Let ${\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}},\mathrm{n}\in N$. If ${\mathrm{P}}_{10}=123,{\mathrm{P}}_{9}=76$, ${\mathrm{P}}_{8}=47$ and ${\mathrm{P}}_{1}=1$, then the quadratic equation having roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is : [JEE Main 2025, 2 Apr (Shift 1)]

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YESPYQ · Accepted Answer

Correct answer: b) ${x}^{2}+x-1=0$. ${P}_{10}={P}_{8}+{P}_{9}\ \Rightarrow {x}^{2}=x+1\mathrm{has}\mathrm{roots}\alpha \mathrm{and}\beta \ \Rightarrow \mathrm{Required}\mathrm{equation}\mathrm{is}\ {\mathrm{x}}^{2}+\mathrm{x}-1=0$

Let \({\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}},\mathrm{n}\in N\). If \({\mathrm{P}}_{10}=1…

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