X2-6x-2-0 is a quadratic equation with the roots - and -Also an-n-n- n-Then the value of a10-2a82a9 is -

Given nbsp;x^2 6x 2=0 α, β are the roots ⇒α^2 6α 2=0 ⇒α^2 2=6α ⇒β^2 6β 2=0 ⇒β^2 2=6β Taking a_10 2a_82a^9 = (α^10 β^10) 2(α^8 β^8)2(α^9 β^9) = α^8(α^2 2) β ...

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Byju's Answer

Standard XII

Mathematics

Relations Between Roots and Coefficients

x2-6x-2=0 is ...

Question

$x^{2} - 6 x - 2 = 0$ is a quadratic equation with the roots $α$ and $β$ .
Also $a_{n} = α^{n} - β^{n}, n \in Z^{+}$ .
Then the value of $\frac{a_{10} - 2 a_{8}}{2 a^{9}}$ is :

6

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Solution

The correct option is D $3$
Given $x^{2} - 6 x - 2 = 0$
$α, β$ are the roots
$\Rightarrow α^{2} - 6 α - 2 = 0 \Rightarrow α^{2} - 2 = 6 α$
$\Rightarrow β^{2} - 6 β - 2 = 0 \Rightarrow β^{2} - 2 = 6 β$

Taking $\frac{a_{10} - 2 a_{8}}{2 a^{9}}$

$= \frac{(α^{10} - β^{10}) - 2 (α^{8} - β^{8})}{2 (α^{9} - β^{9})}$

$= \frac{α^{8} (α^{2} - 2) - β^{8} (β^{2} - 2)}{2 (α^{9} - β^{9})}$

$= \frac{α^{8} (6 α) - β^{8} (6 β)}{2 (α^{9} - β^{9})}$

$= \frac{6 (α^{9} - β^{9})}{2 (α^{9} - β^{9})} = 3$

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Similar questions

Q. Let

α

and

β

be the roots of

x^{2} - 6 x - 2 = 0

with

α > β

. If

a_{n} = α_{n} - β_{n}

for

n \geq 1

, then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

Q. 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}}

Q. Let

α

and

β

be the roots of equation

x^{2} - 6 x - 2 = 0

. If

a_{n} = α^{n} - β^{n}

, for

n ⩾ 1

, then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

is equal to

Q. Let

α

and

β

be the roots of

x^{2} - 6 x - 2 = 0

, with

α > β

. If

a_{n} = α^{n} - β^{n} f o r n \geq 1

, then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

Q. 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}}

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x2−6x−2=0 is a quadratic equation with the roots α and β. Also an=αn−βn,n∈Z+. Then the value of a10−2a82a9 is :

The correct option is D 3Given x2−6x−2=0 α,β are the roots ⇒α2−6α−2=0⇒α2−2=6α ⇒β2−6β−2=0⇒β2−2=6β Taking a10−2a82a9 =(α10−β10)−2(α8−β8)2(α9−β9) =α8(α2−2)−β8(β2−2)2(α9−β9) =α8(6α)−β8(6β)2(α9−β9) =6(α9−β9)2(α9−β9)=3

$x^{2} - 6 x - 2 = 0$ is a quadratic equation with the roots $α$ and $β$ .
Also $a_{n} = α^{n} - β^{n}, n \in Z^{+}$ .
Then the value of $\frac{a_{10} - 2 a_{8}}{2 a^{9}}$ is :