SATHEE: Theory of Equations 1 Question 17

Theory of Equations 1 Question 17

18. Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ , for $n \geq 1$ , then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is

(a) 6

(b) -6

(d) -3

(2015 Main)

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Answer:

Correct Answer: 18. (c)

Solution:

Given, $α$ and $β$ are the roots of the equation $x^{2} - 6 x - 2 = 0$ .

$∴ \begin{aligned} a_{n} & = α^{n} - β^{n} for n \geq 1 \\ a_{10} & = α^{10} - β^{10} \\ a_{8} & = α^{8} - β^{8} \\ a_{9} & = α^{9} - β^{9} \end{aligned}$

Now, consider

$\begin{aligned} \frac{a_{10} - 2 a_{8}}{2 a_{9}} = & \frac{α^{10} - β^{10} - 2 (α^{8} - β^{8})}{2 (α^{9} - α^{9})} \\ = & \frac{α^{8} (α^{2} - 2) - β^{8} (β^{2} - 2)}{2 (α^{9} - β^{9})} \\ = & \frac{α^{8} \cdot 6 α - β^{8} 6 β}{2 (α^{9} - β^{9})} = \frac{6 α^{9} - 6 β^{9}}{2 (α^{9} - 6 β^{9})} = \frac{6}{2} = 3 \\ ∵ α and β are the roots of \\ \Rightarrow x^{2} - 6 x - 2 = 0 or x^{2} = 6 x + 2 \\ and α^{2} = 6 α + 2 \Rightarrow α^{2} - 2 = 6 α + 2 \Rightarrow β^{2} - 2 = 6 β \end{aligned}$

Alternate Solution

Since, $α$ and $β$ are the roots of the equation

$\begin{aligned} x^{2} - 6 x - 2 = 0 . \\ or x^{2} = 6 x + 2 \\ ∴ α^{2} = 6 α + 2 \\ \Rightarrow α^{10} = 6 α^{9} + 2 α^{8} \\ Similarly, β^{10} = 6 β^{9} + 2 β^{8} \end{aligned}$

On subtracting Eq. (ii) from Eq. (i), we get

$\begin{aligned} α^{10} - β^{10} = 6 (α^{9} - β^{9}) + 2 (α^{8} - β^{8}) (∵ a_{n} = α^{n} - β^{n}) \\ \Rightarrow a_{10} & = 6 a_{9} + 2 a_{8} \\ \Rightarrow a_{10} - 2 a_{8} & = 6 a_{9} \Rightarrow \frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3 \end{aligned}$

Theory of Equations 1 Question 17

18. Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ , for $n \geq 1$ , then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is

Answer:

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Theory of Equations 1 Question 17

18. Let α and β be the roots of equation x2−6x−2=0. If an=αn−βn, for n≥1, then the value of a10−2a82a9 is

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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18. Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ , for $n \geq 1$ , then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is