$\left|\begin{array}{rr}\sin 35^{\circ} & -\cos 35^{\circ} \\ \sin 55^{\circ} & \cos 55^{\circ}\end{array}\right|=$ . . . . . .

If $A = \begin{bmatrix} 2-k & 2 \\ 1 & 3-k \end{bmatrix}$ is a singular matrix,then the value of $5k - k^2$ is equal to

For any $a, b, c \in R$,the determinant $\left|\begin{array}{lll}bc & b+c & 1 \\ ca & c+a & 1 \\ ab & a+b & 1\end{array}\right|$ is equal to

If $\alpha, \beta, \gamma$ are the roots of $\left|\begin{array}{ccc} 1-x & -2 & 1 \\ -2 & 4-x & -2 \\ 1 & -2 & 1-x \end{array}\right|=0$,then $\alpha \beta+\beta \gamma+\gamma \alpha=$

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:

$\left| {\begin{array}{ccc} 19 & 17 & 15 \\ 9 & 8 & 7 \\ 1 & 1 & 1 \end{array}} \right| = $