicsNumericalJEE Main 2023

Let

A = \begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}

, where

a, c \in \mathbb{R}

. If

A^3 = A

and the positive value of

a

belongs to the interval

(n - 1, n]

, where

n \in \mathbb{N}

, then

n

is equal to _______ .

Q129·MathematicsSingle correctJEE Main 2023

Let

A

be a

2 \times 2

matrix with real entries such that

A' = \alpha A + 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

(A) $0$
(B) $\frac{3}{2}$
(C) $\frac{5}{2}$
(D) $2$

Q130·MathematicsSingle correctJEE Main 2023

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

(A) $4$
(B) $3$
(C) $5$
(D) $6$

Q131·MathematicsSingle correctJEE Main 2023

If

A=\frac{1}{5!\,6!\,7!}\begin{bmatrix}5! & 6! & 7!\\ 6! & 7! & 8!\\ 7! & 8! & 9!\end{bmatrix}

, then

\left|\mathrm{adj}\left(\mathrm{adj}(2A)\right)\right|

is equal to:

(A) $2^8$
(B) $2^{12}$
(C) $2^{20}$
(D) $2^{16}$

Q132·MathematicsSingle correctJEE Main 2023

If A is a

3\times3

matrix and

|A|=2

, then

\left|3\,adj\left(\left|3A\right|A^2\right)\right|

is equal to

(A) $3^{11}\cdot6^{10}$
(B) $3^{12}\cdot6^{10}$
(C) $3^{10}\cdot6^{11}$
(D) $3^{12}\cdot6^{11}$

Q133·MathematicsNumericalJEE Main 2023

Let S be the set of values of

\lambda

for which the system of equations

6\lambda x-3y+3z=4\lambda^2

,

2x+6\lambda y+4z=1

,

3x+2y+3\lambda z=\lambda

has no solution. Then

12\sum_{\lambda\in S}|\lambda|

is equal to _______ .

Q134·MathematicsSingle correctJEE Main 2023

For the system of linear equations

2x-y+3z=5

,

3x+2y-z=7

,

4x+5y+\alpha z=\beta

, which of the following is NOT correct?

(A)The system has infinitely many solutions for $\alpha=-5$ and $\beta=9$
(B)The system has a unique solution for $\alpha\neq-5$ and $\beta=8$
(C)The system has infinitely many solutions for $\alpha=-6$ and $\beta=9$
(D)The system is inconsistent for $\alpha=-5$ and $\beta=8$

Q135·MathematicsSingle correctJEE Main 2023

Let

A=\begin{bmatrix}2 & 1 & 0\\ 1 & 2 & -1\\ 0 & -1 & 2\end{bmatrix}

. If

|\,\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\,2A))\,|=(16)^{n}

, then

n

is equal to

(A)10
(B)9
(C)12
(D)8

Q136·MathematicsSingle correctJEE Main 2023

Let

P=\begin{bmatrix}\frac{\sqrt{3}}{2} & \frac{1}{2}\\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}

,

A=\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}

and

Q=PAP^{T}

. If

P^{T}Q^{2007}P=\begin{bmatrix}a & b\\ c & d\end{bmatrix}

, then

2a+b-3c-4d

equal to

(A)2007
(B)2005
(C)2006
(D)2004

Q137·MathematicsSingle correctJEE Main 2023

Let S be the set of all values of

\theta\in[-\pi,\pi]

for which the system of linear equations

x+y+\sqrt{3}\,z=0

,

-x+(\tan\theta)y+\sqrt{7}\,z=0

,

x+y+(\tan\theta)z=0

has non-trivial solution. Then

\dfrac{120}{\pi}\sum_{\theta\in S}\theta

is equal to

(A)40
(B)10
(C)20
(D)30

Q138·MathematicsSingle correctJEE Main 2023

If

A=\begin{bmatrix}1&5\\\lambda&10\end{bmatrix}

,

A^{-1}=\alpha A+\beta I

and

\alpha+\beta=-2

, then

4\alpha^{2}+\beta^{2}+\lambda^{2}

is equal to

(A)12
(B)10
(C)19
(D)14

Q139·MathematicsSingle correctJEE Main 2023

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:

(A)23
(B)28
(C)25
(D)20

Q140·MathematicsSingle correctJEE Main 2023

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

(A)18
(B)40
(C)24
(D)22

Q141·MathematicsSingle correctJEE Main 2023

Let

A=\begin{bmatrix} p & q \\ r & s \end{bmatrix}

be such that

A^{2}=bI

, where

b\in\mathbb{R}\setminus\{0\},\,I

is the identity matrix and all entries

a_{ij}

of

A

are non-zero. If

|A|=a,\,a\ne 0

then

3a^{2}+4b^{2}

equals:

(A) $7$
(B) $5$
(C) $8$
(D) $4$

Q142·MathematicsSingle correctJEE Main 2023

For the system of equations

x+y+z=6

,

x+2y+\alpha z=10

,

x+3y+5z=\beta

, which one of the following is NOT true?

(A)System has a unique solution for $\alpha=3,\beta\neq 14$ .
(B)System has no solution for $\alpha=3,\beta=24$ .
(C)System has a unique solution for $\alpha=-3,\beta=14$ .
(D)System has infinitely many solutions for $\alpha=3,\beta=14$ .

Q143·MathematicsSingle correctJEE Main 2023

Let

S

denote the set of all real values of

\lambda

such that the system of equations

\lambda x+y+z=1

,

x+\lambda y+z=1

,

x+y+\lambda z=1

is inconsistent, then

\sum_{\lambda\in S}\left(|\lambda|^2+|\lambda|\right)

is equal to

(A) $4$
(B) $12$
(C) $6$
(D) $2$

Q144·MathematicsSingle correctJEE Main 2023

For the system of linear equations

\alpha x+y+z=1

,

x+\alpha y+z=1

,

x+y+\alpha z=\beta

, which one of the following statements is NOT correct?

(A)It has infinitely many solutions if $\alpha=2$ and $\beta=-1$
(B)It has no solution if $\alpha=-2$ and $\beta=1$
(C) $x+y+z=\dfrac34$ if $\alpha=2$ and $\beta=1$
(D)It has infinitely many solutions if $\alpha=1$ and $\beta=1$

Q145·MathematicsSingle correctJEE Main 2023

Let

f(x)=\begin{vmatrix}1+\sin^2 x & \cos^2 x & \sin 2x\\ \sin^2 x & 1+\cos^2 x & \sin 2x\\ \sin^2 x & \cos^2 x & 1+\sin 2x\end{vmatrix}

,

x\in\left[\frac{\pi}{6},\frac{\pi}{3}\right]

. If

\alpha

and

\beta

respectively are the maximum and the minimum values of

f

, then

(A) $\alpha^2+\beta^2=\frac{9}{2}$
(B) $\beta^2-2\sqrt{\alpha}=\frac{19}{4}$
(C) $\alpha^2-\beta^2=4\sqrt{3}$
(D) $\beta^2+2\sqrt{\alpha}=\frac{19}{4}$

Q146·MathematicsSingle correctJEE Main 2023

If

A=\dfrac12\begin{bmatrix}1&\sqrt3\\-\sqrt3&1\end{bmatrix}

, then:

(A) $A^{30}+A^{25}+A=I$
(B) $A^{30}=A^{25}$
(C) $A^{30}+A^{25}-A=I$
(D) $A^{30}-A^{25}=2I$

Q147·MathematicsNumericalJEE Main 2023

Let

A

be a

n\times n

matrix such that

|A|=2

. If the determinant of the matrix

\mathrm{Adj}\left(2\cdot\mathrm{Adj}(2A^{-1})\right)

is

2^{84}

, then

n

is equal to

Q148·MathematicsSingle correctJEE Main 2023

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 ?

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

Q149·MathematicsSingle correctJEE Main 2023

Let

A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}

. Then the sum of the diagonal elements of the matrix

(A + I)^{11}

is equal to

(A) $6144$
(B) $2050$
(C) $4097$
(D) $4094$

Q150·MathematicsSingle correctJEE Main 2023

If a point

P(\alpha,\beta,\gamma)

satisfying

(\alpha\ \beta\ \gamma)\begin{pmatrix}2&10&8\\9&3&8\\8&4&8\end{pmatrix}=(0\ 0\ 0)

lies on the plane

2x+4y+3z=5

, then

6\alpha+9\beta+7\gamma

is equal to:

(A) $-1$
(B) $\dfrac{11}{5}$
(C) $\dfrac54$
(D) $11$

Q151·MathematicsSingle correctJEE Main 2023

If

P

is a

3 \times 3

real matrix such that

P^T=aP + (a - 1)I

, where

a > 1

, then:

(A) $|\mathrm{Adj}\,P|=\dfrac{1}{2}$
(B) $|\mathrm{Adj}\,P|=1$
(C) $P$ is a singular matrix
(D) $|\mathrm{Adj}\,P| > 1$

Q152·MathematicsSingle correctJEE Main 2023

Let

A=\begin{pmatrix}m&n\\p&q\end{pmatrix}

,

d=|A|\neq0

and

|A-d(\mathrm{Adj}\,A)|=0

. Then

(A) $1+d^2=m^2+q^2$
(B) $1+d^2=(m+q)^2$
(C) $(1+d)^2=m^2+q^2$
(D) $(1+d)^2=(m+q)^2$

Q153·MathematicsSingle correctJEE Main 2023

Let the system of linear equations

x+y+kz=2

,

2x+3y-z=1

,

3x+4y+2z=k

have infinitely many solutions. Then the system

(k+1)x+(2k-1)y=7

,

(2k+1)x+(k+5)y=10

has:

(A)infinitely many solutions
(B)unique solution satisfying $x-y=1$
(C)unique solution satisfying $x+y=1$
(D)no solution

Q154·MathematicsSingle correctJEE Main 2023

For

\alpha, \beta \in \mathbb{R}

, suppose the system of linear equations

x - y + z=1

,

2x + 2y + \alpha z=8

,

3x - y + 4z=\beta

has infinitely many solutions. Then

\alpha

and

\beta

are the roots of:

(A) $x^2 + 14x + 24=0$
(B) $x^2 + 18x + 56=0$
(C) $x^2 - 18x + 56=0$
(D) $x^2 - 10x + 16=0$

Q155·MathematicsNumericalJEE Main 2023

Let

A

be a symmetric matrix such that

|A|=2

and

\begin{bmatrix} 2 & 1 \\ 3 & \tfrac32 \end{bmatrix}A=\begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}

. If the sum of the diagonal elements of

A

is

s

, then

\dfrac{\beta s}{\alpha^2}

is equal to _____.

Q156·MathematicsSingle correctJEE Main 2023

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

(A) $\beta = -8$
(B) $\beta = 8$
(C) $\alpha = 4$
(D) $\alpha = 1$

Q157·MathematicsSingle correctJEE Main 2023

Consider the following system of equations

\alpha x + 2y + z = 1

,

2\alpha x + 3y + z = 1

,

3x + \alpha y + 2z = \beta

for some

\beta \in \mathbb{R}

. Then which of the following is NOT correct.

