The nth term of a sequence is given \(3^{1 - n}\), find the sum of the first three terms of the sequence.

Two binary operations \(\ast\) and \(\oplus\) are defines as m \(\ast\) n = mn - n - 1 and m \(\oplus\) n = mn + n - 2 for all real numbers m, n.

Find the value of 3 \(\oplus\) (4 \(\ast\) 5)

If X \(\ast\) Y = X + Y - XY, find x when (x \(\ast\) 2) + (x \(\ast\) 3) = 68

Determine x + y if \(\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}\) \(\begin{pmatrix} x \\ y \end{pmatrix}\) = \(\begin{pmatrix}-1 \\ 8 \end{pmatrix}\)

Find the non-zero positive value of x which satisfies the equation \(\begin{vmatrix} x & 1 & 0 \\ 1 & x & 1 \\ 0 & 1 & x\end{vmatrix}\) = 0