JAMIA 2021 — Mathematics PYQ

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Question

JAMIA 2021 — Mathematics PYQ

JAMIA | Mathematics | 2021

|f a $_{n} = α^{n} - β^{n}$ and $α, β$ are the roots of th equation x $^{2}$ $- 6 x - 2 = 0$ ,then find the value of $\frac{a _{10} 2 a _{8}}{3 a _{9}} ?$

Choose the correct answer:

Explanation

1. Roots $\alpha$ and $\beta$ satisfy the equation $x^{2} - 6x - 2 = 0$ :

\alpha^{2} - 6\alpha - 2 = 0 \implies \alpha^{2} - 2 = 6\alpha

\beta^{2} - 6\beta - 2 = 0 \implies \beta^{2} - 2 = 6\beta

2. Multiply the first equation by $\alpha^{8}$ and the second by $\beta^{8}$ :

\alpha^{10} - 2\alpha^{8} = 6\alpha^{9}

\beta^{10} - 2\beta^{8} = 6\beta^{9}

3. Subtract the two equations:

(\alpha^{10} - \beta^{10}) - 2(\alpha^{8} - \beta^{8}) = 6(\alpha^{9} - \beta^{9})

4. Substitute $a_{n} = \alpha^{n} - \beta^{n}$ :

a_{10} - 2a_{8} = 6a_{9}

5. Calculate the value of the expression:

\frac{a_{10} - 2a_{8}}{3a_{9}} = \frac{6a_{9}}{3a_{9}}

= 2

Final Answer:

2