(A)It has a solution if $\alpha = -1$ and $\beta \neq 2$
(B)It has a solution for all $\alpha \neq -1$ and $\beta = 2$
(C)It has no solution for $\alpha = 3$ and for all $\beta \neq 2$
(D)It has no solution for $\alpha = -1$ and for all $\beta \in \mathbb{R}$

Q158·MathematicsSingle correctJEE Main 2023

The set of all values of

t\in\mathbb{R}

, for which the matrix

\begin{bmatrix} 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{bmatrix}

is invertible, is:

(A) $\mathbb{R}$
(B) $\{k\pi+\dfrac{\pi}{4},\,k\in\mathbb{Z}\}$
(C) $\{k\pi,\,k\in\mathbb{Z}\}$
(D) $\{(2k+1)\dfrac{\pi}{2},\,k\in\mathbb{Z}\}$

Q159·MathematicsSingle correctJEE Main 2023

Let

A = \begin{bmatrix} \dfrac{1}{\sqrt{10}} & \dfrac{3}{\sqrt{10}} \\ \dfrac{-3}{\sqrt{10}} & \dfrac{1}{\sqrt{10}} \end{bmatrix}

and

B = \begin{bmatrix} 1 & -i \\ 0 & 1 \end{bmatrix}

, where

i = \sqrt{-1}

. If

M = A^{T}BA

, then the inverse of the matrix

AM^{2023}A^{T}

is

(A) $\begin{bmatrix} 1 & 0 \\ -2023i & 1 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & -2023i \\ 0 & 1 \end{bmatrix}$
(C) $\begin{bmatrix} 1 & 0 \\ 2023i & 1 \end{bmatrix}$
(D) $\begin{bmatrix} 1 & 2023i \\ 0 & 1 \end{bmatrix}$

Q160·MathematicsSingle correctJEE Main 2023

Let

x,y,z>1

and

A=\begin{bmatrix}1 & \log_x y & \log_x z\\ \log_y x & 2 & \log_y z\\ \log_z x & \log_z y & 3\end{bmatrix}

. Then

|\mathrm{adj}(\mathrm{adj}\,A^2)|

is equal to:

(A) $2^8$
(B) $4^9$
(C) $6^4$
(D) $2^4$

Q161·MathematicsSingle correctJEE Main 2023

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)

A^{13}B^{26}-B^{26}A^{13}

is symmetric (S2)

A^{26}C^{13}-C^{13}A^{26}

is symmetric Then,

(A)Only S2 is true
(B)Both S1 and S2 are false
(C)Only S1 is true
(D)Both S1 and S2 are true

Q162·MathematicsSingle correctJEE Main 2023

Let

S_1

and

S_2

be respectively the sets of all

a\in\mathbb{R}-\{0\}

for which the system of linear equations

ax+2ay-3az=1

;

(2a+1)x+(2a+3)y+(a+1)z=2

;

(3a+5)x+(a+5)y+(a+2)z=3

has unique solution and infinitely many solutions. Then:

(A) $S_1$ is an infinite set and $n(S_2)=2$
(B) $S_2$ is an infinite set and $n(S_1)=2$
(C) $S_1=\Phi$ and $S_2=\mathbb{R}-\{0\}$
(D) $S_1=\mathbb{R}-\{0\}$ and $S_2=\Phi$

Q163·MathematicsSingle correctJEE Main 2023

If the system of equations

x+2y+3z=3

,

4x+3y-4z=4

,

8x+4y+\lambda z=9+\mu

has infinitely many solutions, then the ordered pair

(\lambda,\mu)

is equal to:

(A) $\left(-\dfrac{72}{5},\dfrac{21}{5}\right)$
(B) $\left(-\dfrac{72}{5},-\dfrac{21}{5}\right)$
(C) $\left(\dfrac{72}{5},-\dfrac{21}{5}\right)$
(D) $\left(\dfrac{72}{5},\dfrac{21}{5}\right)$

Q164·MathematicsSingle correctJEE Main 2023

Let

\alpha

be a root of the equation

(a-c)x^2+(b-a)x+(c-b)=0

where

a,b,c

are distinct real numbers such that the matrix

\begin{bmatrix}\alpha^2 & \alpha & 1\\ 1 & 1 & 1\\ a & b & c\end{bmatrix}

is singular. Then, the value of

\dfrac{(a-c)^2}{(b-a)(c-b)}+\dfrac{(b-a)^2}{(a-c)(c-b)}+\dfrac{(c-b)^2}{(a-c)(b-a)}

is

(A) $12$
(B) $9$
(C) $3$
(D) $6$

Q165·MathematicsSingle correctJEE Main 2023

Let A be a

3\times3

matrix such that

|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\,A))|=12^4

. Then

|A^{-1}\,\mathrm{adj}\,A|

is equal to

(A) $\sqrt6$
(B) $2\sqrt3$
(C) $12$
(D) $1$

Q166·MathematicsSingle correctJEE Main 2023

If

A

and

B

are two non-zero

n\times n

matrices such that

A^2+B=A^2 B

, then

(A) $A^2=I$ or $B=I$
(B) $A^2 B=I$
(C) $AB=I$
(D) $A^2 B=BA^2$

Q167·MathematicsMultiple correctJEE Advanced 2022

Let

|M|

denote the determinant of a square matrix

M

. Let

g : \left[0, \frac{\pi}{2}\right] \rightarrow R

be the function defined by

g(\theta) = \sqrt{f(\theta) - 1} + \sqrt{f\left(\frac{\pi}{2} - \theta\right) - 1}

where

f(\theta) = \frac{1}{2}\begin{vmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{vmatrix} + \begin{vmatrix} \sin\pi & \cos\left(\theta + \frac{\pi}{4}\right) & \tan\left(\theta - \frac{\pi}{4}\right) \\ \sin\left(\theta - \frac{\pi}{4}\right) & -\cos\frac{\pi}{2} & \log_{e}\left(\frac{4}{\pi}\right) \\ \cot\left(\theta + \frac{\pi}{4}\right) & \log_{e}\left(\frac{\pi}{4}\right) & \tan\pi \end{vmatrix}

. Let

p(x)

be a quadratic polynomial whose roots are the maximum and minimum values of the function

g(\theta)

and

p(2) = 2 - \sqrt{2}

. Then, which of the following is/are TRUE?

(A) $p\left(\frac{3 + \sqrt{2}}{4}\right) < 0$
(B) $p\left(\frac{1 + 3\sqrt{2}}{4}\right) > 0$
(C) $p\left(\frac{5\sqrt{2} - 1}{4}\right) > 0$
(D) $p\left(\frac{5 - \sqrt{2}}{4}\right) < 0$

Q168·MathematicsSingle correctJEE Advanced 2022

If M =

\begin{pmatrix} \frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2} \end{pmatrix}

, then which of the following matrices is equal to M

^{2022}

?

(A) $\begin{pmatrix} 3034 & 3033 \\ -3033 & -3032 \end{pmatrix}$
(B) $\begin{pmatrix} 3034 & -3033 \\ 3033 & -3032 \end{pmatrix}$
(C) $\begin{pmatrix} 3033 & 3032 \\ -3032 & -3031 \end{pmatrix}$
(D) $\begin{pmatrix} 3032 & 3031 \\ -3031 & -3030 \end{pmatrix}$

Q169·MathematicsIntegerJEE Advanced 2022

Let β be real number. Consider the matrix A =

\begin{pmatrix} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{pmatrix}

. If

A^{7} - \left(\beta - 1\right)A^{6} - \beta A^{5}

is a singular matrix, then the value of 9β is ________.

Q170·MathematicsSingle correctJEE Advanced 2022

Let

p

,

q

,

r

be nonzero real numbers that are, respectively, the

10^{\text{th}}

,

100^{\text{th}}

and

1000^{\text{th}}

terms of a harmonic progression. Consider the system of linear equation

x + y + z = 1

10x + 100y + 1000z = 0

qrx + pry + pqz = 0

The correct option is:

List-I	List-II
I.If $\frac{q}{r} = 10$ , then the system of linear equations has	P. $x = 0$ , $y = \frac{10}{9}$ , $z = -\frac{1}{9}$
II.If $\frac{p}{r} \neq 100$ , then the system of linear equations has	Q. $x = \frac{10}{9}$ , $y = -\frac{1}{9}$ , $z = 0$ as a solution
III.If $\frac{p}{q} \neq 10$ , then the system of linear equation has	R.infinitely many solutions
IV.If $\frac{p}{q} = 10$ , then the system of linear equations has	S.no solution
	T.at least one solution

(A)(I) → (T); (II) → (R); (III) → (S); (IV) → (T)
(B)(I) → (Q); (II) → (S); (III) → (S); (IV) → (R)
(C)(I) → (Q); (II) → (R); (III) → (P); (IV) → (R)
(D)(I) → (T); (II) → (S); (III) → (P); (IV) → (T)

Q171·MathematicsSingle correctJEE Main 2022

If the system of equations

x + y + z = 6

2x + 5y + \alpha z = \beta

x + 2y + 3z = 14

has infinitely many solutions, then

\alpha + \beta

is equal to :

(A) $8$
(B) $36$
(C) $44$
(D) $48$

Q172·MathematicsSingle correctJEE Main 2022

Let A and B be two

3 \times 3

non-zero real matrices such that AB is a zero matrix. Then

(A)The system of linear equations $AX = 0$ has a unique solution
(B)The system of linear equations $AX = 0$ has infinitely many solutions
(C)B is an invertible matrix
(D)adj (A) is an invertible matrix

Q173·MathematicsNumericalJEE Main 2022

Let p and p + 2 be prime numbers and let

\Delta = \begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \end{vmatrix}

Then the sum of the maximum values of

\alpha

and

\beta

, such that

p^{\alpha}

and

(p+2)^{\beta}

divide

\Delta

, is __________.

Q174·MathematicsNumericalJEE Main 2022

Let

x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}

and

A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}

. For

k \in \mathbb{N}

, if

X' A^k X = 33

, then k is equal to:

Q175·MathematicsSingle correctJEE Main 2022

Which of the following matrices can NOT be obtained from the matrix

\begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix}

by a single elementary row operation?

(A) $\begin{bmatrix} 0 & 1 \\ 1 & -1 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}$
(C) $\begin{bmatrix} -1 & 2 \\ -2 & 7 \end{bmatrix}$
(D) $\begin{bmatrix} -1 & 2 \\ -1 & 3 \end{bmatrix}$

Q176·MathematicsSingle correctJEE Main 2022

Let A and B be any two

3 \times 3

symmetric and skew symmetric matrices respectively. Then which of the following is <b>NOT</b> true?

