Determinants — JEE Main PYQs

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 :

Among the statements :
I : If …

If $\mathrm{f}:\mathrm{N}\to \mathrm{Z}$ is defined by
$f(n)=\left|\begin{array}{ccc}n& -1& -5\\ -2n^2& 3(2k+1)& 2k+1\\ -3n^3& 3k(2k+1)& 3k(k+2)+1\end{array}\right|,k\in N$, and ${\sum }_{n=1}^{k}f(n)=98$, then $k$ is equal to:

The sum of all possible values of $\theta \in [0,2\pi ]$, for which the system of equations :

has a non-trivial solution, is equal to:

Let $\mathrm{A}=\left|\begin{array}{ccc}1& 3& -1\\ 2& 1& \alpha \\ 0& 1& -1\end{array}\right|$ be a singular matrix. Let $f(x)={\int }_0^x\left(t^2+2t+3\right)dt,x\in [1,\alpha ]$. If $M$ and $m$ are respectively the maximum and the minimum values of $f$ in …

Let $\alpha ,\beta (\alpha \ne \beta )$ be the values of $m$, for which the equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=1;\mathrm{x}+2\mathrm{y}+4\mathrm{z}=\mathrm{m}$ and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of …

Let M and m respectively be the maximum and the minimum values of
$f(x)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 4x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 4x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 4x\end{array}\right|,x\in R$
Then …

Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]=\left[\begin{array}{cc}{\log }_{5}128& {\log }_{4}5\\ {\log }_{5}8& {\log }_{4}25\end{array}\right]$.

If ${\mathrm{A}}_{\mathrm{ij}}$ is the cofactor of …

If the system of equations
$x+2y-3z=2$
$2\mathrm{x}+\lambda \mathrm{y}+5\mathrm{z}=5$
$14\mathrm{x}+3\mathrm{y}+\mu \mathrm{z}=33$
has infinitely many solutions, then $\lambda +\mu$ is equal to:

If the system of equations
$2\mathrm{x}-\mathrm{y}+\mathrm{z}=4$
$5\mathrm{x}+\lambda \mathrm{y}+3\mathrm{z}=12$
$100\mathrm{x}-47\mathrm{y}+\mu \mathrm{z}=212$,
has infinitely many solutions, then $\mu -2\lambda$ is equal to

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

Let $\mathrm{A}=\left[\begin{array}{ccc}2& 2+\mathrm{p}& 2+\mathrm{p}+\mathrm{q}\\ 4& 6+2\mathrm{p}& 8+3\mathrm{p}+2\mathrm{q}\\ 6& 12+3\mathrm{p}& 20+6\mathrm{p}+3\mathrm{q}\end{array}\right]$. If …

If the system of equations $x+(\sqrt{2}\sin \alpha )y+(\sqrt{2}\cos \alpha )z=0$, $x+(\cos \alpha )y+(\sin \alpha )z=0$, $x+(\sin \alpha )y-(\cos \alpha )z=0$ has a non-trivial solution, then $\alpha \in \left(0,\frac{\pi }{2}\right)$ is equal to

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

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $\left|A^{T}A(\mathrm{adj}(2A){)}^{-1}(\mathrm{adj}(4B))(\mathrm{adj}(AB){)}^{-1}A{A}^T\right|$ is equal to:

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

Let $\alpha \beta \ne 0$ and $A=\left[\begin{array}{ccc}\beta & \alpha & 3\\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{array}\right]$. If $B=\left[\begin{array}{ccc}3\alpha & -9& 3\alpha \\ -\alpha & 7& -2\alpha \\ -2\alpha & 5& -2\beta \end{array}\right]$ is the matrix of cofactors …

For $\alpha ,\beta \in R$ and a natural number $n$, let $A_r=\left|\begin{array}{ccc}r& 1& \frac{n^2}{2}+\alpha \\ 2r& 2& n^2-\beta \\ 3r-2& 3& \frac{n(3n-1)}{2}\end{array}\right|$. Then $2A_{10}-A_8$ is

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 ______.

If $\alpha \ne a,\beta \ne b,\gamma \ne c$ and $\left|\begin{array}{lll}\alpha & b& c\\ a& \beta & c\\ a& b& \gamma \end{array}\right|=0$, then
$\frac{\mathrm{a}}{\alpha -\mathrm{a}}+\frac{\mathrm{b}}{\beta -\mathrm{b}}+\frac{\gamma }{\gamma -\mathrm{c}}$ is equal to …

Let A be a $3\times 3$ matrix and $\det (\mathrm{A})=2$. If


Then the remainder when n is divided by 9 is …

The values of $\alpha$, for which
$\left|\begin{array}{ccc}1& \frac{3}{2}& \alpha +\frac{3}{2}\\ 1& \frac{1}{3}& \alpha +\frac{1}{3}\\ 2\alpha +3& 3\alpha +1& 0\end{array}\right|=0$, lie in the interval

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& \alpha & \beta \\ 0& \beta & \alpha \end{array}\right]$ and $|2A{|}^3=2^{21}$ where $\alpha ,\beta \in Z$,

