Questions & Answers

Question

Answers

( a ) 3

( b ) -3

( c ) 6

( d ) -6

Answer
Verified

As in question it is clearly given that, $\alpha $ and $\beta $ be the roots of equation ${{x}^{2}}-6x-2=0$ .

then, we can put x = $\alpha $ and x = $\beta $,

so, putting x = $\alpha $ in ${{x}^{2}}-6x-2=0$, we get

${{\alpha }^{2}}-6\alpha -2=0$…..( i )

so, putting x = $\beta $ in ${{x}^{2}}-6x-2=0$, we get

${{\beta }^{2}}-6\beta -2=0$…. ( ii )

Multiplying equation ( i ) by ${{\alpha }^{8}}$, we get

${{\alpha }^{8}}({{\alpha }^{2}}-6\alpha -2)=0({{\alpha }^{8}})$

$({{\alpha }^{10}}-6{{\alpha }^{9}}-2{{\alpha }^{8}})=0$

${{\alpha }^{10}}=6{{\alpha }^{9}}+2{{\alpha }^{8}}$……( iii )

Multiplying equation ( i ) by ${{\beta }^{8}}$, we get

${{\beta }^{8}}({{\beta }^{2}}-6\beta -2)=0({{\beta }^{8}})$

$({{\beta }^{10}}-6{{\beta }^{9}}-2{{\beta }^{8}})=0$

${{\beta }^{10}}=6{{\beta }^{9}}+2{{\beta }^{8}}$…..( iv )

Subtracting ( iv ) from ( iii ), we get

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

On simplifying we get

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

As, in question it is given that ${{a}_{n}}={{\alpha }^{n}}-{{\beta }^{n}}$, so ${{a}_{9}}={{\alpha }^{9}}-{{\beta }^{9}}$and ${{a}_{10}}={{\alpha }^{10}}-{{\beta }^{10}}$ and ${{a}_{8}}={{\alpha }^{8}}-{{\beta }^{8}}$

${{a}_{10}}=6({{a}_{9}})+2({{a}_{8}})$

Moving, $2({{a}_{8}})$ from right hand side to left hand side, we get

${{a}_{10}}-{{a}_{8}}=6{{a}_{9}}$

Re – writing above equation, we get

${{a}_{10}}-{{a}_{8}}=3\cdot 2{{a}_{9}}$

Using cross-multiplication, we get

$\dfrac{{{a}_{10}}-{{a}_{8}}}{2{{a}_{9}}}=3$