If $D = \begin{vmatrix} a^2 + 1 & ab & ac \\ ba & b^2 + 1 & bc \\ ca & cb & c^2 + 1 \end{vmatrix}$,then $D =$

If ${a_1}, {a_2}, {a_3}, \dots, {a_n}, \dots$ are in $G$.$P$. and ${a_i} > 0$ for each $i$,then the value of the determinant $\Delta = \begin{vmatrix} \log {a_n} & \log {a_{n+2}} & \log {a_{n+4}} \\ \log {a_{n+6}} & \log {a_{n+8}} & \log {a_{n+10}} \\ \log {a_{n+12}} & \log {a_{n+14}} & \log {a_{n+16}} \end{vmatrix}$ is equal to

Evaluate the determinant: $\left| \begin{array}{ccc} x & 4 & y + z \\ y & 4 & z + x \\ z & 4 & x + y \end{array} \right|$

The parameter on which the value of the determinant $\left| \begin{array}{ccc} 1 & a & a^2 \\ \cos(p-d)x & \cos px & \cos(p+d)x \\ \sin(p-d)x & \sin px & \sin(p+d)x \end{array} \right|$ does not depend is:

Using properties of determinants,prove that:
$\left|\begin{array}{ccc}\alpha & \alpha^{2} & \beta+\gamma \\ \beta & \beta^{2} & \gamma+\alpha \\ \gamma & \gamma^{2} & \alpha+\beta\end{array}\right|=(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)(\alpha+\beta+\gamma)$

The value of the determinant $\left| \begin{array}{ccc} a & a+b & a+2b \\ a+2b & a & a+b \\ a+b & a+2b & a \end{array} \right|$ is