# Matrices and Determinants · Mathematics · JEE Main

Start Practice## MCQ (Single Correct Answer)

JEE Main 2024 (Online) 9th April Evening Shift

Let $$B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$$ and $$A$$ be a $$2 \times 2$$ matrix such that $$A B^{-1}=A^{-1}$$. If $$B C B^{-1}=...

JEE Main 2024 (Online) 9th April Morning Shift

Let $$\lambda, \mu \in \mathbf{R}$$. If the system of equations
$$\begin{aligned}
& 3 x+5 y+\lambda z=3 \\
& 7 x+11 y-9 z=2 \\
& 97 x+155 y-189 z=\mu
...

JEE Main 2024 (Online) 8th April Evening Shift

If $$\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$$ and $$\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm...

JEE Main 2024 (Online) 8th April Evening Shift

If the system of equations $$x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$$ has infinitely many solutions, then $$(2 \mu+3 \lambda)$$ is equal to :...

JEE Main 2024 (Online) 8th April Morning Shift

Let $$A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$$. If $$A^3=4 A^2-A-21 I$$, where $$I$$ is the identity matrix o...

JEE Main 2024 (Online) 6th April Evening Shift

If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{ad...

JEE Main 2024 (Online) 6th April Morning Shift

For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta...

JEE Main 2024 (Online) 5th April Evening Shift

The values of $$m, n$$, for which the system of equations
$$\begin{aligned}
& x+y+z=4, \\
& 2 x+5 y+5 z=17, \\
& x+2 y+\mathrm{m} z=\mathrm{n}
\end{al...

JEE Main 2024 (Online) 5th April Evening Shift

Let $$\alpha \beta \neq 0$$ and $$A=\left[\begin{array}{rrr}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\rig...

JEE Main 2024 (Online) 5th April Morning Shift

Let A and B be two square matrices of order 3 such that $$\mathrm{|A|=3}$$ and $$\mathrm{|B|=2}$$. Then $$|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operato...

JEE Main 2024 (Online) 5th April Morning Shift

If the system of equations
$$\begin{array}{r}
11 x+y+\lambda z=-5 \\
2 x+3 y+5 z=3 \\
8 x-19 y-39 z=\mu
\end{array}$$
has infinitely many solutions, t...

JEE Main 2024 (Online) 4th April Evening Shift

Let $$A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$$ and $$B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} ...

JEE Main 2024 (Online) 4th April Morning Shift

Let $$\alpha \in(0, \infty)$$ and $$A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$$. If $$\operatorname{det}\le...

JEE Main 2024 (Online) 4th April Morning Shift

If the system of equations
$$\begin{aligned}
& x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\
& x+(\cos \alpha) y+(\sin \alpha) z=0 \\
& x+(...

JEE Main 2024 (Online) 1st February Evening Shift

Let the system of equations $x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :

JEE Main 2024 (Online) 1st February Morning Shift

If $\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\rig...

JEE Main 2024 (Online) 1st February Morning Shift

If the system of equations
$$
\begin{aligned}
& 2 x+3 y-z=5 \\\\
& x+\alpha y+3 z=-4 \\\\
& 3 x-y+\beta z=7
\end{aligned}
$$
has infinitely many solut...

JEE Main 2024 (Online) 31st January Evening Shift

Let $$A$$ be a $$3 \times 3$$ real matrix such that
$$A\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right)=2\left(\begin{array}{l}
1 \\
0 \\
1
\end{...

JEE Main 2024 (Online) 31st January Morning Shift

If the system of linear equations
$$\begin{aligned}
& x-2 y+z=-4 \\
& 2 x+\alpha y+3 z=5 \\
& 3 x-y+\beta z=3
\end{aligned}$$
has infinitely many solu...

JEE Main 2024 (Online) 30th January Evening Shift

Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero $$3 \times 3$$ matrix, where $$x \sin \theta=y \...

JEE Main 2024 (Online) 30th January Evening Shift

Consider the system of linear equations $$x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$$, where $$\lambda, \mu \in \mathbb{R}$$. Then, which of t...

JEE Main 2024 (Online) 30th January Morning Shift

Consider the system of linear equations $$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$$ where $$\lambda, \mu \in \mathbf{R}$$....

JEE Main 2024 (Online) 29th January Evening Shift

Let $$A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]$$ and $$P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 &...

JEE Main 2024 (Online) 29th January Morning Shift

$$\text { Let } A=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & \alpha & \beta \\
0 & \beta & \alpha
\end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{...

JEE Main 2024 (Online) 29th January Morning Shift

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

JEE Main 2024 (Online) 27th January Evening Shift

The values of $$\alpha$$, for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha...

JEE Main 2024 (Online) 27th January Morning Shift

Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$.
Given below are two sta...

JEE Main 2023 (Online) 15th April Morning Shift

Let the determinant of a square matrix A of order $m$ be $m-n$, where $m$ and $n$ satisfy $4 m+n=22$ and $17 m+4 n=93$. If $\operatorname{det}(n \oper...

JEE Main 2023 (Online) 13th April Evening Shift

Let for $$A = \left[ {\matrix{
1 & 2 & 3 \cr
\alpha & 3 & 1 \cr
1 & 1 & 2 \cr
} } \right],|A| = 2$$. If $$\mathrm{|2\,adj\,(2\,adj\,...

JEE Main 2023 (Online) 13th April Evening Shift

If the system of equations
$$2 x+y-z=5$$
$$2 x-5 y+\lambda z=\mu$$
$$x+2 y-5 z=7$$
has infinitely many solutions, then $$(\lambda+\mu)^{2}+(\lambda-\m...

JEE Main 2023 (Online) 13th April Morning Shift

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

JEE Main 2023 (Online) 13th April Morning Shift

Let $$B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$$ be the adjoint of a matrix $$A$$ an...

JEE Main 2023 (Online) 13th April Morning Shift

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 :

JEE Main 2023 (Online) 12th April Morning Shift

Let $$A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$$. If $$\mathrm{B}=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right...

JEE Main 2023 (Online) 11th April Evening Shift

If the system of linear equations
$$
\begin{aligned}
& 7 x+11 y+\alpha z=13 \\\\
& 5 x+4 y+7 z=\beta \\\\
& 175 x+194 y+57 z=361
\end{aligned}
$$
has ...

JEE Main 2023 (Online) 11th April Evening Shift

$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$, then $$\lambda, \frac{\...

JEE Main 2023 (Online) 11th April Morning Shift

Let $$\mathrm{A}$$ be a $$2 \times 2$$ matrix with real entries such that $$\mathrm{A}'=\alpha \mathrm{A}+\mathrm{I}$$, where $$\alpha \in \mathbb{R}-...

JEE Main 2023 (Online) 10th April Evening Shift

If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$, then $$|\opera...

JEE Main 2023 (Online) 10th April Morning Shift

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

JEE Main 2023 (Online) 10th April Morning Shift

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?...

JEE Main 2023 (Online) 8th April Evening Shift

If $$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$$ and $$\alpha+\beta=-2$$, t...

JEE Main 2023 (Online) 8th April Evening Shift

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...

JEE Main 2023 (Online) 8th April Morning Shift

Let $$A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$$. If $$|\operatorname{adj}(\operatorname{adj}(\operatorname{a...

JEE Main 2023 (Online) 8th April Morning Shift

Let $$P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1...

JEE Main 2023 (Online) 6th April Evening Shift

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

JEE Main 2023 (Online) 6th April Evening Shift

For the system of equations
$$x+y+z=6$$
$$x+2 y+\alpha z=10$$
$$x+3 y+5 z=\beta$$, which one of the following is NOT true?

JEE Main 2023 (Online) 6th April Morning Shift

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

JEE Main 2023 (Online) 6th April Morning Shift

Let $$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}$$, where $$\mathrm{a}_{\mathrm{ij}} \neq 0$$ for all $$\mathrm{i}, \mathrm{j}$$ an...

JEE Main 2023 (Online) 1st February Evening Shift

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?...

JEE Main 2023 (Online) 1st February Evening Shift

If $$A = {1 \over 2}\left[ {\matrix{
1 & {\sqrt 3 } \cr
{ - \sqrt 3 } & 1 \cr
} } \right]$$, then :

JEE Main 2023 (Online) 1st February Morning Shift

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...

JEE Main 2023 (Online) 31st January Morning Shift

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

JEE Main 2023 (Online) 31st January Morning Shift

Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
0 & 4 & { - 1} \cr
0 & {12} & { - 3} \cr
} } \right)$$. Then the sum of the diagonal element...

JEE Main 2023 (Online) 30th January Evening Shift

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations
$$
\begin{aligned}
& x-y+z=5 \\
& 2 x+2 y+\alpha z=8 \\
& 3 x-y+4 z=\beta
\...

JEE Main 2023 (Online) 30th January Evening Shift

If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>1$, then :

JEE Main 2023 (Online) 30th January Morning Shift

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

JEE Main 2023 (Online) 30th January Morning Shift

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

JEE Main 2023 (Online) 29th January Evening Shift

The set of all values of $$\mathrm{t\in \mathbb{R}}$$, for which the matrix $$\left[ {\matrix{
{{e^t}} & {{e^{ - t}}(\sin t - 2\cos t)} & {{e^{ - t...

JEE Main 2023 (Online) 29th January Morning Shift

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

JEE Main 2023 (Online) 29th January Morning Shift

Consider the following system of equations
$$\alpha x+2y+z=1$$
$$2\alpha x+3y+z=1$$
$$3x+\alpha y+2z=\beta$$
for some $$\alpha,\beta\in \mathbb{R}$$. ...

JEE Main 2023 (Online) 25th January Evening Shift

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$$^{26}$$ $$-...

JEE Main 2023 (Online) 25th January Evening Shift

Let $$A = \left[ {\matrix{
{{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr
{{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}...

JEE Main 2023 (Online) 25th January Morning Shift

Let $$x,y,z > 1$$ and $$A = \left[ {\matrix{
1 & {{{\log }_x}y} & {{{\log }_x}z} \cr
{{{\log }_y}x} & 2 & {{{\log }_y}z} \cr
{{{\log }_z}...

JEE Main 2023 (Online) 25th January Morning Shift

Let S$$_1$$ and S$$_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{ 0\} $$ for which the system of linear equations
$$ax + 2ay - 3az = 1$$...

JEE Main 2023 (Online) 24th January Evening Shift

Let A be a 3 $$\times$$ 3 matrix such that $$\mathrm{|adj(adj(adj~A))|=12^4}$$. Then $$\mathrm{|A^{-1}~adj~A|}$$ is equal to

JEE Main 2023 (Online) 24th January Evening Shift

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$...

JEE Main 2023 (Online) 24th January Morning Shift

If A and B are two non-zero n $$\times$$ n matrices such that $$\mathrm{A^2+B=A^2B}$$, then :

JEE Main 2023 (Online) 24th January Morning Shift

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 $$\lef...

JEE Main 2022 (Online) 29th July Evening Shift

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

JEE Main 2022 (Online) 29th July Evening Shift

If the system of equations
$$
\begin{aligned}
&x+y+z=6 \\
&2 x+5 y+\alpha z=\beta \\
&x+2 y+3 z=14
\end{aligned}
$$
has infinitely many solutions, the...

JEE Main 2022 (Online) 29th July Morning Shift

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

JEE Main 2022 (Online) 28th July Evening Shift

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

JEE Main 2022 (Online) 28th July Morning Shift

Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix $$B_{0}=A^{49}+2 A^{98}$$. If $$B_{n...

JEE Main 2022 (Online) 27th July Evening Shift

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

JEE Main 2022 (Online) 27th July Morning Shift

Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^{2}+\beta A=2 I$$. T...

JEE Main 2022 (Online) 26th July Evening Shift

$$
\text { Let } A=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
9^{2} & -10^{2} & 11^{2} \\
12^{2} &...

JEE Main 2022 (Online) 26th July Morning Shift

If the system of linear equations.
$$8x + y + 4z = - 2$$
$$x + y + z = 0$$
$$\lambda x - 3y = \mu $$
has infinitely many solutions, then the distance...

JEE Main 2022 (Online) 26th July Morning Shift

Let A be a 2 $$\times$$ 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 :...

JEE Main 2022 (Online) 25th July Evening Shift

The number of real values of $$\lambda$$, such that the system of linear equations
2x $$-$$ 3y + 5z = 9
x + 3y $$-$$ z = $$-$$18
3x $$-$$ y + ($$\lamb...

JEE Main 2022 (Online) 25th July Morning Shift

The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations
$$
\begin{aligned}
&3(\sin 3 \theta) x-y+z=2 \\\\
&3(\cos 2 \theta) x+...

JEE Main 2022 (Online) 30th June Morning Shift

Let $$A = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
} } \right]$$ and $$B = \left[ {\matrix{
2 & \alpha \cr ...

JEE Main 2022 (Online) 30th June Morning Shift

Let A and B be two square matrices of order 2. If $$det\,(A) = 2$$, $$det\,(B) = 3$$ and $$\det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}...

