Let \( \alpha, \beta \) be the roots of the equation \( \mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0....
Let \( \alpha, \beta \) be the roots of the equation \( \mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0 \).
Let \( \mathrm{S}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}} \) for \( \mathrm{n} \geq 1 \) and
P
\[
\Delta=\left|\begin{array}{ccc}
3 & 1+\mathrm{S}_{1} \cdot & 1+\mathrm{S}_{2} \\
1+\mathrm{S}_{1} & 1+\mathrm{S}_{2} & 1+\mathrm{S}_{3} \\
1+\mathrm{S}_{2} & 1+\mathrm{S}_{3} & 1+\mathrm{S}_{4}
\end{array}\right|
\]
If \( a, b, c \) are rational and one of the roots of the equations is \( 1+\sqrt{2} \), then the value of \( \Delta \) is
(A) 8
(B) 12
(C) 30
(D) 32
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