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JEE Main 202125 FebruaryEvening Shift
Exam
JEE Main
Year
2021
Date
25 February
Shift
Evening Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2021 · 25 February · Evening Shift · Mathematics · Quadratic Equation

Let α and β be the roots of x ^ 2 - 6x - 2 = 0 . If a _ n = α ^ n -...

Let α\alpha and β\beta be the roots of x2−6x−2=0{x}^{2} - 6x - 2 = 0. If an=αn−βn{a}_{n} = {\alpha}^{n} - {\beta}^{n} for n⩾1n ⩾ 1, then the value of a10−2a83a9\dfrac{{a}_{10} - 2{a}_{8}}{3{a}_{9}} is:

Options

  1. A11
  2. B33
  3. C22
  4. D44

Correct answer

C. 22

Step-by-step solution

α2−6α−2=0{\alpha}^{2} - 6\alpha - 2 = 0

α10−6α9−2α8=0β10−6β9−2β8=0(α10−β10)−6(α9−β9)−2(α8−β8)=0\begin{array}{ll}{\alpha}^{10} - 6{\alpha}^{9} - 2{\alpha}^{8} = 0 \\ {\beta}^{10} - 6{\beta}^{9} - 2{\beta}^{8} = 0 \\ \left({\alpha}^{10} - {\beta}^{10}\right) - 6\left({\alpha}^{9} - {\beta}^{9}\right) - 2\left({\alpha}^{8} - {\beta}^{8}\right) = 0\end{array}

⇒a10−6a9−2a8=0⇒ {a}_{10} - 6{a}_{9} - 2{a}_{8} = 0

⇒a10−2a83a9=2⇒ \dfrac{{a}_{10} - 2{a}_{8}}{3{a}_{9}} = 2

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