Evaluation of Determinants

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 _________.

If $∆\left(k\right)=\left|\begin{array}{ccc}1& 2k-1& 2k\\ n& n^2& n\left(n+1\right)\\ {\cos }^{2}n& {\cos }^2\left(n+1\right)& \left(n+2\right)\end{array}\right|$, then find the value of $\sum _{k=1}^n∆\left(k\right)=$

If $\omega$ is a cube root of unity, then a factor of $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

Let $p$ and $p+2$ be prime numbers and let $∆=\left|\begin{array}{ccc}p!& \left(p+1\right)!& \left(p+2\right)!\\ \left(p+1\right)!& \left(p+2\right)!& \left(p+3\right)!\\ \left(p+2\right)!& \left(p+3\right)!& \left(p+4\right)!\end{array}\right|$, then the sum of the maximum values of $\alpha$ and $\beta$, such that $p^{\alpha }$ and ${\left(p+2\right)}^{\beta }$ divide $∆$ is _____

Let $a, b, c$ be sides of triangle satisfying$\left|\begin{array}{ccc}b+c-2a& c+a-2b& a+b-2c\\ bc& ac& ab\\ a^2& b^2& c^2\end{array}\right|=0$, the the triangle has to be

If $p, q, r, s$ are in arithmetic progression and $f\left(x\right)=\left|\begin{array}{ccc}p+\sin x& q+\sin x& p-r+\sin x\\ q+\sin x& r+\sin x& -1+\sin x\\ r+\sin x& s+\sin x& s-q+\sin x\end{array}\right|$ such that ${\int }_0^{2}f\left(x\right)dx=-4$, then the common difference of the progression is

If $\left|\begin{array}{ccc}0& \cos x& -\sin x\\ \sin x& 0& \cos x\\ \cos x& \sin x& 0\end{array}\right|=\left|\begin{array}{ccc}1& -a& a\\ -a& 1& a\\ a& a& 1\end{array}\right|$, then $a=$

The sum of the values of $x$ so that the matrix $\left[\begin{array}{ccc}2& 2& 1\\ 1& 3& 1\\ 1& 2& 2\end{array}\right]-x\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ is singular, is-

If $A=\left[\begin{array}{ccc}1+a^2+a^4& 1+ab+a^2{b}^2& 1+ac+a^2{c}^2\\ 1+ab+a^2{b}^2& 1+b^2+b^4& 1+bc+b^2{c}^2\\ 1+ac+a^2{c}^2& 1+bc+b^2{c}^2& 1+c^2+c^4\end{array}\right]$ 

and $det\left(A\right)=det\left(4I\right)$, where I is $3\times 3$ Identity matrix, then ${\left(a-b\right)}^3+{\left(b-c\right)}^3+{\left(c-a\right)}^3$ can be equal to -

If $\left|\begin{array}{cccc}\alpha & x& x& x\\ x& \beta & x& x\\ x& x& \gamma & x\\ x& x& x& \delta \end{array}\right|=f\left(x\right)-x{f}^{\text{'}}\left(x\right)$, then $f\left(x\right)$ is

If $2s=a+b+c$, then $\left|\begin{array}{ccc}{a}^2& {\left(s-a\right)}^2& {\left(s-a\right)}^2\\ {\left(s-b\right)}^2& b^2& {\left(s-b\right)}^2\\ {\left(s-c\right)}^2& {\left(s-c\right)}^2& c^2\end{array}\right|=$

If the equation $\left|\begin{array}{ccc}4& 3& 4x+3\\ 3& c& 3x+c\\ 4x+3& 3x+c& 0\end{array}\right|=0$ has real roots (where $c\ne \frac{9}{4}$), then maximum integral value of $c$ is

$A$ is a $3\times 5$ matrix and $B$ is a $5\times 3$ matrix. If $\left|AB\right|=2021$ then $\left|BA\right|=$______

For all real numbers $x, y$ and $z$, the determinant $\left|\begin{array}{ccc}2x& \left(xy-xz\right)& y\\ \left(2x+z+1\right)& \left(xy-xz+yz-z^2\right)& \left(1+y\right)\\ \left(3x+1\right)& \left(2xy-2xz\right)& \left(1+y\right)\end{array}\right|$ is equal to

If $P={\left[p_{ij}\right]}_{2\times 3}$, $Q={\left[q_{ij}\right]}_{3\times 2}$ and the determinant of $PQ$ is equal to $4$, then the determinant of $QP$ is equal to

The number of solution of the equation $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin x& 1\\ 1& 1& 1+\cot x\end{array}\right|=\left|\begin{array}{ccc}1& 15& 7\\ 2& 26& 12\\ 3& 51& 24\end{array}\right|$ in $x\in \left[0, 2\mathrm{\pi }\right]$ is equal to

If $\alpha ,\beta ,\gamma$ are the roots of equation $x^3+p{x}^2+x+q^3+p^2+p=0$, then the value of $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$ is equal to

Solve for $x$, $\left|\begin{array}{ccc}x+2& x+6& x-1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0$.

Which among the below given determinants is same as determinant $\left|\begin{array}{ccc}1& 0& 2\\ 3& -2& -1\\ 2& 5& 4\end{array}\right|$ ?