Determinants — NDA Mathematics PYQs

If $\Delta =\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$ and $A,B,C,D,G$ are the cofactors of the elements $a,b,c,d,g$ respectively, then what is $bB+cC-dD-gG$ equal to?

Consider the following statements in respect of the determinant $\Delta =\left|\begin{array}{ccc}k(k+2)& 2k+1& 1\\ 2k+1& k+2& 1\\ 3& 3& 1\end{array}\right|$
I. $\Delta$ is positive if $k>0$.
II. $\Delta$ is negative if $k<0$.
III. $\Delta$ is zero if $k=0$.
How many of the statements given above are correct?

If $\left|\begin{array}{ccc}2& 3+i& -1\\ 3-i& 0& i-1\\ -1& -1-i& 1\end{array}\right|=A+iB$, where $i=\sqrt{-1}$, then what is $A+B$ equal to?

If $A^2+B^2+C^2=0$, then what is the value of $\left|\begin{array}{ccc}1& \cos C& \cos B\\ \cos C& 1& \cos A\\ \cos B& \cos A& 1\end{array}\right|$?

If $\omega$ is a non-real cube root of unity, then what is a root of the following equation? $\left|\begin{array}{ccc}x+1& \omega & {\omega }^2\\ \omega & x+{\omega }^2& 1\\ {\omega }^2& 1& x+\omega \end{array}\right|=0$

What is the value of the determinant of the inverse of the matrix $\left[\begin{array}{cc}-4& -5\\ 2& 2\end{array}\right]$?

The system of equations $2x-3y-5=0$, $15y-10x+50=0$

In obtaining the solution of the system of equations $x+y+z=7$, $x+2y+3z=16$ and $x+3y+4z=22$ by Cramer's rule, the value of $y$ is obtained by dividing $D$ by $D_2$, where
$D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 4\end{array}\right|$
What is the value of the determinant $D_2$?

The value of the determinant $\left|\begin{array}{ccc}a& b& c\\ l& m& n\\ p& q& r\end{array}\right|$ is equal to

If $a,b,c$ are the sides of a triangle $ABC$ and $p$ is the perimeter of the triangle, then what is $\left|\begin{array}{ccc}p+c& a& b\\ c& p+a& b\\ c& a& p+b\end{array}\right|$ equal to?