The value of the determinant $\left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1+y \end{array} \right|$ is

$\begin{aligned} & \text { If }\left|\begin{array}{ccc}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ (n+2)^2 & (n+3)^2 & (n+4)^2\end{array}\right|=\Delta \text { and } \\ & \left|\begin{array}{ccc}1 & -4 & 7 \\ -2 & 3 & -5 \\ 3 & x & -3\end{array}\right|=2 \Delta+1, \text { then } x=\end{aligned}$

The constant term in the expansion of $ \left|\begin{array}{ccc} 3x+1 & 2x-1 & x+2 \\ 5x-1 & 3x+2 & x+1 \\ 7x-2 & 3x+1 & 4x-1 \end{array}\right| $ is

If $a_i^2 + b_i^2 + c_i^2 = 1$ for $(i = 1, 2, 3)$ and $a_i a_j + b_i b_j + c_i c_j = 0$ for $(i \ne j, i, j = 1, 2, 3)$,then the value of $\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|^2$ is

The values of $\theta, \lambda$ for which the following equations $\sin \theta x - \cos \theta y + (\lambda + 1)z = 0$; $\cos \theta x + \sin \theta y - \lambda z = 0$; $\lambda x + (\lambda + 1)y + \cos \theta z = 0$ have a non-trivial solution are:

$A$ determinant is chosen at random from the set of all determinants of order $2 \times 2$ with elements $0$ or $1$ only. The probability that the determinant chosen is non-zero is .........