Then a value of $\alpha$ is

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 $\frac{120}{\pi }{\sum }_{\theta \in \mathrm{s}}\theta$ is equal …

If $A=\frac{1}{5!6!7!}\left[\begin{array}{ccc}5!& 6!& 7!\\ 6!& 7!& 8!\\ 7!& 8!& 9!\end{array}\right]$, then $|\mathrm{adj}(\mathrm{adj}(2A))|$ is equal to :

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

Let for $A=\left[\begin{array}{ccc}1& 2& 3\\ a& 3& 1\\ 1& 1& 2\end{array}\right]$, $|A|=2$. If $|2\mathrm{adj}(2\mathrm{adj}(2A))|=32^n$, then $3n+\alpha$ is equal to

If $\left|\begin{array}{ccc}\mathrm{x}+1& \mathrm{x}& \mathrm{x}\\ \mathrm{x}& \mathrm{x}+\lambda & \mathrm{x}\\ \mathrm{x}& \mathrm{x}& \mathrm{x}+{\lambda }^2\end{array}\right|=\frac{9}{8}(103\mathrm{x}+81)$, then $\lambda$, $\frac{\lambda }{3}$ are the roots of the …

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$\mathrm{x}+\mathrm{y}+\mathrm{z}=1$
$2\mathrm{x}+\mathrm{Ny}+2\mathrm{z}=2$
$3\mathrm{x}+3\mathrm{y}+\mathrm{Nz}=3$
has unique solution is $\frac{k}{6}$, …

Let $\alpha$ be a root of the equation
$(\mathrm{a}-\mathrm{c}){\mathrm{x}}^2+(\mathrm{b}-\mathrm{a})\mathrm{x}+(\mathrm{c}-\mathrm{b})=0$ where $\mathrm{a},\mathrm{b},\mathrm{c}$ are distinct real numbers such that the matrix …

Let $f(\mathrm{x})=\left|\begin{array}{ccc}1+{\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& \sin 2\mathrm{x}\\ {\sin }^2\mathrm{x}& 1+{\cos }^2\mathrm{x}& \sin 2\mathrm{x}\\ {\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 1+\sin 2\mathrm{x}\end{array}\right|$, …

Let $x,y,z>1$ and …

The number of $\theta \in (0,4\pi )$ for which the system of linear equations
$3(\sin 3\theta )\mathrm{x}-\mathrm{y}+\mathrm{z}=2$
$3(\cos 2\theta )x+4y+3z=3$
$6x+7y+7z=9$
has no solution is :

Consider a matrix $A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If …

Let $f(x)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|,a\in R$. Then the sum of the squares of all the values of $a$ for which $2f^{\prime }(10)-f^{\prime }(5)+100=0$ is :

Let for some real numbers $\alpha$ and $\beta$, $a=\alpha -i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)x+\bar{a}y=0$ has more than one solution then $\frac{\alpha }{\beta }$ is equal to :

Let $p$ and $p+2$ be prime numbers and let $\Delta =\left|\begin{array}{ccc}p!& (p+1)!& (p+2)!\\ (p+1)!& (p+2)!& (p+3)!\\ (p+2)!& (p+3)!& (p+4)!\end{array}\right|$

Then the sum of the maximum values of $\alpha$ and $\beta$, such that ${\mathrm{p}}^{\alpha }$ and …

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

Let $\mathrm{a}\in \mathrm{S}$ and $\mathrm{A}=\left[\begin{array}{ccc}1& 0& \mathrm{a}\\ -1& 1& 0\\ -\mathrm{a}& 0& 1\end{array}\right]$
If …

Let the system of linear equations
$x+y+\alpha z=2$
$3x+y+z=4$
$x+2z=1$
have a unique solution $(x^{\ast },y^{\ast },z^{\ast })$. If $(\alpha ,x^{\ast })$, $(y^{\ast },\alpha )$ and $(x^{\ast },-y^{\ast })$ are collinear points, then the sum of absolute values of all possible values of …

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

Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a $3\times 3$ matrix, where …

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

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

The value of $\left|\begin{array}{lll}(a+1)(a+2)& a+2& 1\\ (a+2)(a+3)& a+3& 1\\ (a+3)(a+4)& a+4& 1\end{array}\right|$ is :

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

Let $M=\left\{A=\left(\begin{array}{ll}a& b\\ c& d\end{array}\right):a,b,c,d\in \{\pm 3,\pm 2,\pm 1,0\}\right\}$. Define $f:M\to Z$, as $f(A)=\det (A),$ for all $A\in M$, where $Z$ is set of all integers. Then the number of …

The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is:

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

The maximum value of $f(x)=\left|\begin{array}{ccc}{\sin }^{2}x& 1+{\cos }^{2}x& \cos 2x\\ 1+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& \sin 2x\end{array}\right|,x\in R$ is :

If $x,y,z$ are in arithmetic progression with common difference $d,x\ne 3d$, and the determinant of the matrix $\left[\begin{array}{ccc}3& 4\sqrt{2}& x\\ 4& 5\sqrt{2}& y\\ 5& k& z\end{array}\right]$ is zero, then the value of $k^2$ is: