If $k = p + q + r$,then the value of $\left|\begin{array}{ccc} k+r & p & q \\ r & k+p & q \\ r & p & k+q \end{array}\right|$ is equal to:

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$,$2x+Ny+2z=2$,and $3x+3y+Nz=3$ has a unique solution is $\frac{k}{6}$,then the sum of the value of $k$ and all possible values of $N$ is:

If $f(x) = \left| \begin{array}{ccc} x-3 & 2x^2-18 & 3x^3-81 \\ x-5 & 2x^2-50 & 4x^3-500 \\ 1 & 2 & 3 \end{array} \right|$,then $f(1)f(3) + f(3)f(5) + f(5)f(1)$ is equal to

If $a+x=b+y=c+z+1,$ where $a, b, c, x, y, z$ are non-zero distinct real numbers,then $\left|\begin{array}{lll}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{array}\right|$ is equal to

The value of the determinant $\left| \begin{array}{ccc} 0 & b^3 - a^3 & c^3 - a^3 \\ a^3 - b^3 & 0 & c^3 - b^3 \\ a^3 - c^3 & b^3 - c^3 & 0 \end{array} \right|$ is equal to:

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1, 0, 1\}$. Then,the maximum possible value of the determinant of $P$ is: