Let \( \alpha, \beta \), be the roots of the equation \( a x^{2}+b x+c=0 \).
Let \( S_{n}=\alpha... VIDEO
Let \( \alpha, \beta \), be the roots of the equation \( a x^{2}+b x+c=0 \).
Let \( S_{n}=\alpha^{n}+\beta^{n} \) for \( n \geq 1 \) and \( \Delta=\left|\begin{array}{ccc}3 & 1+S_{1} & 1+S_{2} \\ 1+S_{1} & 1+S_{2} & 1+S_{3} \\ 1+S_{2} & 1+S_{3} & 1+S_{4}\end{array}\right| \)
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
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