(A) $A^{4}-B^{4}$ is a symmetric matrix
(B) $AB-BA$ is a symmetric matrix
(C) $B^{5}-A^{5}$ is a skew-symmetric matrix
(D) $AB+BA$ is a skew-symmetric matrix

Q177·MathematicsSingle correctJEE Main 2022

Let the matrix

A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}

and the matrix

B_{0} = A^{49} + 2A^{98}

. If

B_{n} = Adj(B_{n-1})

for all

n \geq 1

, then

\det(B_{4})

is equal to :

(A) $3^{28}$
(B) $3^{30}$
(C) $3^{32}$
(D) $3^{36}$

Q178·MathematicsNumericalJEE Main 2022

Let

A = \begin{bmatrix} 1 & -1 \\ 2 & \alpha \end{bmatrix}

and

B = \begin{bmatrix} \beta & 1 \\ 1 & 0 \end{bmatrix}

,

\alpha, \beta \in R

. Let

\alpha_{1}

be the value of

\alpha

which satisfies

(A + B)^{2} = A^{2} + \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}

and

\alpha_{2}

be the value of

\alpha

which satisfies

(A + B)^{2} = B^{2}

. Then

\left| \alpha_{1} - \alpha_{2} \right|

is equal to _______.

Q179·MathematicsNumericalJEE Main 2022

Consider a matrix

A = \begin{bmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{bmatrix}

, where

\alpha, \beta, \gamma

are three distinct natural numbers. If

\frac{\det\left(\operatorname{adj}\left(\operatorname{adj}\left(\operatorname{adj}\left(\operatorname{adj}A\right)\right)\right)\right)}{\left(\alpha-\beta\right)^{16}\left(\beta-\gamma\right)^{16}\left(\gamma-\alpha\right)^{16}} = 2^{32} \times 3^{16}

, then the number of such 3 – tuples

(\alpha, \beta, \gamma)

is ________.

Q180·MathematicsNumericalJEE Main 2022

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 __________.

Q181·MathematicsSingle correctJEE Main 2022

Let

A = \begin{pmatrix} 4 & -2 \\ \alpha & \beta \end{pmatrix}

. If

A^{2}+\gamma A + 18I = O

, then det (A) is equal to ______.

(A) $-18$
(B) $18$
(C) $-50$
(D) $50$

Q182·MathematicsSingle correctJEE Main 2022

Let

A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}

. Let

\alpha, \beta \in \mathbb{R}

be such that

\alpha A^2 + \beta A = 2I

. Then

\alpha + \beta

is equal to -

(A)-10
(B)-6
(C)6
(D)10

Q183·MathematicsNumericalJEE Main 2022

The number of matrices

A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}

, where

a,b,c,d\in\{-1,0,1,2,3,\ldots\ldots,10\}

, such that

A=A^{-1}

, is______.

Q184·MathematicsSingle correctJEE Main 2022

Let

A=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}

and

B=\begin{bmatrix} 9^{2} & -10^{2} & 11^{2} \\ 12^{2} & 13^{2} & -14^{2} \\ -15^{2} & 16^{2} & 17^{2} \end{bmatrix}

, then the value of

A'BA

is:

(A)1224
(B)1042
(C)540
(D)539

Q185·MathematicsSingle correctJEE Main 2022

Let A be a 2 × 2 matrix with det (A) = –1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :

(A)–1
(B)2
(C)1
(D) $-\sqrt{2}$

Q186·MathematicsSingle correctJEE Main 2022

If the system of linear equations. 8x + y + 4z = –2 x + y + z = 0 λx – 3y = μ has infinitely many solutions, then the distance of the point

\left(\lambda,\mu,-\dfrac{1}{2}\right)

from the plane 8x + y + 4z + 2 = 0 is :

(A) $3\sqrt{5}$
(B)4
(C) $\dfrac{26}{9}$
(D) $\dfrac{10}{3}$

Q187·MathematicsNumericalJEE Main 2022

Let

A = \begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{bmatrix}

and

B = A - I

. If

\omega = \frac{\sqrt{3}i - 1}{2}

, then the number of elements in the set

\{n \in \{1, 2, \ldots, 100\} : A^{n} + (\omega B)^{n} = A + B\}

is equal to ________.

Q188·MathematicsSingle correctJEE Main 2022

The number of

\theta \in (0, 4\pi)

for which the system of linear equations

3 (\sin 3\theta) x - y + z = 2

3 (\cos 2\theta) x + 4y + 3z = 3

6x + 7y + 7z = 9

has no solution is :

(A)6
(B)7
(C)8
(D)9

Q189·MathematicsNumericalJEE Main 2022

Let

M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}

, where

\alpha

is a non-zero real number an

N = \sum_{k=1}^{49} M^{2k}

. If

(I - M^2)N = -2I

, then the positive integral value of

\alpha

is ____.

Q190·MathematicsSingle correctJEE Main 2022

Let

A = [a_{ij}]

be a square matrix of order 3 such that

a_{ij} = 2^{\,j-i}

, for all i, j = 1, 2, 3. Then, the matrix

A^{2} + A^{3} + \ldots + A^{10}

is equal to :

(A) $\left(\frac{3^{10} - 3}{2}\right)A$
(B) $\left(\frac{3^{10} - 1}{2}\right)A$
(C) $\left(\frac{3^{10} + 1}{2}\right)A$
(D) $\left(\frac{3^{10} + 3}{2}\right)A$

Q191·MathematicsSingle correctJEE Main 2022

Let

A = \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix}

. If

B = I - {}^{5}C_{1}\,(\mathrm{adj}A) + {}^{5}C_{2}\,(\mathrm{adj}A)^{2} - ... - {}^{5}C_{5}\,(\mathrm{adj}A)^{5}

, then the sum of all elements of the matrix B is:

(A) $-5$
(B) $-6$
(C) $-7$
(D) $-8$

Q192·MathematicsSingle correctJEE Main 2022

If the system of linear equations

2x + y - z = 7

x - 3y + 2z = 1

x + 4y + \delta z = k

, where δ, k ∈ R has infinitely many solutions, then δ + k is equal to:

(A)−3
(B)3
(C)6
(D)9

Q193·MathematicsNumericalJEE Main 2022

Let

A = \begin{pmatrix} 1+i & 1 \\ -i & 0 \end{pmatrix}

where

i = \sqrt{-1}

. Then, the number of elements in the set

\{n \in \{1, 2, \ldots, 100\} : A^n = A\}

is

Q194·MathematicsSingle correctJEE Main 2022

Let A be a matrix of order 3 × 3 and det (A) = 2. Then det (det (A) adj (5 adj (

\mathrm{A}^{3}

))) is equal to ____.

(A) $512 \times 10^{6}$
(B) $256 \times 10^{6}$
(C) $1024 \times 10^{6}$
(D) $256 \times 10^{11}$

Q195·MathematicsSingle correctJEE Main 2022

If the system of linear equations

2x + 3y - z = -2

x + y + z = 4

x - y + |\lambda| z = 4\lambda - 4

where

\lambda \in \mathbb{R},

has no solution, then

(A) $\lambda = 7$
(B) $\lambda = -7$
(C) $\lambda = 8$
(D) $\lambda^{2} = 1$

Q196·MathematicsNumericalJEE Main 2022

If the system of linear equations

2x - 3y = \gamma + 5

,

\alpha x + 5y = \beta + 1

, where

\alpha, \beta, \gamma \in \mathbf{R}

has infinitely many solutions, then the value of

\left|9\alpha + 3\beta + 5\gamma\right|

is equal to

Q197·MathematicsNumericalJEE Main 2022

The positive value of the determinant of the matrix A, whose

Adj\left(Adj(A)\right)=\begin{pmatrix} 14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14 \end{pmatrix}

, is __________.

Q198·MathematicsSingle correctJEE Main 2022

Let the system of linear equations

x + 2y + z = 2

,

\alpha x + 3y - z = \alpha

,

-\alpha x + y + 2z = -\alpha

be inconsistent. Then

\alpha

is equal to :

(A) $\frac{5}{2}$
(B) $-\frac{5}{2}$
(C) $\frac{7}{2}$
(D) $-\frac{7}{2}$

Q199·MathematicsSingle correctJEE Main 2022

Let

f(x)=\begin{vmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^{2} & ax & a \end{vmatrix}

,

a \in \mathbb{R}

. Then the sum of which the squares of all the values of a for

2f'(10)-f'(5)+100=0

is :

(A) $117$
(B) $106$
(C) $125$
(D) $136$

Q200·MathematicsSingle correctJEE Main 2022

Let A and B be two

3\times3

matrices such that

AB=I

and

|A|=\dfrac{1}{8}

then

\left|\mathrm{adj}(B\,\mathrm{adj}(2A))\right|

is equal to

(A) $16$
(B) $32$
(C) $64$
(D) $128$

Q201·MathematicsSingle correctJEE Main 2022

Let A be a

3 \times 3

invertible matrix. If

|adj (24A)| = adj(3adj(2A))|

, then

|A|^{2}

is equal to :

(A) $6^{6}$
(B) $2^{12}$
(C) $2^{6}$
(D)1

Q202·MathematicsSingle correctJEE Main 2022

If the system of equations αx + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β, has infinitely many solutions, then the ordered pair (α, β) is equal to :

(A)(1,–3)
(B)(–1, 3)
(C)(1, 3)
(D)(–1, –3)

Q203·MathematicsSingle correctJEE Main 2022

The ordered pair (a, b), for which the system of linear equations

3x - 2y + z = b

5x - 8y + 9z = 3

2x + y + az = -1

has no solution, is :

(A) $\left(3, \frac{1}{3}\right)$
(B) $\left(-3, \frac{1}{3}\right)$
(C) $\left(-3, -\frac{1}{3}\right)$
(D) $\left(3, -\frac{1}{3}\right)$

Q204·MathematicsNumericalJEE Main 2022

Let

X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}

,

Y=\alpha I+\beta X+\gamma X^{2}

and

Z=\alpha^{2}I-\alpha\beta X+\left(\beta^{2}-\alpha\gamma\right)X^{2}

,

\alpha,\beta,\gamma\in\mathbb{R}

. If

Y^{-1}=\begin{bmatrix} \frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{-2}{5} \\ 0 & 0 & \frac{1}{5} \end{bmatrix}

, then

(\alpha-\beta+\gamma)^{2}

is equal to ______ .