JEE Main 2022 (Online) 29th June Evening Shift

Let $$A = \left( {\matrix{
2 & { - 1} \cr
0 & 2 \cr
} } \right)$$. If $$B = I - {}^5{C_1}(adj\,A) + {}^5{C_2}{(adj\,A)^2} - \,\,.....\,\, ...

JEE Main 2022 (Online) 29th June Morning Shift

If the system of linear equations
2x + y $$-$$ z = 7
x $$-$$ 3y + 2z = 1
x + 4y + $$\delta$$z = k, where $$\delta$$, k $$\in$$ R has infinitely many s...

JEE Main 2022 (Online) 29th June Morning Shift

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 A2 + A3 + ...... +...

JEE Main 2022 (Online) 28th June Morning Shift

If the system of linear equations
$$2x + 3y - z = - 2$$
$$x + y + z = 4$$
$$x - y + |\lambda |z = 4\lambda - 4$$
where, $$\lambda$$ $$\in$$ R, has n...

JEE Main 2022 (Online) 28th June Morning Shift

Let A be a matrix of order 3 $$\times$$ 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________.

JEE Main 2022 (Online) 27th June Evening Shift

Let $$f(x) = \left| {\matrix{
a & { - 1} & 0 \cr
{ax} & a & { - 1} \cr
{a{x^2}} & {ax} & a \cr
} } \right|,\,a \in R$$. Then the sum ...

JEE Main 2022 (Online) 27th June Evening Shift

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

JEE Main 2022 (Online) 27th June Morning Shift

Let the system of linear equations $$x + 2y + z = 2$$, $$\alpha x + 3y - z = \alpha $$, $$ - \alpha x + y + 2z = - \alpha $$ be inconsistent. Then $$...

JEE Main 2022 (Online) 26th June Evening Shift

If the system of equations
$$\alpha$$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $$\beta$$
has infinitely many solutions, then the ordered pair ($$\...

JEE Main 2022 (Online) 26th June Morning Shift

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

JEE Main 2022 (Online) 26th June Morning Shift

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 :...

JEE Main 2022 (Online) 25th June Evening Shift

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

JEE Main 2022 (Online) 25th June Morning Shift

Let A be a 3 $$\times$$ 3 real matrix such that
$$A\left( {\matrix{
1 \cr
1 \cr
0 \cr
} } \right) = \left( {\matrix{
1 \cr
1 ...

JEE Main 2022 (Online) 25th June Morning Shift

Let $$A = \left[ {\matrix{
0 & { - 2} \cr
2 & 0 \cr
} } \right]$$. If M and N are two matrices given by $$M = \sum\limits_{k = 1}^{10} {{A...

JEE Main 2022 (Online) 24th June Evening Shift

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 ($$\...

JEE Main 2022 (Online) 24th June Morning Shift

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$$\alpha$$...

JEE Main 2022 (Online) 24th June Morning Shift

Let S = {$$\sqrt{n}$$ : 1 $$\le$$ n $$\le$$ 50 and n is odd}.
Let a $$\in$$ S and $$A = \left[ {\matrix{
1 & 0 & a \cr
{ - 1} & 1 & 0 \cr
...

JEE Main 2021 (Online) 1st September Evening Shift

Consider the system of linear equations$$-$$x + y + 2z = 03x $$-$$ ay + 5z = 12x $$-$$ 2y $$-$$ az = 7Let S1 be the set of all a$$\in$$R for which the...

JEE Main 2021 (Online) 31st August Evening Shift

If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations x + (cos $$\gamma$$)y + (cos $$\beta$$)z = 0(cos $$\gamma$$)x + y + (c...

JEE Main 2021 (Online) 31st August Morning Shift

If the following system of linear equations2x + y + z = 5x $$-$$ y + z = 3x + y + az = bhas no solution, then :

JEE Main 2021 (Online) 31st August Morning Shift

If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then the determinant $$\left| {\matrix{
...

JEE Main 2021 (Online) 27th August Evening Shift

Let $$A = \left( {\matrix{
{[x + 1]} & {[x + 2]} & {[x + 3]} \cr
{[x]} & {[x + 3]} & {[x + 3]} \cr
{[x]} & {[x + 2]} ...

JEE Main 2021 (Online) 27th August Evening Shift

Let A(a, 0), B(b, 2b + 1) and C(0, b), b $$\ne$$ 0, |b| $$\ne$$ 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all pos...

JEE Main 2021 (Online) 27th August Evening Shift

Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equat...

JEE Main 2021 (Online) 27th August Morning Shift

If the matrix $$A = \left( {\matrix{
0 & 2 \cr
K & { - 1} \cr
} } \right)$$ satisfies $$A({A^3} + 3I) = 2I$$, then the value of K ...

JEE Main 2021 (Online) 26th August Evening Shift

Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
0 & 1 & 1 \cr
1 & 0 & 0 \cr
} } \right)$$. Then A2025 $$-$$ A2020 is...

JEE Main 2021 (Online) 26th August Morning Shift

Let $$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations$$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\thet...

JEE Main 2021 (Online) 26th August Morning Shift

If $$A = \left( {\matrix{
{{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr
{{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \c...

JEE Main 2021 (Online) 27th July Evening Shift

Let A and B be two 3 $$\times$$ 3 real matrices such that (A2 $$-$$ B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determ...

JEE Main 2021 (Online) 27th July Morning Shift

Let $$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right]$$. If A$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$...

JEE Main 2021 (Online) 25th July Evening Shift

The number of distinct real roots of $$\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr...

JEE Main 2021 (Online) 25th July Evening Shift

If $$P = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]$$, then P50 is :

JEE Main 2021 (Online) 25th July Morning Shift

The values of a and b, for which the system of equations 2x + 3y + 6z = 8x + 2y + az = 53x + 5y + 9z = bhas no solution, are :...

JEE Main 2021 (Online) 22th July Evening Shift

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...

JEE Main 2021 (Online) 22th July Evening Shift

Let A = [aij] be a real matrix of order 3 $$\times$$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix...

JEE Main 2021 (Online) 20th July Evening Shift

The value of k $$\in$$R, for which the following system of linear equations3x $$-$$ y + 4z = 3,x + 2y $$-$$ 3z = $$-$$26x + 5y + kz = $$-$$3,has infin...

JEE Main 2021 (Online) 20th July Morning Shift

Let $$A = \left[ {\matrix{
2 & 3 \cr
a & 0 \cr
} } \right]$$, a$$\in$$R be written as P + Q where P is a symmetric matrix and Q is...

JEE Main 2021 (Online) 18th March Evening Shift

Let the system of linear equations 4x + $$\lambda$$y + 2z = 02x $$-$$ y + z = 0$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\mu$$$$\in$$R.has a non-trivial ...

JEE Main 2021 (Online) 18th March Morning Shift

The solutions of the equation $$\left| {\matrix{
{1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr
{{{\cos }^2}x} & {1 + {{...

JEE Main 2021 (Online) 18th March Morning Shift

Let $$\alpha$$, $$\beta$$, $$\gamma$$ be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c $$\in$$ R and a, b $$\ne$$ 0). If the system ...

JEE Main 2021 (Online) 18th March Morning Shift

Let $$A + 2B = \left[ {\matrix{
1 & 2 & 0 \cr
6 & { - 3} & 3 \cr
{ - 5} & 3 & 1 \cr
} } \right]$$ and $$2A - ...

JEE Main 2021 (Online) 17th March Evening Shift

If x, y, z are in arithmetic progression with common difference d, x $$\ne$$ 3d, and the determinant of the matrix $$\left[ {\matrix{
3 & {4\sq...

JEE Main 2021 (Online) 17th March Morning Shift

The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to :

JEE Main 2021 (Online) 17th March Morning Shift

If $$A = \left( {\matrix{
0 & {\sin \alpha } \cr
{\sin \alpha } & 0 \cr
} } \right)$$ and $$\det \left( {{A^2} - {1 \over 2}I} \ri...

JEE Main 2021 (Online) 16th March Morning Shift

Let $$A = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right],i = \sqrt { - 1} $$. Then, the system of linear equations $${...

JEE Main 2021 (Online) 26th February Evening Shift

Consider the following system of equations :x + 2y $$-$$ 3z = a2x + 6y $$-$$ 11z = bx $$-$$ 2y + 7z = c,where a, b and c are real constants. Then the ...

JEE Main 2021 (Online) 26th February Morning Shift

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices...

JEE Main 2021 (Online) 26th February Morning Shift

The value of $$\left| {\matrix{
{(a + 1)(a + 2)} & {a + 2} & 1 \cr
{(a + 2)(a + 3)} & {a + 3} & 1 \cr
{(a + 3)(a + 4)} &a...

JEE Main 2021 (Online) 25th February Evening Shift

Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 $$ \to $$ 2...

JEE Main 2021 (Online) 25th February Evening Shift

If for the matrix, $$A = \left[ {\matrix{
1 & { - \alpha } \cr
\alpha & \beta \cr
} } \right]$$, $$A{A^T} = {I_2}$$, then the va...

JEE Main 2021 (Online) 25th February Evening Shift

The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x $$-$$ y + 4z = 8

JEE Main 2021 (Online) 24th February Evening Shift

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 (A2B2 ...

JEE Main 2021 (Online) 24th February Evening Shift

For the system of linear equations:$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,consider the following statements :(A) The system has unique s...

JEE Main 2021 (Online) 24th February Morning Shift

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

JEE Main 2020 (Online) 6th September Evening Slot

Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \c...

JEE Main 2020 (Online) 6th September Morning Slot

Let m and M be respectively the minimum and maximum values of
$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{...

JEE Main 2020 (Online) 6th September Morning Slot

The values 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 i...

JEE Main 2020 (Online) 5th September Evening Slot

If a + x = b + y = c + z + 1, where a, b, c, x, y, z
are non-zero distinct real numbers, then
$$\left| {\matrix{
x & {a + y} & {x + a} \cr...

JEE Main 2020 (Online) 5th September Evening Slot

If the system of linear equations
x + y + 3z = 0
x + 3y + k2z = 0
3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k $$ \in $$ R,
then x + $...

JEE Main 2020 (Online) 5th September Morning Slot

If the minimum and the maximum values of the function $$f:\left[ {{\pi \over 4},{\pi \over 2}} \right] \to R$$, defined by
$$f\left( \theta \right...

JEE Main 2020 (Online) 5th September Morning Slot

Let $$\lambda \in $$ R . The system of linear equations
2x1
- 4x2 + $$\lambda $$x3 = 1
x1 - 6x2 + x3 = 2
$$\lambda $$x1 - 10x2 + 4x3 = 3
is inconsist...

JEE Main 2020 (Online) 4th September Evening Slot

Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 a...

JEE Main 2020 (Online) 4th September Evening Slot

If the system of equations
x+y+z=2
2x+4y–z=6
3x+2y+$$\lambda $$z=$$\mu $$
has infinitely many solutions, then

JEE Main 2020 (Online) 4th September Morning Slot

If $$A = \left[ {\matrix{
{\cos \theta } & {i\sin \theta } \cr
{i\sin \theta } & {\cos \theta } \cr
} } \right]$$, $$\left( {\thet...

JEE Main 2020 (Online) 3rd September Evening Slot

Let A be a 3 $$ \times $$ 3 matrix such that
adj A = $$\left[ {\matrix{
2 & { - 1} & 1 \cr
{ - 1} & 0 & 2 \cr
1 & { -...

JEE Main 2020 (Online) 3rd September Morning Slot

If $$\Delta $$ = $$\left| {\matrix{
{x - 2} & {2x - 3} & {3x - 4} \cr
{2x - 3} & {3x - 4} & {4x - 5} \cr
{3x - 5} & {...

JEE Main 2020 (Online) 2nd September Evening Slot

Let a, b, c $$ \in $$ R be all non-zero and satisfy
a3 + b3 + c3 = 2. If the matrix
A = $$\left( {\matrix{
a & b & c \cr
b & c &am...

JEE Main 2020 (Online) 2nd September Evening Slot

Let A = {X = (x, y, z)T: PX = 0 and
x2 + y2 + z2 = 1} where
$$P = \left[ {\matrix{
1 & 2 & 1 \cr
{ - 2} & 3 & { - 4} \cr
...

JEE Main 2020 (Online) 2nd September Morning Slot

Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system
of linear equations
2x – y + 2z = 2
x – 2y +
$$\lambda $$z = –4
x +
$$\lambda $$...

JEE Main 2020 (Online) 2nd September Morning Slot

Let A be a 2 $$ \times $$ 2 real matrix with entries from
{0, 1} and |A|
$$ \ne $$ 0. Consider the following two
statements :
(P) If A $$ \ne $$ I2
, ...

JEE Main 2020 (Online) 9th January Evening Slot

The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
x – 2y – 6z = 0, has

JEE Main 2020 (Online) 9th January Morning Slot

If the matrices A = $$\left[ {\matrix{
1 & 1 & 2 \cr
1 & 3 & 4 \cr
1 & { - 1} & 3 \cr
} } \right]$$,
B = adjA...

JEE Main 2020 (Online) 9th January Morning Slot

If for some $$\alpha $$ and $$\beta $$ in R, the intersection of the
following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + $$\alpha $$z = 5
...

