Question

Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y\end{array}\right]$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval $(a, b)$. Then $8 \int_a^b|A| d m$ is equal to_________

JEE Main 2024 (09 Apr Shift 2)

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Question

Let $D_k=\left|\begin{array}{c}\begin{array}{ccc}1& 2k& 2k-1\\ n& n^2+n+2& n^2\\ n& n^2+n& n^2+n+2\end{array}\end{array}\right|$. If $\sum _{k=1}^n{D}_k=96$, then $n$ is equal to _________.

JEE Main 2023 (12 Apr Shift 1)

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Question

Let $A=\left[\begin{array}{ccc}[x+1]& [x+2]& [x+3]\\ \left[x\right]& [x+3] & [x+3]\\ \left[x\right]& [x+2]& [x+4]\end{array}\right]\begin{array}{}\\ \\ \end{array}$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$. If $det\left(A\right)$$=192$, then the set of values of $x$ is in the interval:

JEE Main 2021 (27 Aug Shift 2)

Options

• A: $[62,63)$
• B: $[65,66)$
• C: $[60,61)$
• D: $[68,69)$

Question

If $f\left(x\right)=\left|\begin{array}{ccc}2{\cos }^{4}x& 2{\sin }^{4}x& 3+{\sin }^{2}2x\\ 3+2{\cos }^{4}x& 2{\sin }^{4}x& {\sin }^{2}2x\\ 2{\cos }^{4}x& 3+2{\sin }^{4}x& {\sin }^{2}2x\end{array}\right|$ then $\frac{1}{5}{f}^{\text{'}}\left(0\right)$ is equal to ________.

JEE Main 2024 (30 Jan Shift 1)

Options

• A: $0$
• B: $1$
• C: $2$
• D: $6$

Question

If ${\mathrm{a}}_{\mathrm{r}}=\cos \frac{2\mathrm{r}\pi }{9}+i\sin \frac{2\mathrm{r}\pi }{9},\mathrm{r}=1,2,3,\dots ,i=\sqrt{-1}$, then the determinant $\left|\begin{array}{lll}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ is equal to :

JEE Main 2021 (31 Aug Shift 1)

Options

• A: ${\mathrm{a}}_9$
• B: $a_1{a}_9-a_3{a}_7$
• C: $a_5$
• D: $a_2{a}_6-a_4{a}_8$

Question

Let $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|, x\in \left[0,\pi \right].$ Then the maximum value of  $f\left(x\right)$ is equal to

JEE Main 2021 (27 Jul Shift 1)

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Question

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

JEE Main 2024 (31 Jan Shift 2)

Options

• A: unique solution
• B: exactly two solutions
• C: no solution
• D: infinitely many solutions

Question

If $\alpha +\beta +\gamma =2\pi ,$ then the system of equations

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

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

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

has :

JEE Main 2021 (31 Aug Shift 2)

Options

• A: infinitely many solutions
• B: a unique solution
• C: no solution
• D: exactly two solutions

Question

If $\Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 2x-3& 3x-4& 4x-5\\ 3x-5& 5x-8& 10x-17\end{array}\right|=A{x}^3+B{x}^2+Cx+D$, then $B+C$  is equal to :

JEE Main 2020 (03 Sep Shift 1)

Options

• A: $-1$
• B: $1$
• C: $-3$
• D: $9$

Question

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 $x+2 y+3 z=4$, has infinitely many solutions. Then $(PQ{)}^2$ is equal to _______.

JEE Main 2024 (29 Jan Shift 2)

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Question

Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda$. If $\left|\begin{array}{lll}x+a-c& x+b& x+a\\ x-1& x+c& x+b\\ x-b+d& x+d& x+c\end{array}\right|=2$, then value of ${\lambda }^2$ is equal to________.

JEE Main 2021 (20 Jul Shift 1)

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Question

Suppose the vectors $x_1,x_2$ and $x_3$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to $b_1,b_2$ and $b_3$ respectively. If $x_1=\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right], x_2=\left[\begin{array}{l}0\\ 2\\ 1\end{array}\right], x_3=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right]$; $b_1=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right], b_2=\left[\begin{array}{l}0\\ 2\\ 0\end{array}\right], b_3=\left[\begin{array}{l}0\\ 0\\ 2\end{array}\right]$, then the determinant of $A$ is equal to

JEE Main 2020 (04 Sep Shift 2)

Options

• A: $4$
• B: $2$
• C: $\frac{1}{2}$
• D: $\frac{3}{2}$

Question

If the minimum and the maximum values of the function $f:\left[\frac{\pi }{4},\frac{\pi }{2}\right]\to R,$ defined by $f\left(\theta \right)=\left|\begin{array}{ccc}-{\sin }^2\theta & -1-{\sin }^2\theta & 1\\ -{\cos }^2\theta & -1-{\cos }^2\theta & 1\\ 12& 10& -2\end{array}\right|$ are $m$ and $M$respectively, then the ordered pair $(\mathrm{m}, \mathrm{M})$ is equal to :

JEE Main 2020 (05 Sep Shift 1)

Options

• A: $\left(0,2\sqrt{2}\right)$
• B: $\left(-4,0\right)$
• C: $\left(-4, 4\right)$
• D: $\left(0,4\right)$

Question

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

${\cos }^2\theta x+\left(1+{\sin }^2\theta \right)y+4\sin 3\theta z=0$

${\cos }^2\theta x+{\sin }^2\theta y+(1+4\sin 3\theta )z=0$
has a non-trivial solution, then the value of $\theta$ is:

JEE Main 2021 (26 Aug Shift 1)

Options

• A: $\frac{4\pi }{9}$
• B: $\frac{5\pi }{18}$
• C: $\frac{7\pi }{18}$
• D: $\frac{\pi }{18}$

Question

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

JEE Main 2024 (01 Feb Shift 1)

Options

• A: $1110$
• B: $1120$
• C: $1210$
• D: $1220$

Question

Let $S$ be the set of all values of $\theta \in [-\pi ,\pi ]$ for which the system of linear equations
$x+y+\sqrt{3}z=0$
$-x+\left(\tan \theta \right)y+\sqrt{7}z=0$
$x+y+\left(\tan \theta \right)z=0$
has non-trivial solution.Then $\frac{120}{\pi }\sum _{0\in S}\theta$ is equal to

JEE Main 2023 (08 Apr Shift 2)

Options

• A: $20$
• B: $40$
• C: $30$
• D: $10$