Q205·MathematicsSingle correctJEE Main 2022

Let

A = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}

. If M and N are two matrices given by

M = \sum_{k=1}^{10} A^{2k}

and

N = \sum_{k=1}^{10} A^{2k-1}

then

MN^2

is

(A)a non-identity symmetric matrix
(B)a skew-symmetric matrix
(C)neither symmetric nor skew-symmetric matrix
(D)an identify matrix

Q206·MathematicsSingle correctJEE Main 2022

The system of equations

-kx + 3y - 14z = 25

-15x + 4y - kz = 3

-4x + y + 3z = 4

is consistent for all k in the set

(A) $R$
(B) $R - \{-11,13\}$
(C) $R - \{13\}$
(D) $R - \{-11,11\}$

Q207·MathematicsNumericalJEE Main 2022

Let

A = \begin{pmatrix} 2 & -2 \\ 1 & -1 \end{pmatrix}

and

B = \begin{pmatrix} -1 & 2 \\ -1 & 2 \end{pmatrix}

. Then the number of elements in the set

\{(n, m) : n, m \in \{1, 2, ....., 10\}

and

nA^{n} + mB^{m} = I\}

is ______

Q208·MathematicsSingle correctJEE Main 2022

Let A be a

3 \times 3

real matrix such that

A\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}

;

A\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}

and

A\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}

. If

X = (x_1, x_2, x_3)^T

and I is an identity matrix of order 3, then the system

(A - 2I)X = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix}

has

(A)no solution
(B)infinitely many solutions
(C)unique solution
(D)exactly two solutions

Q209·MathematicsSingle correctJEE Main 2022

Let the system of linear equations

x + y + \alpha z = 2

3x + y + z = 4

x + 2z = 1

have a unique solution

(x^*, y^*, z^*)

. If

(\alpha, x^*)

,

(y^*, \alpha)

and

(x^*, -y^*)

are collinear points, then the sum of absolute values of all possible values of

\alpha

is :

(A)4
(B)3
(C)2
(D)1

Q210·MathematicsSingle correctJEE Main 2022

Let

S=\{\sqrt{n}:1\le n\le 50\ \text{and n is odd}\}

. Let

a\in S

and

A=\begin{bmatrix}1&0&a\\-1&1&0\\-a&0&1\end{bmatrix}

. If

\sum_{a\in S}\det(\mathrm{adj}A)=100\lambda

, then

\lambda

is equal to

(A)218
(B)221
(C)663
(D)1717

Q211·MathematicsNumericalJEE Main 2022

Let

S = \left\{\begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix}; a, b \in \{1, 2, 3, \ldots 100\}\right\}

and let

T_n = \{A \in S : A^{n(n+1)} = I\}

. Then the number of elements in

\bigcap_{n=1}^{100} T_n

is ______.

Q212·MathematicsSingle correctJEE Main 2022

The number of values of

\alpha

for which the system of equations :

x + y + z = \alpha

\alpha x + 2\alpha y + 3z = -1

x + 3\alpha y + 5z = 4

is inconsistent, is

(A)0
(B)1
(C)2
(D)3

Q213·MathematicsMultiple correctJEE Advanced 2021

For any

3 \times 3

matrix M, let

\left|M\right|

denote the determinant of M. Let I be the

3 \times 3

identity matrix. Let E and F be two

3 \times 3

matrices such that

(I - EF)

is invertible. If

G = (I - EF)^{-1}

, then which of the following statements is (are) TRUE ?

(A) $\left|FE\right| = \left|I - FE\right|\left|FGE\right|$
(B) $(I - FE)(I + FGE) = I$
(C) $EFG = GEF$
(D) $\left(I - FE\right)\left(I - FGE\right) = I$

Q214·MathematicsMultiple correctJEE Advanced 2021

For any

3 \times 3

matrix M, let

\left|M\right|

denote the determinant of M. Let

E = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}

,

P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}

and

F = \begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}

If Q is a nonsingular matrix of order

3 \times 3

, then which of the following statements is (are) TRUE ?

(A) $F = PEP$ and $P^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
(B) $\left|EQ + PFQ^{-1}\right| = \left|EQ\right| + \left|PFQ^{-1}\right|$
(C) $\left|(EF)^3\right| > \left|EF\right|^2$
(D)Sum of the diagonal entries of $P^{-1}EP + F$ is equal to the sum of diagonal entries of $E + P^{-1}FP$

Q215·MathematicsNumericalJEE Advanced 2021

Let

\alpha

,

\beta

and

\gamma

be real numbers such that the system of linear equations

x + 2y + 3z = \alpha

4x + 5y + 6z = \beta

7x + 8y + 9z = \gamma - 1

is consistent. Let

|M|

represent the determinant of the matrix

M = \begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}

Let P be the plane containing all those

(\alpha, \beta, \gamma)

for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P. The value of

\left|M\right|

is ________.

Q216·MathematicsSingle correctJEE Main 2021

Let

J_{n,m} = \int_{0}^{\frac{1}{2}} \frac{x^{n}}{x^{m} - 1}dx

, ∀ n > m and n, m ∈ N . Consider a matrix

A = [a_{ij}]_{3 \times 3}

where

a_{ij} = \begin{cases} J_{6+i,3} - J_{i+3,3}, & i \le j \\ 0, & i > j \end{cases}

. Then

\left|\mathrm{adj}A^{-1}\right|

is :

(A) $(15)^{2} \times 2^{42}$
(B) $(15)^{2} \times 2^{34}$
(C) $(105)^{2} \times 2^{38}$
(D) $(105)^{2} \times 2^{36}$

Q217·MathematicsSingle correctJEE Main 2021

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 ∈ ℝ for which the system is inconsistent and

S_{2}

be the set of all a ∈ 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

(A) $n(S_{1}) = 2$ , $n(S_{2}) = 2$
(B) $n(S_{1}) = 1$ , $n(S_{2}) = 0$
(C) $n(S_{1}) = 2$ , $n(S_{2}) = 0$
(D) $n(S_{1}) = 0$ , $n(S_{2}) = 2$

Q218·MathematicsSingle correctJEE Main 2021

If the following system of linear equations

2x + y + z = 5

x - y + z = 3

x + y + az = b

has no solution, then :

(A) $a = -\frac{1}{3},\ b \ne \frac{7}{3}$
(B) $a \ne \frac{1}{3},\ b = \frac{7}{3}$
(C) $a \ne -\frac{1}{3},\ b = \frac{7}{3}$
(D) $a = \frac{1}{3},\ b \ne \frac{7}{3}$

Q219·MathematicsSingle correctJEE Main 2021

If

\alpha + \beta + \gamma = 2\pi

, then the system of equations

x + (\cos \gamma)y + (\cos \beta)z = 0

(\cos \gamma)x + y + (\cos \alpha)z = 0

(\cos \beta)x + (\cos \alpha)y + z = 0

has :

(A)no solution
(B)infinitely many solution
(C)exactly two solutions
(D)a unique solution

Q220·MathematicsNumericalJEE Main 2021

The number of elements in the set

\left\{ A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\} \text{ and } (I - A)^{3} = I - A^{3} \right\}

, where I is

2 \times 2

identity matrix, is :

Q221·MathematicsSingle correctJEE Main 2021

Let [λ] be the greatest integer less than or equal to λ. The set of all values of λ for which the system of linear equations x + y + z = 4, 3x + 2y+ 5z = 3, 9x + 4y + (28+ [λ])z = [λ] has a solution is:

(A)ℝ
(B) $(-\infty, -9) \cup (-9, \infty)$
(C)[−9, −8)
(D) $(-\infty, -9) \cup [-8, \infty)$

Q222·MathematicsSingle correctJEE Main 2021

Let

A = \begin{pmatrix} [x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4] \end{pmatrix}

, where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval:

(A)[68, 69)
(B)[62, 63)
(C)[65, 66)
(D)[60, 61)

Q223·MathematicsNumericalJEE Main 2021

If the system of linear equations

2x + y - z = 3

x - y - z = \alpha

3x + 3y + \beta z = 3

has infinitely many solution, then

\alpha + \beta - \alpha\beta

is equal to __________.

Q224·MathematicsSingle correctJEE Main 2021

If the matrix

A = \begin{pmatrix} 0 & 2 \\ K & -1 \end{pmatrix}

satisfies

A(A^3 + 3I) = 2I

, then the value of K is :

(A) $\frac{1}{2}$
(B) $-\frac{1}{2}$
(C) $-1$
(D) $1$

Q225·MathematicsSingle correctJEE Main 2021

Let

A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{pmatrix}

. Then

A^{2025} - A^{2020}

is equal to :

(A) $A^{6} - A$
(B) $A^{5}$
(C) $A^{5} - A$
(D) $A^{6}$

Q226·MathematicsNumericalJEE Main 2021

Let A be a

3 \times 3

real matrix. If

\det(2\mathrm{Adj}(2\ \mathrm{Adj}(\mathrm{Adj}(2A)))) = 2^{41}

, then the value of

\det(A^{2})

equal ________.

Q227·MathematicsSingle correctJEE Main 2021

Two fair dice are thrown. The numbers on them are taken as

\lambda

and

\mu

, and a system of linear equations

x + y + z = 5

x + 2y + 3z = \mu

x + 3y + \lambda z = 1

is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then :

(A) $p = \frac{1}{6}$ and $q = \frac{1}{36}$
(B) $p = \frac{5}{6}$ and $q = \frac{5}{36}$
(C) $p = \frac{5}{6}$ and $q = \frac{1}{36}$
(D) $p = \frac{1}{6}$ and $q = \frac{5}{36}$

Q228·MathematicsSingle correctJEE Main 2021

Let

\theta \in \left(0, \frac{\pi}{2}\right)

. If the system of linear equations

(1 + \cos^2\theta)x + \sin^2\theta\, y + 4\sin 3\theta\, z = 0

\cos^2\theta\, x + (1 + \sin^2\theta)\, y + 4\sin 3\theta\, z = 0

\cos^2\theta\, x + \sin^2\theta\, y + (1 + 4\sin 3\theta)\, z = 0

has a non-trivial solution, then the value of

\theta

is :

(A) $\frac{4\pi}{9}$
(B) $\frac{7\pi}{18}$
(C) $\frac{\pi}{18}$
(D) $\frac{5\pi}{18}$

Q229·MathematicsNumericalJEE Main 2021

Let

f\left(x\right)=\begin{vmatrix}\sin^2 x & -2+\cos^2 x & \cos 2x \\ 2+\sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 x & \cos^2 x & 1+\cos 2x\end{vmatrix}

,

x \in \left[0,\pi\right]

. Then the maximum value of

f\left(x\right)

is equal to ____.

Q230·MathematicsSingle correctJEE Main 2021

Let A and B be two 3 x 3 real matrices such that

(A^2 - B^2)

is invertible matrix. If

A^5 = B^5

and

A^3B^2 = A^2B^3

, then the value of the determinant of the matrix

A^3 + B^3

is equal to :

(A)2
(B)1
(C)0
(D)4

Q231·MathematicsNumericalJEE Main 2021

For real numbers

\alpha

and

\beta

, consider the following system of linear equations:

x+y-z=2

,

x+2y+\alpha z=1

,

2x-y+z=\beta

. If the system has infinite solutions, then

\alpha+\beta

is equal to ____.