JEE Main 2020 (Online) 8th January Evening Slot

The system of linear equations
$$\lambda $$x + 2y + 2z = 5
2$$\lambda $$x + 3y + 5z = 8
4x + $$\lambda $$y + 6z = 10 has

JEE Main 2020 (Online) 8th January Evening Slot

If $$A = \left( {\matrix{
2 & 2 \cr
9 & 4 \cr
} } \right)$$ and $$I = \left( {\matrix{
1 & 0 \cr
0 & 1 \cr
} }...

JEE Main 2020 (Online) 8th January Morning Slot

For which of the following ordered pairs ($$\mu $$, $$\delta $$),
the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = $$\mu $$
4x + 4y + 4z ...

JEE Main 2020 (Online) 7th January Evening Slot

Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3.
If the determinant of B is 81, then the de...

JEE Main 2020 (Online) 7th January Morning Slot

Let $$\alpha $$ be a root of the equation x2 + x + 1 = 0 and the matrix A = $${1 \over {\sqrt 3 }}\left[ {\matrix{
1 & 1 & 1 \cr
1 &am...

JEE Main 2020 (Online) 7th January Morning Slot

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 distinct; has a non-ze...

JEE Main 2019 (Online) 12th April Evening Slot

A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which
$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } &...

JEE Main 2019 (Online) 12th April Morning Slot

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \rig...

JEE Main 2019 (Online) 12th April Morning Slot

If $$B = \left[ {\matrix{
5 & {2\alpha } & 1 \cr
0 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
} } \right]$$ is the ...

JEE Main 2019 (Online) 10th April Evening Slot

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 + ...

JEE Main 2019 (Online) 10th April Evening Slot

The sum of the real roots of the equation
$$\left| {\matrix{
x & { - 6} & { - 1} \cr
2 & { - 3x} & {x - 3} \cr
{ - 3} &am...

JEE Main 2019 (Online) 10th April Morning Slot

If $${\Delta _1} = \left| {\matrix{
x & {\sin \theta } & {\cos \theta } \cr
{ - \sin \theta } & { - x} & 1 \cr
{\cos \the...

JEE Main 2019 (Online) 10th April Morning Slot

If the system of linear equations
x + y + z = 5
x + 2y + 2z = 6
x + 3y + $$\lambda $$z = $$\mu $$, ($$\lambda $$, $$\mu $$ $$ \in $$ R), has infinitel...

JEE Main 2019 (Online) 9th April Evening Slot

If the system of equations 2x + 3y – z = 0, x + ky
– 2z = 0 and 2x – y + z = 0 has a non-trival solution
(x, y, z), then $${x \over y} + {y \over z} +...

JEE Main 2019 (Online) 9th April Evening Slot

The total number of matrices
$$A = \left( {\matrix{
0 & {2y} & 1 \cr
{2x} & y & { - 1} \cr
{2x} & { - y} & 1 \cr...

JEE Main 2019 (Online) 9th April Morning Slot

Let $$\alpha $$ and $$\beta $$ be the roots of the equation
x2 + x + 1 = 0. Then for y $$ \ne $$ 0 in R,
$$$\left| {\matrix{
{y + 1} & \alpha ...

JEE Main 2019 (Online) 9th April Morning Slot

If $$\left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
0 & 1 \cr
} } \right]$$$$\left...

JEE Main 2019 (Online) 8th April Evening Slot

Let the number 2,b,c be in an A.P. and
A = $$\left[ {\matrix{
1 & 1 & 1 \cr
2 & b & c \cr
4 & {{b^2}} & {{c^2}} ...

JEE Main 2019 (Online) 8th April Morning Slot

Let $$A = \left( {\matrix{
{\cos \alpha } & { - \sin \alpha } \cr
{\sin \alpha } & {\cos \alpha } \cr
} } \right)$$, ($$\alpha $$ ...

JEE Main 2019 (Online) 8th April Morning Slot

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 solutio...

JEE Main 2019 (Online) 12th January Evening Slot

If A = $$\left[ {\matrix{
1 & {\sin \theta } & 1 \cr
{ - \sin \theta } & 1 & {\sin \theta } \cr
{ - 1} & ...

JEE Main 2019 (Online) 12th January Evening Slot

The set of all values of $$\lambda $$ for which the system of linear equations
x – 2y – 2z = $$\lambda $$x
x + 2y + z = $$\lambda $$y
– x – y = $$\lam...

JEE Main 2019 (Online) 12th January Morning Slot

An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations
(1 + $$\alpha $$) x + $$\beta $$y + z = 2
$$\alpha $$x + (1 + $$\...

JEE Main 2019 (Online) 12th January Morning Slot

Let P = $$\left[ {\matrix{
1 & 0 & 0 \cr
3 & 1 & 0 \cr
9 & 3 & 1 \cr
} } \right]$$ and Q = [qij] be two 3 $$ ...

JEE Main 2019 (Online) 11th January Evening Slot

If $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a...

JEE Main 2019 (Online) 11th January Evening Slot

Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABAT) = 8 and det(AB–1) = 8,
then det (BA–1 BT) is equal to :
...

JEE Main 2019 (Online) 11th January Morning Slot

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 one solution, the...

JEE Main 2019 (Online) 11th January Morning Slot

Let A = $$\left( {\matrix{
0 & {2q} & r \cr
p & q & { - r} \cr
p & { - q} & r \cr
} } \right).$$ ...

JEE Main 2019 (Online) 10th January Evening Slot

Let A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ where b > 0...

JEE Main 2019 (Online) 10th January Evening Slot

The number of values of $$\theta $$ $$ \in $$ (0, $$\pi $$) for which the system of linear equations
x + 3y + 7z = 0
$$-$$ x + 4y + 7z = 0
(sin3$$\the...

JEE Main 2019 (Online) 10th January Morning Slot

If the system of equations
x + y + z = 5
x + 2y + 3z = 9
x + 3y + az = $$\beta $$
has infinitely many solutions, then $$\beta $$ $$-$$ $$\alpha $$ equ...

JEE Main 2019 (Online) 10th January Morning Slot

Let d $$ \in $$ R, and
$$A = \left[ {\matrix{
{ - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr
1 & {\le...

JEE Main 2019 (Online) 9th January Evening Slot

If the system of linear equations
x $$-$$ 4y + 7z = g
3y $$-$$ 5z = h
$$-$$2x + 5y $$-$$ 9z = k
is consiste...

JEE Main 2019 (Online) 9th January Evening Slot

If $$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - ...

JEE Main 2019 (Online) 9th January Morning Slot

If $$A = \left[ {\matrix{
{\cos \theta } & { - \sin \theta } \cr
{\sin \theta } & {\cos \theta } \cr
} } \right]$$, then the matri...

JEE Main 2019 (Online) 9th January Morning Slot

The system of linear equations
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a2 – 1) z = a + 1 then

JEE Main 2018 (Online) 16th April Morning Slot

Let A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$ and B = A20. Then the sum ...

JEE Main 2018 (Online) 16th April Morning Slot

The number of values of k for which the system of linear equations,
(k + 2)x + 10y = k
kx + (k +3)y = k -1
has no solution, is :

JEE Main 2018 (Offline)

If $$\left| {\matrix{
{x - 4} & {2x} & {2x} \cr
{2x} & {x - 4} & {2x} \cr
{2x} & {2x} & {x - 4} \cr
} } \righ...

JEE Main 2018 (Offline)

If the system of linear equations
x + ky + 3z = 0
3x + ky - 2z = 0
2x + 4y - 3z = 0
has a non-zero solution (x, y, z), then $${{xz} \over {{y^2}}}$$ i...

JEE Main 2018 (Online) 15th April Evening Slot

Suppose A is any 3$$ \times $$ 3 non-singular matrix and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I3 and O = O3. If $$\alpha $$A + $$\beta $$A-1 = 4I,...

JEE Main 2018 (Online) 15th April Evening Slot

If the system of linear equations
x + ay + z = 3
x + 2y + 2z = 6
x + 5y + 3z = b
has no solution, then :

JEE Main 2018 (Online) 15th April Morning Slot

Let $$A$$ be a matrix such that $$A.\left[ {\matrix{
1 & 2 \cr
0 & 3 \cr
} } \right]$$ is a scalar matrix and |3A| = 108.
Then A2...

JEE Main 2018 (Online) 15th April Morning Slot

Let S be the set of all real values of k for which the systemof linear equations
x + y + z = 2
2x + y $$-$$ z = 3
3x + 2y + kz = 4
has a unique soluti...

JEE Main 2017 (Online) 9th April Morning Slot

For two 3 × 3 matrices A and B, let A + B = 2BT and 3A + 2B = I3, where BT is
the transpose of B and I3 is 3 × 3 identity matrix. Then :...

JEE Main 2017 (Online) 8th April Morning Slot

Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?

JEE Main 2017 (Online) 8th April Morning Slot

The number of real values of $$\lambda $$ for which the system of linear equations
2x + 4y $$-$$ $$\lambda $$z = 0
4x + $$\lambda $$y + 2z = 0
$$\lamb...

JEE Main 2017 (Online) 8th April Morning Slot

If
$$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{
0 & {\cos x} & { - \sin x} \cr
{\sin x} & 0 & {\cos x} \cr...

JEE Main 2017 (Offline)

If $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$,
then adj(3A2 + 12A) is equal to

JEE Main 2017 (Offline)

If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solu...

JEE Main 2016 (Online) 10th April Morning Slot

Let A be a 3 $$ \times $$ 3 matrix such that A2 $$-$$ 5A + 7I = 0
Statement - I :
A$$-$$1 = $${1 \over 7}$$ (5I $$-$$ A).
Statement - II ...

JEE Main 2016 (Online) 10th April Morning Slot

If A = $$\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$,
then the determinant of the matrix (A2016 − 2...

JEE Main 2016 (Online) 9th April Morning Slot

If P = $$\left[ {\matrix{
{{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr
{ - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr
} } \right],A =...

JEE Main 2016 (Online) 9th April Morning Slot

The number of distinct real roots of the equation,
$$\left| {\matrix{
{\cos x} & {\sin x} & {\sin x} \cr
{\sin x} & {\cos x} &am...

JEE Main 2016 (Offline)

If $$A = \left[ {\matrix{
{5a} & { - b} \cr
3 & 2 \cr
} } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :

JEE Main 2016 (Offline)

The system of linear equations
$$\matrix{
{x + \lambda y - z = 0} \cr
{\lambda x - y - z = 0} \cr
{x + y - \lambda z = 0} \cr
} $$
...

JEE Main 2015 (Offline)

The set of all values of $$\lambda $$ for which the system of linear equations:
$$\matrix{
{2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr
{2{x_...

JEE Main 2015 (Offline)

If $$A = \left[ {\matrix{
1 & 2 & 2 \cr
2 & 1 & { - 2} \cr
a & 2 & b \cr
} } \right]$$ is a matrix satisfying...

JEE Main 2014 (Offline)

If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and
$$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1...

JEE Main 2014 (Offline)

If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and
$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:

JEE Main 2013 (Offline)

The number of values of $$k$$, for which the system of equations : $$$\matrix{
{\left( {k + 1} \right)x + 8y = 4k} \cr
{kx + \left( {k + 3} \r...

JEE Main 2013 (Offline)

If $$P = \left[ {\matrix{
1 & \alpha & 3 \cr
1 & 3 & 3 \cr
2 & 4 & 4 \cr
} } \right]$$ is the adjoint of a $...

AIEEE 2012

Let $$P$$ and $$Q$$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If $${P^3} = {Q^3}$$ and
$${P^2}Q = {Q^2}P$$ then determinant of $$\left( {{P^2} + {Q^2}}...

AIEEE 2012

Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
2 & 1 & 0 \cr
3 & 2 & 1 \cr
} } \right)$$. If $${u_1}$$ and $${u_2}$...

AIEEE 2011

Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$.
Statement - 1 : $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric matrices.
Statement - 2 : $...

AIEEE 2011

The number of values of $$k$$ for which the linear equations
$$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is :...

AIEEE 2010

Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$
where $$I$$ is $$2 \times 2$$ identity matrix. Define
$$Tr$$$$(A...

AIEEE 2010

Consider the system of linear equations;
$$$\matrix{
{{x_1} + 2{x_2} + {x_3} = 3} \cr
{2{x_1} + 3{x_2} + {x_3} = 3} \cr
{3{x_1} + 5{x_2}...

AIEEE 2010

The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is :

AIEEE 2009

Let $$A$$ be a $$\,2 \times 2$$ matrix
Statement - 1 : $$adj\left( {adj\,A} \right) = A$$
Statement - 2 :$$\left| {adj\,A} \right| = \left| A \right|$...

AIEEE 2009

Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if
$$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} &a...

AIEEE 2008

Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal ent...

AIEEE 2008

Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$ $$y=az+cx,$$ and $$z=bx+ay.$$...

AIEEE 2008

Let $$A$$ be a square matrix all of whose entries are integers.
Then which one of the following is true?

AIEEE 2007

Let $$A = \left| {\matrix{
5 & {5\alpha } & \alpha \cr
0 & \alpha & {5\alpha } \cr
0 & 0 & 5 \cr
} } \right...

AIEEE 2007

If $$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + x} & 1 \cr
1 & 1 & {1 + y} \cr
} } \right|$$ for $$x \ne 0,y...

AIEEE 2006

If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that
$${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then wh...

AIEEE 2006

Let $$A = \left( {\matrix{
1 & 2 \cr
3 & 4 \cr
} } \right)$$ and $$B = \left( {\matrix{
a & 0 \cr
0 & b \cr
} ...

AIEEE 2005

If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is :

AIEEE 2005

The system of equations
$$\matrix{
{\alpha \,x + y + z = \alpha - 1} \cr
{x + \alpha y + z = \alpha - 1} \cr
{x + y + \alpha \,z = \al...

AIEEE 2005

If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant
$$$\Delta = \left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}}...

AIEEE 2005

If $${a^2} + {b^2} + {c^2} = - 2$$ and
f$$\left( x \right) = \left| {\matrix{
{1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 +...

AIEEE 2004

Let $$A = \left( {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right)$$. The only c...

AIEEE 2004

Let $$A = \left( {\matrix{
1 & { - 1} & 1 \cr
2 & 1 & { - 3} \cr
1 & 1 & 1 \cr
} } \right).$$ and $$10$$ $$B...

AIEEE 2004

If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant
$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n +...

AIEEE 2003

If $$A = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$ and $${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta ...

AIEEE 2003

If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then
$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \c...

AIEEE 2003

If the system of linear equations
$$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
has a non - zero solution, then $$a, b, c$$....

AIEEE 2002

If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then
$$\left| {\matrix{
a & b & {ax + b} \cr
b & c & {bx ...

## Numerical

JEE Main 2024 (Online) 9th April Evening Shift

Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\lef...

JEE Main 2024 (Online) 9th April Morning Shift

Let $$A$$ be a non-singular matrix of order 3. If $$\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \...

JEE Main 2024 (Online) 8th April Morning Shift

Let $$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$$. If the sum of the diagonal elements of $$A^{13}$$ is $$3^n$$, then $$n$$ is equal ...

JEE Main 2024 (Online) 6th April Evening Shift

If the system of equations
$$\begin{aligned}
& 2 x+7 y+\lambda z=3 \\
& 3 x+2 y+5 z=4 \\
& x+\mu y+32 z=-1
\end{aligned}$$
has infinitely many solutio...

JEE Main 2024 (Online) 4th April Evening Shift

Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\rig...

JEE Main 2024 (Online) 4th April Morning Shift

Let $$A$$ be a square matrix of order 2 such that $$|A|=2$$ and the sum of its diagonal elements is $$-$$3 . If the points $$(x, y)$$ satisfying $$\ma...

JEE Main 2024 (Online) 4th April Morning Shift

Let $$A$$ be a $$3 \times 3$$ matrix of non-negative real elements such that $$A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{arra...

JEE Main 2024 (Online) 1st February Evening Shift

Let $A=I_2-2 M M^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M=I_1$ holds. If $\lambda$ is a real number such tha...

JEE Main 2024 (Online) 31st January Evening Shift

Let A be a $$3 \times 3$$ matrix and $$\operatorname{det}(A)=2$$. If $$n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots ....

JEE Main 2024 (Online) 29th January Evening Shift

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$a x+b y+c=0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$...

JEE Main 2024 (Online) 27th January Evening Shift

Let $$A$$ be a $$2 \times 2$$ real matrix and $$I$$ be the identity matrix of order 2. If the roots of the equation $$|\mathrm{A}-x \mathrm{I}|=0$$ be...

JEE Main 2024 (Online) 27th January Morning Shift

Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], B=\left[B_1, B_2, B_3\right]$, where $B_1, B_2, B_3$ are column ...

JEE Main 2023 (Online) 12th April Morning Shift

Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\
n & n^{2}+n+2 & n^{2} \\
n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If $$\s...

JEE Main 2023 (Online) 11th April Morning Shift

Let $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$$, where $$a, c \in \mathbb{R}$$. If $$A^{3}=A$$ and the positiv...

JEE Main 2023 (Online) 10th April Evening Shift

Let $$\mathrm{S}$$ be the set of values of $$\lambda$$, for which the system of equations $$6 \lambda x-3 y+3 z=4 \lambda^{2}$$, $$2 x+6 \lambda y+4 z...

JEE Main 2023 (Online) 31st January Evening Shift

Let A be a $n \times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix
$\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(...

JEE Main 2023 (Online) 29th January Evening Shift

Let A be a symmetric matrix such that $$\mathrm{|A|=2}$$ and $$\left[ {\matrix{
2 & 1 \cr
3 & {{3 \over 2}} \cr
} } \right]A = \left[ {\ma...

JEE Main 2023 (Online) 25th January Morning Shift

Let $$\mathrm{A_1,A_2,A_3}$$ be the three A.P. with the same common difference d and having their first terms as $$\mathrm{A,A+1,A+2}$$, respectively....

JEE Main 2022 (Online) 29th July Evening Shift

Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$...

JEE Main 2022 (Online) 29th July Morning Shift

Let p and p + 2 be prime numbers and let
$$
\Delta=\left|\begin{array}{ccc}
\mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\
(\mathrm{p}+1) ! & (...

JEE Main 2022 (Online) 28th July Morning Shift

Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \...

JEE Main 2022 (Online) 27th July Evening Shift

Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+...

JEE Main 2022 (Online) 27th July Morning Shift

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 s...

JEE Main 2022 (Online) 26th July Evening Shift

The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such ...

JEE Main 2022 (Online) 25th July Evening Shift

Let $$A=\left[\begin{array}{lll}
1 & a & a \\
0 & 1 & b \\
0 & 0 & 1
\end{array}\right], a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}, A^{n}=...

JEE Main 2022 (Online) 25th July Morning Shift

Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\frac{\sqrt{3} i-1}{2}$$, then ...

JEE Main 2022 (Online) 29th June Evening Shift

Let $$M = \left[ {\matrix{
0 & { - \alpha } \cr
\alpha & 0 \cr
} } \right]$$, where $$\alpha$$ is a non-zero real number an $$N = \sum\li...

JEE Main 2022 (Online) 28th June Evening Shift

If the system of linear equations $$2x - 3y = \gamma + 5$$, $$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has inf...

JEE Main 2022 (Online) 28th June Evening Shift

Let $$A = \left( {\matrix{
{1 + i} & 1 \cr
{ - i} & 0 \cr
} } \right)$$ where $$i = \sqrt { - 1} $$. Then, the number of elements in the s...

JEE Main 2022 (Online) 27th June Morning Shift

The positive value of the determinant of the matrix A, whose
Adj(Adj(A)) = $$\left( {\matrix{
{14} & {28} & { - 14} \cr
{ - 14} & {14} & {28} ...

JEE Main 2022 (Online) 26th June Evening Shift

Let $$X = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$ and $$Z = ...

JEE Main 2022 (Online) 25th June Evening Shift

Let $$A = \left( {\matrix{
2 & { - 2} \cr
1 & { - 1} \cr
} } \right)$$ and $$B = \left( {\matrix{
{ - 1} & 2 \cr
{ - 1} & 2 \cr
...

JEE Main 2022 (Online) 24th June Evening Shift

Let $$S = \left\{ {\left( {\matrix{
{ - 1} & a \cr
0 & b \cr
} } \right);a,b \in \{ 1,2,3,....100\} } \right\}$$ and let $${T_n} = \{ A \i...

JEE Main 2021 (Online) 31st August Evening Shift

The number of elements in the set $$\left\{ {A = \left( {\matrix{
a & b \cr
0 & d \cr
} } \right):a,b,d \in \{ - 1,0,1\} \,and\,{...

JEE Main 2021 (Online) 27th August Morning Shift

If the system of linear equations2x + y $$-$$ z = 3x $$-$$ y $$-$$ z = $$\alpha$$3x + 3y + $$\beta$$z = 3has infinitely many solution, then $$\alpha$$...

JEE Main 2021 (Online) 26th August Evening Shift

Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.

JEE Main 2021 (Online) 27th July Evening Shift

If $$A = \left[ {\matrix{
1 & 1 & 1 \cr
0 & 1 & 1 \cr
0 & 0 & 1 \cr
} } \right]$$ and M = A + A2 + A3 + ........

JEE Main 2021 (Online) 27th July Morning Shift

For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y ...

JEE Main 2021 (Online) 27th July Morning Shift

Let $$f(x) = \left| {\matrix{
{{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr
{2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\c...

JEE Main 2021 (Online) 25th July Morning Shift

Let $$M = \left\{ {A = \left( {\matrix{
a & b \cr
c & d \cr
} } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}$$. Defi...

JEE Main 2021 (Online) 22th July Evening Shift

Let $$A = \left[ {\matrix{
0 & 1 & 0 \cr
1 & 0 & 0 \cr
0 & 0 & 1 \cr
} } \right]$$. Then the number of 3 $$\t...

JEE Main 2021 (Online) 20th July Evening Shift

Let $$A = \{ {a_{ij}}\} $$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{
{{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr ...

JEE Main 2021 (Online) 20th July Morning Shift

Let $$A = \left( {\matrix{
1 & { - 1} & 0 \cr
0 & 1 & { - 1} \cr
0 & 0 & 1 \cr
} } \right)$$ and B = 7A20 $$-...

JEE Main 2021 (Online) 20th July Morning Shift

Let a, b, c, d in arithmetic progression with common difference $$\lambda$$. If $$\left| {\matrix{
{x + a - c} & {x + b} & {x + a} \cr
...

JEE Main 2021 (Online) 18th March Evening Shift

Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$. Then t...

JEE Main 2021 (Online) 17th March Evening Shift

Let $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ and $$B = \left[ {\matrix{
\alpha \cr
\beta \cr
} } \ri...

JEE Main 2021 (Online) 17th March Evening Shift

If 1, log10(4x $$-$$ 2) and log10$$\left( {{4^x} + {{18} \over 5}} \right)$$ are in arithmetic progression for a real number x, then the value of the ...

JEE Main 2021 (Online) 17th March Morning Shift

If $$A = \left[ {\matrix{
2 & 3 \cr
0 & { - 1} \cr
} } \right]$$, then the value of det(A4) + det(A10 $$-$$ (Adj(2A))10) is equal ...

JEE Main 2021 (Online) 16th March Evening Shift

Let $$A = \left[ {\matrix{
{{a_1}} \cr
{{a_2}} \cr
} } \right]$$ and $$B = \left[ {\matrix{
{{b_1}} \cr
{{b_2}} \cr
} } \right...

JEE Main 2021 (Online) 16th March Morning Shift

Let $$P = \left[ {\matrix{
{ - 30} & {20} & {56} \cr
{90} & {140} & {112} \cr
{120} & {60} & {14} \cr
} } \ri...

JEE Main 2021 (Online) 16th March Morning Shift

The total number of 3 $$\times$$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is e...

JEE Main 2021 (Online) 26th February Evening Shift

If the matrix $$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]$$ satisfies t...

JEE Main 2021 (Online) 25th February Morning Shift

If $$A = \left[ {\matrix{
0 & { - \tan \left( {{\theta \over 2}} \right)} \cr
{\tan \left( {{\theta \over 2}} \right)} & 0 \cr
}...

JEE Main 2021 (Online) 25th February Morning Shift

Let $$A = \left[ {\matrix{
x & y & z \cr
y & z & x \cr
z & x & y \cr
} } \right]$$, where x, y and z are real...

JEE Main 2021 (Online) 25th February Morning Shift

If the system of equationskx + y + 2z = 13x $$-$$ y $$-$$ 2z = 2$$-$$2x $$-$$2y $$-$$4z = 3has infinitely many solutions, then k is equal to _________...

JEE Main 2021 (Online) 24th February Morning Shift

Let P = $$\left[ {\matrix{
3 & { - 1} & { - 2} \cr
2 & 0 & \alpha \cr
3 & { - 5} & 0 \cr
} } \right]$$, wher...

JEE Main 2021 (Online) 24th February Morning Shift

Let M be any 3 $$ \times $$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements ...

JEE Main 2020 (Online) 6th September Evening Slot

The sum of distinct values of $$\lambda $$ for which the
system of equations$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lam...

JEE Main 2020 (Online) 4th September Morning Slot

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............

JEE Main 2020 (Online) 3rd September Evening Slot

Let S be the set of all integer solutions, (x, y, z),
of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 $$ \...

JEE Main 2020 (Online) 3rd September Morning Slot

Let A = $$\left[ {\matrix{
x & 1 \cr
1 & 0 \cr
} } \right]$$, x $$ \in $$ R and A4 = [aij].
If
a11 = 109, then a22 is equal to ___...

JEE Main 2020 (Online) 8th January Morning Slot

The number of all 3 × 3 matrices A, with
enteries from the set {–1, 0, 1} such that the sum
of the diagonal elements of AAT is 3, is

JEE Main 2020 (Online) 7th January Evening Slot

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 $$\mu $$ - $$\l...