Let $\alpha ,\beta ;\alpha \beta$, be the roots of the equation ${x}^{2}-\sqrt{2}x-\sqrt{3}=0$. Let ${P}_{n}={\alpha }^{n}-{\beta }^{n},n\in N$. Then $(11\sqrt{3}-10\sqrt{2}){P}_{10}+(11\sqrt{2}+10){P}_{11}-11{P}_{12}$ is equal to [JEE Main 2024, 9 Apr (Shift 2)]

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Let $\alpha ,\beta ;\alpha >\beta$, be the roots of the equation ${x}^{2}-\sqrt{2}x-\sqrt{3}=0$. Let ${P}_{n}={\alpha }^{n}-{\beta }^{n},n\in N$. Then $(11\sqrt{3}-10\sqrt{2}){P}_{10}+(11\sqrt{2}+10){P}_{11}-11{P}_{12}$ is equal to [JEE Main 2024, 9 Apr (Shift 2)]

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Let \(\alpha ,\beta ;\alpha >\beta\), be the roots of the equation \({x}^{2}-\sqrt{2}x-\sqrt{3}=0\). Let \({P}_{n}={\alp…

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