Determinant of Order 3 and Its Expansion

The value of determinant $\left|\begin{array}{ccc}18& 40& 89\\ 40& 89& 198\\ 89& 198& 440\end{array}\right|$ is

For what value of $k$ is the matrix 

$\left[\begin{array}{ccc}2\cos 2\theta & 2\cos 2\theta & 6\\ 1-2{\sin }^2\theta & 2{\cos }^2\theta -1& 3\\ k& 2k& 1\end{array}\right]$ singular $?$

If $f\left(x\right)=\left|\begin{array}{ccc}1& 1& 1\\ 1& x& x^2\\ 1& x^2& x^3\end{array}\right|,$ then $f^{\text{'}}\left(x\right)$ vanishes at

If $A_{ij}$ and $M_{ij}$ are the cofactors and minors of $△=\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|,$ then

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

If the matrix $\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$ is singular then $\theta =$

If $\left|\begin{array}{ccc}\alpha & -\beta & 0\\ 0& \alpha & \beta \\ \beta & 0& \alpha \end{array}\right|=0$, then 

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

If $f\left(x\right)=\left|\begin{array}{ccc}1& 1& 1\\ 1& x& x^2\\ 1& x^2& x^3\end{array}\right|,$ then $f\text{'}\left(x\right)$ vanishes at

$\left|\begin{array}{cc}{\sin }^2\theta & {\cos }^2\theta \\ -{\cos }^2\theta & {\sin }^2\theta \end{array}\right|=$

If $f\left(x\right)=\left|\begin{array}{ccc}x-2& {\left(x-1\right)}^2& x^3\\ x-1& x^2& {\left(x+1\right)}^3\\ x& {\left(x+1\right)}^2& {\left(x+2\right)}^2\end{array}\right|$, then coefficient of $x$ in $f\left(x\right)$ is

If $\left|\begin{array}{cc}4& 0\\ 0& 1\end{array}\right|=\left|\begin{array}{cc}x& 0\\ 1& 2x\end{array}\right|$, then possible values of $x$ are

If $\left|\begin{array}{ccc}a& a^3& a^4\\ b& b^3& b^4\\ c& c^3& c^4\end{array}\right|=k\left(a-b\right)\left(b-c\right)\left(c-a\right) \text{then} k=?$

If $z=\left|\begin{array}{ccc}3-2i& 5+i& 7+3i\\ i& 2i& -3i\\ 3+2i& 5-i& 7-3i\end{array}\right|$, then

Write the value of determinant $\left|\begin{array}{cc}3& -1\\ 2& 1\end{array}\right|$

Let $0<a<b<\frac{\mathrm{\pi }}{2}$, If $f\left(x\right)=\left|\begin{array}{ccc}\tan x& \tan a& \tan b\\ \sin x& \sin a& \sin b\\ \cos x& \cos a& \cos b\end{array}\right|$, then find the minimum possible number of roots of $f^{\text{'}}\left(x\right)=0$ in $\left(a,b\right)$.

If $a,b,c$ are the roots of the equation $x^3+2x^2+2x+1=0$ and $∆=\left|\begin{array}{ccc}3& a+b+c& a^3+b^3+c^3\\ a+b+c& a^2+b^2+c^2& a^4+b^4+c^4\\ a^2+b^2+c^2& a^3+b^3+c^3& a^5+b^5+c^5\end{array}\right|$, then