Q232·MathematicsSingle correctJEE Main 2021

Let

A = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}

. If

A^{-1} = \alpha I + \beta A

,

\alpha, \beta \in R

. I is a 2 x 2 identify matrix, then

4\left(\alpha - \beta\right)

is equal to :

(A)2
(B)5
(C)4
(D) $\frac{8}{3}$

Q233·MathematicsSingle correctJEE Main 2021

If

P=\begin{bmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{bmatrix}

, then

P^{50}

is :

(A) $\begin{bmatrix} 1 & 0 \\ 25 & 1 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & 25 \\ 0 & 1 \end{bmatrix}$
(C) $\begin{bmatrix} 1 & 0 \\ 50 & 1 \end{bmatrix}$
(D) $\begin{bmatrix} 1 & 50 \\ 0 & 1 \end{bmatrix}$

Q234·MathematicsNumericalJEE Main 2021

Let

M = \left\{ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a,b,c,d \in \left\{ \pm 3, \pm 2, \pm 1, 0 \right\} \right\}

. Define

f : M \rightarrow Z

, as

f\left( A \right) = \det\left( A \right)

, for all

A \in M

, where

Z

is set of all integers. Then the number of

A \in M

such that

f\left( A \right) = 15

is equal to...........

Q235·MathematicsSingle correctJEE Main 2021

The number of distinct real roots of

\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0

in the interval

-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}

is:

(A)2
(B)4
(C)1
(D)3

Q236·MathematicsSingle correctJEE Main 2021

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 :

(A) $a \neq 3$ , b = 3
(B) $a \neq 3$ , b $\neq$ 13
(C)a = 3, b $\neq$ 13
(D)a = 3, b = 13

Q237·MathematicsSingle correctJEE Main 2021

The values of

\lambda

and

\mu

such that the system of equations

x + y + z = 6

,

3x + 5y + 5z = 26

,

x + 2y + \lambda z = \mu

has no solution, are :

(A) $\lambda = 2, \mu \neq 10$
(B) $\lambda = 3, \mu \neq 10$
(C) $\lambda = 3, \mu = 5$
(D) $\lambda \neq 2, \mu = 10$

Q238·MathematicsSingle correctJEE Main 2021

Let

A = \left|a_{ij}\right|

be a real matrix of order 3 x 3, such that

a_{11} + 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 :

(A) $9$
(B) $3$
(C) $1$
(D) $2$

Q239·MathematicsNumericalJEE Main 2021

Let

A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}

. Then the number of 3 x 3 matrices B with entries from the set

\{1,2,3,4,5\}

and satisfying AB = BA is...........

Q240·MathematicsSingle correctJEE Main 2021

The value of

k\in R

, for which the following system of linear equations

3x-y+4z=3,

x+2y-3z=-2,

6x+5y+kz=-3,

has infinitely many solutions, is :

(A) $3$
(B) $-3$
(C) $-5$
(D) $5$

Q241·MathematicsNumericalJEE Main 2021

Let

A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix}

and

B = 7A^{20} - 20A^{7} + 2I

, where

I

is an identity matrix of order 3 x 3. If

B = \left[ b_{ij} \right]

, then

b_{13}

is equal to..........

Q242·MathematicsSingle correctJEE Main 2021

Let

A = \begin{bmatrix} 2 & 3 \\ a & 0 \end{bmatrix}

,

a \in 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 :

(A)18
(B)36
(C)24
(D)45

Q243·MathematicsNumericalJEE Main 2021

Let a, b, c, d be in arithmetic progression with common difference

\lambda

. If

\begin{vmatrix} x + a - c & x + b & x + a \\ x - 1 & x + c & x + b \\ x - b + d & x + d & x + c \end{vmatrix} = 2

, then the value of

\lambda^2

is equal to.........

Q244·MathematicsNumericalJEE Main 2021

Let

A=\{a_{ij}\}

be a 3 x 3 matrix, where

a_{ij}=\begin{cases}(-1)^{j-i} & \text{if } i<j, \\ 2 & \text{if } i=j, \\ (-1)^{i+j} & \text{if } i>j,\end{cases}

then

\det\left(3\,\text{Adj}\left(2A^{-1}\right)\right)

is equal to…….

Q245·MathematicsSingle correctJEE Main 2021

The solutions of the equation

\begin{vmatrix} 1+\sin^{2}x & \sin^{2}x & \sin^{2}x \\ \cos^{2}x & 1+\cos^{2}x & \cos^{2}x \\ 4\sin 2x & 4\sin 2x & 1+4\sin 2x \end{vmatrix} = 0,\left(0 < x < \pi\right)

, are

(A) $\dfrac{\pi}{12}, \dfrac{\pi}{6}$
(B) $\dfrac{\pi}{6}, \dfrac{5\pi}{6}$
(C) $\dfrac{5\pi}{12}, \dfrac{7\pi}{12}$
(D) $\dfrac{7\pi}{12}, \dfrac{11\pi}{12}$

Q246·MathematicsSingle correctJEE Main 2021

Let

A + 2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}

and

2A - B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}

. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A)

-

Tr(B) has value equal to

(A) $1$
(B) $2$
(C) $0$
(D) $3$

Q247·MathematicsSingle correctJEE Main 2021

Let the system of linear equations 4x + λy + 2z = 0 2x − y + z = 0 μx + 2y + 3z = 0, λ, μ ∈ R. has a non-trivial solution. Then which of the following is true ?

(A)μ = 6, λ ∈ R
(B)λ = 2, μ ∈ R
(C)λ = 3, μ ∈ R
(D)μ = −6, λ ∈ R

Q248·MathematicsNumericalJEE Main 2021

Let I be an identity matrix of order 2 × 2 and

P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}

. Then the value of n ∈ N for which

P^{n} = 5I - 8P

is equal to ________ .

Q249·MathematicsSingle correctJEE Main 2021

Let

\alpha, \beta, \gamma

be the real roots of the equation,

x^{3} + ax^{2} + bx + c = 0

,

(a, b, c \in R

and

a, b \neq 0)

. If the system of equations (in, u, v, w) given by

\alpha u + \beta v + \gamma w = 0

,

\beta u + \gamma v + \alpha w = 0

;

\gamma u + \alpha v + \beta w = 0

has non-trivial solution, then the value of

\dfrac{a^{2}}{b}

is

(A) $5$
(B) $3$
(C) $1$
(D) $0$

Q250·MathematicsNumericalJEE Main 2021

If

1

,

\log_{10}(4^{x}-2)

and

\log_{10}\left(4^{x}+\frac{18}{5}\right)

are in arithmetic progression for a real number

x

, then the value of the determinant

\begin{vmatrix} 2\left(x-\frac{1}{2}\right) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0 \end{vmatrix}

is equal to :

Q251·MathematicsNumericalJEE Main 2021

Let

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

and

B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}

such that

AB = B

and

a + d = 2021

, then the value of

ad - bc

is equal to ________.

Q252·MathematicsSingle correctJEE Main 2021

If

A = \begin{pmatrix} 0 & \sin\alpha \\ \sin\alpha & 0 \end{pmatrix}

and

\det\left(A^2 - \frac{1}{2}I\right) = 0

, then a possible value of

\alpha

is

(A) $\frac{\pi}{2}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{4}$
(D) $\frac{\pi}{6}$

Q253·MathematicsSingle correctJEE Main 2021

If

x, y, z

are in arithmetic progression with common difference

d

,

x \neq 3d

, and the determinant of the matrix

\begin{bmatrix} 3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z \end{bmatrix}

is zero, then the value of

k^{2}

is

(A) $72$
(B) $12$
(C) $36$
(D) $6$

Q254·MathematicsNumericalJEE Main 2021

If

A = \begin{bmatrix} 2 & 3 \\ 0 & -1 \end{bmatrix}

, then the value of

\det(A^4) + \det\left(A^{10} - (\text{Adj}(2A))^{10}\right)

is _________ .

Q255·MathematicsSingle correctJEE Main 2021

The system of equations

kx + y + z = 1

,

x + ky + z = k

and

x + y + zk = k^2

has no solution if k is equal to :

(A)0
(B)1
(C) $-1$
(D) $-2$

Q256·MathematicsSingle correctJEE Main 2021

The maximum value of

f(x)=\begin{vmatrix} \sin^{2}x & 1+\cos^{2}x & \cos 2x \\ 1+\sin^{2}x & \cos^{2}x & \cos 2x \\ \sin^{2}x & \cos^{2}x & \sin 2x \end{vmatrix}

,

x \in R

is:

(A) $\sqrt{7}$
(B) $\frac{3}{4}$
(C) $\sqrt{5}$
(D) $5$

Q257·MathematicsNumericalJEE Main 2021

The total number of 3 × 3 matrices A having enteries from the set (0, 1, 2, 3) such that the sum of all the diagonal entries of

AA^{T}

is 9, is equal to _____.

Q258·MathematicsNumericalJEE Main 2021

Let

A=\begin{bmatrix} a_{1} \\ a_{2} \end{bmatrix}

and

B=\begin{bmatrix} b_{1} \\ b_{2} \end{bmatrix}

be two

2 \times 1

matrices with real entries such that

A = XB

, where

X=\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}

, and

k \in R

. If

a_{1}^{2}+a_{2}^{2}=\frac{2}{3}\left(b_{1}^{2}+b_{2}^{2}\right)

and

(k^{2}+1)b_{2}^{2} \neq -2b_{1}b_{2}

, then the value of k is ________.

Q259·MathematicsSingle correctJEE Main 2021

Let

A = \begin{bmatrix} i & -i \\ -i & i \end{bmatrix}

,

i = \sqrt{-1}

. Then, the system of linear equations

A^{8}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 64 \end{bmatrix}

has :

(A)A unique solution
(B)Infinitely many solutions
(C)No solution
(D)Exactly two solutions

Q260·MathematicsNumericalJEE Main 2021

Let

P = \begin{bmatrix} -30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}

and

A = \begin{bmatrix} 2 & 7 & \omega^{2} \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1 \end{bmatrix}

where

\omega = \frac{-1+i\sqrt{3}}{2}

, and I₃ 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 α is equal to ______ .

Q261·MathematicsNumericalJEE Main 2021

If the matrix

A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}

satisfies the equation

A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}

for some real numbers

\alpha

and

\beta

, then

\beta - \alpha

is equal to________________.

Q262·MathematicsSingle correctJEE Main 2021

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

(A)has a unique solution when 5a = 2b + c
(B)has infinite number of solutions when 5a = 2b +c
(C)has no solution for all a, b and c
(D)has a unique solution for all a, b and c

Q263·MathematicsSingle correctJEE Main 2021

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:

(A)6
(B)1
(C)4
(D)12

Q264·MathematicsSingle correctJEE Main 2021

The value of

\begin{vmatrix} (a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1 \end{vmatrix}

is

(A)−2
(B)(a+1) (a+2) (a+3)
(C)0
(D)(a+2) (a+3) (a+4)

Q265·MathematicsNumericalJEE Main 2021

If

A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}

and

(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}

, then

13(a^2 + b^2)

is equal to _______.

Q266·MathematicsNumericalJEE Main 2021

Let

A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}

, where

x

,

y

and

z

are real numbers such that

x + y + z > 0

and

xyz = 2

. If

A^2 = I_3

, then the value of

x^3 + y^3 + z^3

is ________.

Q267·MathematicsSingle correctJEE Main 2021

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

(A)does not have any solution
(B)has a unique solution
(C)has a solution (α, β, γ) satisfying α + β $^{2}$ + γ $^{3}$ = 12
(D)has infinitely many solutions

Q268·MathematicsSingle correctJEE Main 2021

If for the matrix, A = [1α–βα], AA

^{T}

= I2, then the value of α

^{4}

+ β

^{4}

is:

(A)1
(B)3
(C)2
(D)4

Q269·MathematicsNumericalJEE Main 2021

If the system of equations

kx + y + 2z = 1

3x - y - 2z = 2

-2x - 2y - 4z = 3

has infinitely many solutions, then

k

is equal to ___________.

Q270·MathematicsSingle correctJEE Main 2021

Let A be a 3 × 3 matrix with det(A) = 4. Let Ri denote the i

^{th}

row of A. If a matrix B is obtained by performing the operation R2→ 2R2 + 5R3 on 2A, then det(B) is equal to:

(A)64
(B)16
(C)80
(D)128

Q271·MathematicsSingle correctJEE Main 2021

For the system of linear equations:

x-2y=1,\ x-y+kz=-2,\ ky+4z=6,\ k\in \mathbf{R}

consider the following statements: (A) The system has unique solution if

k\neq 2, k\neq -2

. (B) The system has unique solution if

k=-2

. (C) The system has unique solution if

k=2

. (D) The system has no-solution if

k=2

. (E) The system has infinite number of solutions if

k\neq -2

. Which of the following statements are correct?

(A)(B) and (E) only
(B)(C) and (D) only
(C)(A) and (D) only
(D)(A) and (E) only

Q272·MathematicsSingle correctJEE Main 2021

The system of linear equations

3x - 2y - kz = 10

2x - 4y - 2z = 6

x + 2y - z = 5m

is inconsistent if :

(A) $k = 3, m = \frac{4}{5}$
(B) $k \neq 3, m \in R$
(C) $k \neq 3, m \neq \frac{4}{5}$
(D) $k = 3, m \neq \frac{4}{5}$

Q273·MathematicsNumericalJEE Main 2021

Let

P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}

, where

\alpha \in R

. Suppose

Q = [q_{ij}]

is a matrix satisfying

PQ = kI_3

for some non-zero

k \in R

. If

q_{23} = -\frac{k}{8}

and

|Q| = \frac{k^2}{2}

, then

\alpha^2 + k^2

is equal to _______

Q274·MathematicsSingle correctJEE Main 2021

Let A and B be

3\times 3

real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations

\left(\mathrm{A}^{2}\mathrm{B}^{2}-\mathrm{B}^{2}\mathrm{A}^{2}\right)\mathrm{X}=\mathrm{O}

, where X is a

3\times 1

column matrix of unknown variables and O is a

3\times 1

null matrix, has :

(A)a unique solution
(B)exactly two solutions
(C)infinitely many solutions
(D)no solution

Q275·MathematicsMultiple correctJEE Advanced 2020

Let M be a

3 \times 3

invertible matrix with real entries and let I denote the

3 \times 3

identity matrix. If

M^{-1} = \operatorname{adj}(\operatorname{adj} M)

, then which of the following statement is/are ALWAYS TRUE ?

(A) $M = I$
(B) $\det M = 1$
(C) $M^{2} = I$
(D) $(\operatorname{adj} M)^{2} = I$

Q276·MathematicsIntegerJEE Advanced 2020

The trace of a square matrix is defined to be the sum of its diagonal entries. If

A

is a

2 \times 2

matrix such that the trace of

A

is 3 and the trace of

A^{3}

is

-18

, then the value of the determinant of

A

is ________

Q277·MathematicsSingle correctJEE Main 2020

The value of

\lambda

and

\mu

for which the system of linear equations

x+y+z=2

,

x+2y+3z=5

,

x+3y+\lambda z=\mu

has infinitely many solutions are, respectively:

(A)6 and 8
(B)5 and 7
(C)5 and 8
(D)4 and 9

Q278·MathematicsSingle correctJEE Main 2020

Let

\theta = \frac{\pi}{5}

and

A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}

. If

B = A + A^{4}

, then det (B):

(A)is one.
(B)lies in (2, 3).
(C)is zero.
(D)lies in (1, 2).

Q279·MathematicsSingle correctJEE Main 2020

Let m and M be respectively the minimum and maximum values of

\begin{vmatrix}\cos^{2}x & 1+\sin^{2}x & \sin 2x\\ 1+\cos^{2}x & \sin^{2}x & \sin 2x\\ \cos^{2}x & \sin^{2}x & 1+\sin 2x\end{vmatrix}

. Then the ordered pair (m, M) is equal to:

(A) $(-3,3)$
(B) $(-3,-1)$
(C) $(-4,-1)$
(D) $(1,3)$

Q280·MathematicsNumericalJEE Main 2020

The sum of distinct values of

\lambda

for which the system of equations

(\lambda - 1)x + (3\lambda + 1)y + 2\lambda z = 0

(\lambda - 1)x + (4\lambda - 2)y + (\lambda + 3)z = 0

2x + (3\lambda + 1)y + 3(\lambda - 1)z = 0,

has non-zero solutions, is __________.

Q281·MathematicsSingle correctJEE Main 2020

If the system of linear equations

x+y+3z=0

x+3y+k^2z=0

3x+y+3z=0

has a non-zero solution

(x,y,z)

for some

k\in R

, then

x+\left(\dfrac{y}{z}\right)

is equal to:

(A) $3$
(B) $9$
(C) $-3$
(D) $-9$

Q282·MathematicsSingle correctJEE Main 2020

Let

\lambda \in R

. The system of linear equations.

2x_1 - 4x_2 + \lambda x_3 = 1

x_1 - 6x_2 + x_3 = 2

\lambda x_1 - 10x_2 + 4x_3 = 3

Is inconsistent for:

(A)exactly one positive value of $\lambda$
(B)exactly one negative value of $\lambda$
(C)every value of $\lambda$
(D)exactly two values of $\lambda$

Q283·MathematicsSingle correctJEE Main 2020

If a + x = b + y = c + z + 1, where a, b, c, x, y, x are non-zero distinct real numbers then

\begin{vmatrix} x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c \end{vmatrix}

is equal to:

(A)y(a - b)
(B)y(b - a)
(C)0
(D)y(a - c)

Q284·MathematicsSingle correctJEE Main 2020

If the system of equations x + y + z = 2 2x + 4y − z = 6

3x + 2y + \lambda z = \mu

has infinitely many solutions, then:

(A) $\lambda + 2\mu = 14$
(B) $2\lambda + \mu = 14$
(C) $2\lambda - \mu = 5$
(D) $\lambda - 2\mu = -5$

Q285·MathematicsSingle correctJEE Main 2020

If

A = \begin{bmatrix} \cos\theta & i\sin\theta \\ i\sin\theta & \cos\theta \end{bmatrix}

,

\left( \theta = \frac{\pi}{24} \right)

and

A^{5} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

, where

i = \sqrt{-1}

, then, which one of the following is not true?

(A) $a^{2} - c^{2} = 1$
(B) $a^{2} - b^{2} = \frac{1}{2}$
(C) $0 \le a^{2} + b^{2} \le 1$
(D) $a^{2} - d^{2} = 0$

Q286·MathematicsNumericalJEE Main 2020

If the system of equations

x - 2y + 3z = 9

2x + y + z = b

x - 7y + az = 24

, has infinitely many solutions, then

a - b

is equal to __________.

Q287·MathematicsNumericalJEE Main 2020

Let

A = \begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}

,

x \in R

and

A^{4} = [a_{ij}]

. If

a_{11} = 109

, then

a_{22}

is equal to __________.

Q288·MathematicsSingle correctJEE Main 2020

If

\Delta=\begin{vmatrix} x-2 & 2x-3 & 3x-4 \\ 2x-3 & 3x-4 & 4x-5 \\ 3x-5 & 5x-8 & 10x-17 \end{vmatrix}=Ax^{3}+Bx^{2}+Cx+D

, then B + C is equal to:

(A)−3
(B)9
(C)−1
(D)1

Q289·MathematicsNumericalJEE Main 2020

Let S be the set of all integer solutions, (x, y, z), of the system of equation

x - 2y + 5z = 0

-2x + 4y + z = 0

-7x + 14y + 9z = 0

such that

15 \le x^2 + y^2 + z^2 \le 150

. Then, the number of elements in the set S is equal to _______.

Q290·MathematicsSingle correctJEE Main 2020

Let A be a

3 \times 3

matrix such that

\mathrm{adj}\,A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{bmatrix}

and

B = \mathrm{adj}\left( \mathrm{adj}\,A \right)

. If

\left| A \right| = \lambda

and

\left| \left( B^{-1} \right)^{T} \right| = \mu

, then the ordered pair,

\left( \left| \lambda \right| . \mu \right)

is equal to:

(A) $\left( 9, \frac{1}{9} \right)$
(B) $(3, 81)$
(C) $\left( 9, \frac{1}{81} \right)$
(D) $\left( 3, \frac{1}{81} \right)$

Q291·MathematicsSingle correctJEE Main 2020

If the matrices

A=\begin{bmatrix}1 & 1 & 2\\1 & 3 & 4\\1 & -1 & 3\end{bmatrix}

,

B=\text{adj}\,A

and

C=3A

, then

\dfrac{|\text{adj}\,B|}{|C|}

is equal to:

(A)72
(B)8
(C)16
(D)2

Q292·MathematicsSingle correctJEE Main 2020

If for some

\alpha

and

\beta

in R, the intersection of the following three planes

x+4y-2z=1

x+7y-5z=\beta

x+5y+\alpha z=5

is a line in

R^{3}

, then

\alpha+\beta

is equal to:

(A) $0$
(B) $2$
(C) $10$
(D) $-10$

Q293·MathematicsSingle correctJEE Main 2020

Let

a - 2b + c = 1

. If

f(x) = \begin{vmatrix} x + a & x + 2 & x + 1 \\ x + b & x + 3 & x + 2 \\ x + c & x + 4 & x + 3 \end{vmatrix}

, then:

(A) $f(-50) = 501$
(B) $f(50) = 1$
(C) $f(50) = -501$
(D) $f(-50) = -1$

Q294·MathematicsNumericalJEE Main 2020

The number of all

3\times3

matrices, A, with enteries from the set

\{-1,0,1\}

such that the sum of the diagonal elements of

AA^{T}

is 3, is

Q295·MathematicsSingle correctJEE Main 2020

The system of linear equations

\lambda x+2y+2z=5

,

2\lambda x+3y+5z=8

,

4x+\lambda y+6z=10

has:

(A)no solution when $\lambda=8$
(B)infinitely many solutions when $\lambda=2$
(C)no solution when $\lambda=2$
(D)a unique solution when $\lambda=-8$

Q296·MathematicsSingle correctJEE Main 2020

If

A=\begin{pmatrix}2 & 2\\9 & 4\end{pmatrix}

and

I=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}

, then

10A^{-1}

is equal to:

(A) $4I-A$
(B) $A-6I$
(C) $A-4I$
(D) $6I-A$

Q297·MathematicsSingle correctJEE Main 2020

For which of the following ordered pairs

(\mu,\delta)

, the system of linear equations

x+2y+3z=1

3x+4y+5z=\mu

is inconsistent?

