$\left|\begin{array}{ccc}x & 3x+2 & 2x-1 \\ 2x-1 & 4x & 3x+1 \\ 7x-2 & 17x+6 & 12x-1\end{array}\right|=0$ is true for

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:

Let $\left|\begin{array}{ccc}x^2+x+1 & x+1 & 2x-3 \\ 3x^2-1 & x+2 & x-1 \\ x^2+5x+1 & 2x+3 & x+4\end{array}\right| = ax^4+bx^3+cx^2+dx+e$ be an identity in $x$. If $a, b, c, d$ are known,then the value of $e$ is

If $\alpha, \beta, \gamma$ $(\alpha < \beta < \gamma)$ are the values of $x$ such that $\begin{vmatrix} x-2 & 0 & 1 \\ 1 & x+3 & 2 \\ 2 & 0 & 2x-1 \end{vmatrix} = 0$ is a singular matrix,then $2\alpha + 3\beta + 4\gamma = $

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 $\left| \begin{matrix} 1 & a & a^2 \\ 1 & x & x^2 \\ b^2 & ab & a^2 \end{matrix} \right| = 0$,then: