NDA Mathematics Determinants Previous Year Questions

If $M=\left(\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right)$, then what is the value of $|M||\mathrm{adj}M|$ ?

Let $f(x)=\left|\begin{array}{ccc}3x^2& \cos x& -\sin x\\ 6& -1& 0\\ q& q^2& q^3\end{array}\right|$ where $q$ is any constant. Then what is $\frac{d^2}{d{x}^2}(f(x))$ at $x=0$ equal to?

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 the value of $\left|\begin{array}{ccc}{p}^2+q^2& r^2& r^2\\ p^2& q^2+r^2& p^2\\ q^2& q^2& r^2+p^2\end{array}\right|$ ?

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 $M^T$ is the transpose of a $2\times 2$ matrix $M$, then which of the following is/are correct?

I. $|M+M^T|=|M|+|M^T|$ if $M$ is symmetric.

II. $|M+M^T|=0$ if $M$ is anti-symmetric.

Select the answer using the code given below:

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?