$\left|\begin{array}{lll}10 & 11 & 12 \\ 11 & 12 & 13 \\ 12 & 13 & 14\end{array}\right|=$ . . . . . . .

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$ and $|2A|^3 = 2^{21}$,where $\alpha, \beta \in \mathbb{Z}$. Then a value of $\alpha$ is:

If $a, b$ and $c$ are real numbers such that $a^2+b^2+c^2-ab-bc-ac \leq 0$,then the value of the determinant $\left|\begin{array}{ccc} (a-b+1)^5 & b^7-c^7 & c^9-a^9 \\ a^{11}-b^{11} & (b-c+2)^3 & c^{13}-a^{13} \\ a^{15}-b^{15} & b^{17}-c^{17} & (c-a+3)^1 \end{array}\right|$ is:

If $a, b, c$ are respectively the $p^{th}, q^{th}, r^{th}$ terms of an $A.P.$,then $\left| \begin{array}{ccc} a & p & 1 \\ b & q & 1 \\ c & r & 1 \end{array} \right| = $

If $\left|\begin{array}{ccc}2 a & x_{1} & y_{1} \\ 2 b & x_{2} & y_{2} \\ 2 c & x_{3} & y_{3}\end{array}\right|=\frac{a b c}{2} \neq 0$,then the area of the triangle whose vertices are $\left(\frac{x_{1}}{a}, \frac{y_{1}}{a}\right), \left(\frac{x_{2}}{b}, \frac{y_{2}}{b}\right), \left(\frac{x_{3}}{c}, \frac{y_{3}}{c}\right)$ is:

Sum of the positive roots of the equation $\left|\begin{array}{ccc}x^2+2x & x+2 & 1 \\ 2x+1 & x-1 & 1 \\ x+2 & -1 & 1\end{array}\right|=0$