Let α, β be roots of x 2 +√(2) x-8=0 . If U n =α n +β n , then frac U 10 +√(2) U 9 2 U 8 is equal to

Mathematics · Quadratic Equations

JEE Main 2024 — 6 April, Shift 2 — Question 28

Let $\alpha, \beta$ be roots of $x^{2}+\sqrt{2} x-8=0$ . If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$ , then $\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_{9}}{2 \mathrm{U}_{8}}$ is equal to $\qquad$

Answer: 4

Numerical answer — enter this value.

Step-by-step solution

$\frac{\alpha^{10}+\beta^{10}+\sqrt{2}\left(\alpha^{9}+\beta^{9}\right)}{2\left(\alpha^{8}+\beta^{8}\right)}$

$\frac{\alpha^{8}\left(\alpha^{2}+\sqrt{2} \alpha\right)+\beta^{8}\left(\beta^{2}+\sqrt{2} \beta\right)}{2\left(\alpha^{8}+\beta^{8}\right)}$

$\frac{8 \alpha^{8}+8 \beta^{8}}{2\left(\alpha^{8}+\beta^{8}\right)}=4$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 6 April, Shift 2
Subject: Mathematics
Chapter: Quadratic Equations
Topic: Theory of Quadratic Equations