Determinants — NDA Mathematics PYQs

If $\left|\begin{array}{ccc}a-b& p-q& x-y\\ b-c& q-r& y-z\\ c-a& r-p& z-x\end{array}\right|=k\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$, then what is the value of $k$ ?

If $p$, $q$, $r$ are the cube roots of unity, then what is $\left|\begin{array}{ccc}{p}^2+q^2& r^2& r^2\\ p^2& q^2+r^2& p^2\\ q^2& q^2& r^2+p^2\end{array}\right|$ equal to ?

If $A$ is a square matrix such that $|A|=-2$, then $|A{A}^T|$, where $A^T$ is the transpose of $A$, is equal to

If $M_k=\left[\begin{array}{cc}k& k-1\\ k-1& k\end{array}\right]$ where $k$ is a natural number, then what is $|M_1|+|M_2|+|M_3|+\cdots +|M_{50}|$ equal to ?

If $M$ is a square matrix such that $M^3=M$, then how many values of $|M|$ are possible ?

Let $p=(x+y+z)$ and $q=xyz$. If $\left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|$ is positive, then which one of the following is correct ?

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 $\left|\begin{array}{ccc}1& \cos C& \cos B\\ \cos C& 1& \cos A\\ \cos B& \cos A& 1\end{array}\right|$?