If $a\, -\, 2b + c = 1$ , then value of $\left| {\begin{array}{*{20}{c}}
  {x + 1}&{x + 2}&{x + a} \\ 
  {x + 2}&{x + 3}&{x + b} \\ 
  {x + 3}&{x + 4}&{x + c} 
\end{array}} \right|$ is

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

If $a \ne b \ne c,$ the value of $x$ which satisfies the equation $\left| {\,\begin{array}{*{20}{c}}0&{x - a}&{x - b}\\{x + a}&0&{x - c}\\{x + b}&{x + c}&0\end{array}\,} \right| = 0$, is

If $x, y, z$ are in arithmetic progression with common difference $d , x \neq 3 d ,$ and the
determinant of the matrix $\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]$ is zero, then the value of $k ^{2}$ is ..... .

The number of distinct real roots of $\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

Evaluate the determinants : $\left|\begin{array}{cc}2 & 4 \\ -5 & -1\end{array}\right|$