4x+4y+4z=\delta

(A)(3, 4)
(B)(1, 0)
(C)(4, 3)
(D)(4, 6)

Q298·MathematicsSingle correctJEE Main 2020

Let

\alpha

be a root of the equation

x^{2}+x+1=0

and the matrix

A=\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4} \end{bmatrix}

, then the matrix

A^{31}

is equal to:

(A) $A^{3}$
(B)A
(C) $I_{3}$
(D) $A^{2}$

Q299·MathematicsNumericalJEE Main 2020

If the system of linear equations,

x+y+z=6

,

x+2y+3z=10

,

3x+2y+\lambda z=\mu

has more than two solutions, then

\mu-\lambda^{2}

is equal to __________.

Q300·MathematicsSingle correctJEE Main 2020

Let

A=[a_{ij}]

and

B=[b_{ij}]

be two

3\times 3

real matrices such that

b_{ij}=(3)^{(i+j-2)}a_{ji}

, where

i,j=1,2,3

. If the determinant of B is 81, then the determinant of A is:

(A) $\dfrac{1}{3}$
(B) $\dfrac{1}{9}$
(C) $\dfrac{1}{81}$
(D)3

Q301·MathematicsSingle correctJEE Main 2020

If the system of linear equations

2x+2ay+az=0

,

2x+3by+bz=0,2x+4cy+cz=0

where a,b,c

\in R

are non-zero and distinct; has a non-zero solution, then:

(A) $a+b+c=0$
(B)a, b, c are in A.P.
(C) $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
(D)a, b, c are in G.P.

Q302·MathematicsMultiple correctJEE Advanced 2019

Let

M = \begin{bmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{bmatrix}

and adj

M = \begin{bmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}

where a and b area real numbers. Which of the following options is/are correct?

(A) $(\mathrm{adj}\ M)^{-1} + \mathrm{adj}\ M^{-1} = -M$
(B)If $M \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ , then $\alpha - \beta + \gamma = 3$
(C) $\det(\mathrm{adj}\ M^2) = 81$
(D) $a + b = 3$

Q303·MathematicsSingle correctJEE Advanced 2019

Let

M = \begin{bmatrix} \sin^4\theta & -1-\sin^2\theta \\ 1+\cos^2\theta & \cos^4\theta \end{bmatrix} = \alpha I + \beta M^{-1}

, where

\alpha = \alpha(\theta)

and

\beta = \beta(\theta)

are real numbers, and I is the

2 \times 2

identity matrix. If

\alpha^*

is the minimum of the set

\{\alpha(\theta) : \theta \in [0, 2\pi)\}

and

\beta^*

is the minimum of the set

\{\beta(\theta) : \theta \in [0, 2\pi)\}

, then the value of

\alpha^* + \beta^*

is

(A) $-\frac{37}{16}$
(B) $-\frac{31}{16}$
(C) $-\frac{17}{16}$
(D) $-\frac{29}{16}$

Q304·MathematicsMultiple correctJEE Advanced 2019

Let

x \in \mathbb{R}

and let

P = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{bmatrix}

,

Q = \begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{bmatrix}

and

R = PQP^{-1}

Then which of the following options is/are correct?

(A)For $x = 0$ , if $R\begin{bmatrix} 1 \\ a \\ b \end{bmatrix} = 6\begin{bmatrix} 1 \\ a \\ b \end{bmatrix}$ , then $a + b = 5$
(B)For $x = 1$ , there exists a unit vector $\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ for which $R\begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
(C) $\det R = \det\begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 5 \end{bmatrix} + 8$ , for all $x \in \mathbb{R}$
(D)There exists a real number $x$ such that $PQ = QP$

Q305·MathematicsMultiple correctJEE Advanced 2019

Let

P_1 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

,

P_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}

P_3 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}

P_4 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}

,

P_5 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}

,

P_6 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}

and

X = \displaystyle\sum_{k=1}^{6} P_k \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{bmatrix} P_k^{T}

where

P_k^{T}

denotes the transpose of the matrix

P_k

. Then which of the following options is/are correct?

(A) $X - 30I$ is an invertible matrix
(B)X is a symmetric matrix
(C)The sum of diagonal entries of X is 18
(D)If $X\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \alpha\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ , then $\alpha = 30$

Q306·MathematicsSingle correctJEE Main 2019

A value of

\theta \in (0, \pi/3)

, for which

\begin{vmatrix} 1+\cos^{2}\theta & \sin^{2}\theta & 4\cos 6\theta \\ \cos^{2}\theta & 1+\sin^{2}\theta & 4\cos 6\theta \\ \cos^{2}\theta & \sin^{2}\theta & 1+4\cos 6\theta \end{vmatrix} = 0

, is:

(A) $\dfrac{\pi}{18}$
(B) $\dfrac{\pi}{9}$
(C) $\dfrac{7\pi}{36}$
(D) $\dfrac{7\pi}{24}$

Q307·MathematicsSingle correctJEE Main 2019

If A is a symmetric matrix and B is a skew-symmetrix matrix such that A + B =

\begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}

, then AB is equal to :

(A) $\begin{bmatrix} 4 & -2 \\ 1 & -4 \end{bmatrix}$
(B) $\begin{bmatrix} 4 & -2 \\ -1 & -4 \end{bmatrix}$
(C) $\begin{bmatrix} -4 & 2 \\ 1 & 4 \end{bmatrix}$
(D) $\begin{bmatrix} -4 & -2 \\ -1 & 4 \end{bmatrix}$

Q308·MathematicsSingle correctJEE Main 2019

If B =

\begin{bmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{bmatrix}

is the inverse of a

3 \times 3

matrix A, then the sum of all values of

\alpha

for which det(A) + 1 = 0, is :

(A)0
(B) $-1$
(C)1
(D)2

Q309·MathematicsSingle correctJEE Main 2019

If [x] denotes the greatest integer

\leq

x, then the system of linear equations [sin

\theta

] x + [

-\cos\theta

] y = 0 [cot

\theta

] x + y = 0

(A)have infinitely many solutions if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ and has a unique solution if $\theta \in \left(\pi, \dfrac{7\pi}{6}\right)$
(B)have infinitely many solutions if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right) \cup \left(\pi, \dfrac{7\pi}{6}\right)$
(C)has a unique solution if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ and have infinitely many solutions if $\theta \in \left(\pi, \dfrac{7\pi}{6}\right)$
(D)has a unique solution if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right) \cup \left(\pi, \dfrac{7\pi}{6}\right)$

Q310·MathematicsSingle correctJEE Main 2019

Let

\lambda

be a real number for which the system of linear equations

x + y + z = 6

4x + \lambda y - \lambda z = \lambda - 2

3x + 2y - 4z = -5

Has indefinitely many solutions. Then

\lambda

is a root of the quadratic equation

(A) $\lambda^{2} - \lambda - 6 = 0$
(B) $\lambda^{2} - 3\lambda - 4 = 0$
(C) $\lambda^{2} + 3\lambda - 4 = 0$
(D) $\lambda^{2} + \lambda - 6 = 0$

Q311·MathematicsSingle correctJEE Main 2019

The sum of the real roots of the equation

\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x - 3 \\ -3 & 2x & x = 2 \end{vmatrix} = 0

is equal to

(A)-4
(B)0
(C)6
(D)1

Q312·MathematicsSingle correctJEE Main 2019

If the system of equations

2x + 3y - z = 0

,

x + ky - 2z = 0

and

2x - y + z = 0

has a non-trivial solution

(x, y, z)

, then

\dfrac{x}{y} + \dfrac{y}{z} + \dfrac{z}{x} + k

is equal to

(A) $\dfrac{3}{4}$
(B) $-4$
(C) $\dfrac{1}{2}$
(D) $-\dfrac{1}{4}$

Q313·MathematicsSingle correctJEE Main 2019

Let

\alpha

and

\beta

be the roots of the equation

x^{2}+x+1=0

. Then for

y\neq 0

in R,

\begin{vmatrix}y+1 & \alpha & \beta\\ \alpha & y+\beta & 1\\ \beta & 1 & y+\alpha\end{vmatrix}

is equal to:

(A) $y(y^{2}-3)$
(B) $y^{3}-1$
(C) $y^{3}$
(D) $y(y^{2}-1)$

Q314·MathematicsSingle correctJEE Main 2019

If

\begin{bmatrix}1&1\\0&1\end{bmatrix}\begin{bmatrix}1&2\\0&1\end{bmatrix}\begin{bmatrix}1&3\\0&1\end{bmatrix}\ldots\ldots\ldots\ldots\begin{bmatrix}1&n-1\\0&1\end{bmatrix}=\begin{bmatrix}1&78\\0&1\end{bmatrix}

, then the inverse of

\begin{bmatrix}1&n\\0&1\end{bmatrix}

is:

(A) $\begin{bmatrix}1&-12\\0&1\end{bmatrix}$
(B) $\begin{bmatrix}1&0\\13&1\end{bmatrix}$
(C) $\begin{bmatrix}1&0\\12&1\end{bmatrix}$
(D) $\begin{bmatrix}1&-13\\0&1\end{bmatrix}$

Q315·MathematicsSingle correctJEE Main 2019

Let

A=\begin{pmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{pmatrix}

,

(\alpha \in R)

such that

A^{32}=\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}

. Then a value of

\alpha

is:

(A) $0$
(B) $\frac{\pi}{16}$
(C) $\frac{\pi}{32}$
(D) $\frac{\pi}{64}$

Q316·MathematicsSingle correctJEE Main 2019

If the system of linear equations

x-2y+kz=1

,

2x+y+z=2

,

3x-y-kz=3

has a solution

(x,y,z)\ne 0

, then

(x,y)

lies on the straight line whose equation is:

(A) $3x-4y-1=0$
(B) $4x-3y-4=0$
(C) $4x-3y-1=0$
(D) $3x-4y-4=0$

Q317·MathematicsSingle correctJEE Main 2019

The greatest value of

c\in R

for which the system of linear equations

x-cy-cz=0

cx-y+cz=0

cx+cy-z=0

has a non-trivial solution, is:

(A)-1
(B)2
(C) $\dfrac{1}{2}$
(D)0

Q318·MathematicsSingle correctJEE Main 2019

Let

P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}

and

Q = [q_{ij}]

be two

3 \times 3

matrices such that

Q - P^{5} = I_{3}

. Then

\dfrac{q_{21} + q_{31}}{q_{32}}

is equal to:

(A)10
(B)135
(C)15
(D)9

Q319·MathematicsSingle correctJEE Main 2019

If

A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}

; then for all

\theta \in \left(\frac{3\pi}{4}, \frac{5\pi}{4}\right)

, det (A) lies in the interval :

(A) $\left(1, \frac{5}{2}\right]$
(B) $\left[\frac{5}{2}, 4\right)$
(C) $\left(0, \frac{3}{2}\right]$
(D) $\left(\frac{3}{2}, 3\right]$

Q320·MathematicsSingle correctJEE Main 2019

An ordered pair

(\alpha, \beta)

for which the system of linear equations

(1 + \alpha)x + \beta y + z = 2

\alpha x + (1 + \beta)y + z = 3

\alpha x + \beta y + 2z = 2

has a unique solution, is:

(A) $(2, 4)$
(B) $(-3, 1)$
(C) $(-4, 2)$
(D) $(1, -3)$

Q321·MathematicsSingle correctJEE Main 2019

Let A and B be two invertible matrices of order

3\times 3

. If

\det(ABA^{T})=8

and

\det(AB^{-1})=8

, then

\det(BA^{-1}B^{T})

is equal to:

(A) $\frac{1}{4}$
(B)1
(C) $\frac{1}{16}$
(D)16

Q322·MathematicsSingle correctJEE Main 2019

If the system of linear equations

2x + 2y + 3z = a

3x - y + 5z = b

x - 3y + 2z = c

Where a, b, c are non zero real numbers, has more than one solution, then:

(A) $b-c+a=0$
(B) $b-c-a=0$
(C) $a+b+c=0$
(D) $b+c-a=0$

Q323·MathematicsSingle correctJEE Main 2019

Let

A=\begin{pmatrix}0 & 2q & r\\ p & q & -r\\ p & -q & r\end{pmatrix}

. If

AA^{T}=I_{3}

,

|P|

then

|p|

is:

(A) $\dfrac{1}{\sqrt{5}}$
(B) $\dfrac{1}{\sqrt{3}}$
(C) $\dfrac{1}{\sqrt{2}}$
(D) $\dfrac{1}{\sqrt{6}}$

Q324·MathematicsSingle correctJEE Main 2019

If the system of equations

x+y+z=5

,

x+2y+3z=9

,

x+3y+\alpha z=\beta

has infinitely many solutions, then

\beta-\alpha

equals:

(A)21
(B)8
(C)18
(D)5

Q325·MathematicsSingle correctJEE Main 2019

If the system of linear equation

x-4y+7z=g, 3y-5z=h, -2x+5y-9z=k

is consistent, then:

(A) $g+h+k=0$
(B) $2g+h+k=0$
(C) $g+h+2k=0$
(D) $g+2h+k=0$

Q326·MathematicsSingle correctJEE Main 2019

If

A = \begin{bmatrix} e^{t} & e^{-t}\cos t & e^{-t}\sin t \\ e^{t} & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^{t} & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}

Then A is

(A)Invertible only if $t = \frac{\pi}{2}$
(B)not invertible for any $t \in R$
(C)invertible for all $t \in R$
(D)invertible only if $t = \pi$

Q327·MathematicsSingle correctJEE Main 2019

If

A=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}

, then the matrix

A^{-50}

when

\theta=\dfrac{\pi}{12}

, is equal to

(A) $\begin{bmatrix}\dfrac{1}{2} & -\dfrac{\sqrt{3}}{2}\\[4pt] \dfrac{\sqrt{3}}{2} & \dfrac{1}{2}\end{bmatrix}$
(B) $\begin{bmatrix}\dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}\\[4pt] \dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{bmatrix}$
(C) $\begin{bmatrix}\dfrac{\sqrt{3}}{2} & \dfrac{1}{2}\\[4pt] -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{bmatrix}$
(D) $\begin{bmatrix}\dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\\[4pt] -\dfrac{\sqrt{3}}{2} & \dfrac{1}{2}\end{bmatrix}$

Q328·MathematicsSingle correctJEE Main 2019

The system of linear equation

x + y + z = 2

,

2x + 3y + 2z = 5

,

2x + 3y + (a^2 - 1)z = a + 1

then

(A)is inconsistent when $a = 4$
(B)has a unique solution for $|a| = \sqrt{3}$
(C)has infinitely many solutions for $a = 4$
(D)inconsistent when $|a| = \sqrt{3}$

Q329·MathematicsNumericalJEE Advanced 2018

Let P be a matrix of order

3 \times 3

such that all the entries in P are from the set

\{-1, 0, 1\}

. Then, the maximum possible value of the determinant of P is ______ .

Q330·MathematicsMultiple correctJEE Advanced 2018

Let S be the set of all column matrices

\begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix}

such that

b_{1}

,

b_{2}

,

b_{3} \in R

and the system of equations (in real variables)

-x + 2y + 5z = b_{1}

2x - 4y + 3z = b_{2}

x - 2y + 2z = b_{3}

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each

\begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix} \in S

?

(A) $x + 2y + 3z = b_{1}$ , $4y + 5z = b_{2}$ and $x + 2y + 6z = b_{3}$
(B) $x + y + 3z = b_{1}$ , $5x + 2y + 6z = b_{2}$ and $-2x - y - 3z = b_{3}$
(C) $-x + 2y - 5z = b_{1}$ , $2x - 4y + 10z = b_{2}$ and $x - 2y + 5z = b_{3}$
(D) $x + 2y + 5z = b_{1}$ , $2x + 3z = b_{2}$ and $x + 4y - 5z = b_{3}$

Q331·MathematicsMultiple correctJEE Advanced 2017

Which of the following is(are) NOT the square of a

3 \times 3

matrix with real entries ?

(A) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
(B) $\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
(C) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
(D) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$

Q332·MathematicsMultiple correctJEE Advanced 2017

If

f(x) = \begin{vmatrix} \cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{vmatrix}

, then

(A) $f'(x) = 0$ at exactly three points in $(-\pi, \pi)$
(B) $f'(x) = 0$ at more than three points in $(-\pi, \pi)$
(C) $f(x)$ attains its maximum at $x = 0$
(D) $f(x)$ attains its minimum at $x = 0$

Q333·MathematicsIntegerJEE Advanced 2017

For a real number

\alpha

, if the system

\begin{bmatrix} 1 & \alpha & \alpha^{2} \\ \alpha & 1 & \alpha \\ \alpha^{2} & \alpha & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}

of linear equations, has infinitely many solutions, then

1 + \alpha + \alpha^{2} =

Q334·MathematicsSingle correctJEE Advanced 2016

Let

P = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}

and I be the identity matrix of order 3. If

Q = \left[ q_{ij} \right]

is a matrix such that

P^{50} - Q = I

, then

\frac{q_{31} + q_{32}}{q_{21}}

equals

(A)52
(B)103
(C)201
(D)205

Q335·MathematicsIntegerJEE Advanced 2016

The total number of distinct

x \in \mathbb{R}

for which

\begin{vmatrix} x & x^{2} & 1 + x^{3} \\ 2x & 4x^{2} & 1 + 8x^{3} \\ 3x & 9x^{2} & 1 + 27x^{3} \end{vmatrix} = 10

is

Q336·MathematicsMultiple correctJEE Advanced 2016

Let

P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}

, where

\alpha \in \mathbb{R}

. Suppose

Q = [q_{ij}]

is a matrix such that

PQ = kI

, where

k \in \mathbb{R}

,

k \neq 0

and

I

is the identity matrix of order 3. If

q_{23} = -\frac{k}{8}

and

\det(Q) = \frac{k^{2}}{2}

, then

(A) $\alpha = 0, k = 8$
(B) $4\alpha - k + 8 = 0$
(C) $\det(P\,\mathrm{adj}(Q)) = 2^{9}$
(D) $\det(Q\,\mathrm{adj}(P)) = 2^{13}$

Q337·MathematicsMultiple correctJEE Advanced 2016

Let

a, \lambda, \mu \in \mathbb{R}

. Consider the system of linear equations

ax + 2y = \lambda

3x - 2y = \mu

Which of the following statement(s) is(are) correct?

(A)If $a = -3$ , then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
(B)If $a \neq -3$ , then the system has a unique solution for all values of $\lambda$ and $\mu$
(C)If $\lambda + \mu = 0$ , then the system has infinitely many solutions for $a = -3$
(D)If $\lambda + \mu \neq 0$ , then the system has no solution for $a = -3$

Q338·MathematicsMultiple correctJEE Advanced 2015

Which of the following values of

\alpha

satisfy the equation

\begin{vmatrix} (1+\alpha)^{2} & (1+2\alpha)^{2} & (1+3\alpha)^{2} \\ (2+\alpha)^{2} & (2+2\alpha)^{2} & (2+3\alpha)^{2} \\ (3+\alpha)^{2} & (3+2\alpha)^{2} & (3+3\alpha)^{2} \end{vmatrix} = -648\alpha

?

(A) $-4$
(B) $9$
(C) $-9$
(D) $4$

Q339·MathematicsMultiple correctJEE Advanced 2014

Let

M

be a

2 \times 2

symmetric matrix with integer entries. Then

M

is invertible if

(A)the first column of $M$ is the transpose of the second row of $M$
(B)the second row of $M$ is the transpose of the first column of $M$
(C) $M$ is a diagonal matrix with non-zero entries in the main diagonal
(D)the product of entries in the main diagonal of $M$ is not the square of an integer

Q340·MathematicsMultiple correctJEE Advanced 2014

Let

M

and

N

be two

3 \times 3

matrices such that

MN = NM

. Further, if

M \neq N^2

and

M^2 = N^4

, then

(A)determinant of $(M^2 + MN^2)$ is $0$
(B)there is a $3 \times 3$ non-zero matrix $U$ such that $(M^2 + MN^2)U$ is the zero matrix
(C)determinant of $(M^2 + MN^2) \geq 1$
(D)for a $3 \times 3$ matrix $U$ , if $(M^2 + MN^2)U$ equals the zero matrix then $U$ is the zero matrix

Q341·MathematicsMultiple correctJEE Advanced 2013

Let

\omega

be a complex cube root of unity with

\omega \neq 1

and

P = [p_{ij}]

be a

n \times n

matrix with

p_{ij} = \omega^{i+j}

. Then

P^{2} \neq 0

, when

n =

(A) $57$
(B) $55$
(C) $58$
(D) $56$

Q342·MathematicsMultiple correctJEE Advanced 2013

For

3\times 3

matrices M and N, which of the following statement(s) is (are) NOT correct ?

(A) $N^TMN$ is symmetric or skew symmetric, according as M is symmetric or skew symmetric
(B) $MN-NM$ is skew symmetric for all symmetric matrices M and N
(C) $MN$ is symmetric for all symmetric matrices M and N
(D) $(\text{adj }M)(\text{adj }N)=\text{adj}(MN)$ for all invertible matrices M and N

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