Question

Let $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0.$ Let $S_n={\alpha }^n+{\beta }^n$ for $n\ge 1$ 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 equation is $1+\sqrt{2}$, then the value of $\Delta$ is

• A. $32$
• B. $24$
• C. $18$
• D. $16$

Answer 1

The correct option is A $32$

$\Delta =\left|\begin{array}{ccc}1+1+1& 1+\alpha +\beta & 1+{\alpha }^2+{\beta }^2\\ 1+\alpha +\beta & 1+{\alpha }^2+{\beta }^2& 1+{\alpha }^3+{\beta }^3\\ 1+{\alpha }^2+{\beta }^2& 1+{\alpha }^3+{\beta }^3& 1+{\alpha }^4+{\beta }^4\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ 1& \alpha & \beta \\ 1& {\alpha }^2& {\beta }^2\end{array}\right|\times \left|\begin{array}{ccc}1& 1& 1\\ 1& \alpha & \beta \\ 1& {\alpha }^2& {\beta }^2\end{array}\right|\text{[multiplying row by row]}=(1-\alpha )(\alpha -\beta )(\beta -1)\times (1-\alpha )(\alpha -\beta )(\beta -1)$
$={\left[\right(1-\alpha \left)\right(\alpha -\beta \left)\right(\beta -1\left)\right]}^2$
As, one root is $1-\sqrt{2},$ hence second root will be $1+\sqrt{2}$
Let $\alpha =1+\sqrt{2},\beta =1-\sqrt{2}$
$\Rightarrow \Delta ={\left[\right(-\sqrt{2}\left)\right(2\sqrt{2}\left)\right(\sqrt{2}\left)\right]}^2=32$