If - and - be the coefficients of x4 and x2 respectively in the expansion of x-x2-16-x-x2-16- then -A- -132- -30C- -60D- -60

The correct option is A α β= 132(x+√(x^2 1))^6+(x √(x^2 1))^6 =2[^6C_0x^6+ ^6C_2x^4(x^2 1)+ ^6C_4x^2(x^2 1)^2+ ^6C_6(x^2 1)^3] =2[32x^6 48x^4+18x^2 1] ⇒ ...

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Byju's Answer

Standard XII

Mathematics

Binomial Expression

If α and β be...

Question

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $(x + \sqrt{x^{2} - 1})^{6} + (x - \sqrt{x^{2} - 1})^{6}$ , then :

α + β = - 30

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α - β = - 132

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α + β = 60

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α - β = 60

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Solution

The correct option is B $α - β = - 132$
$(x + \sqrt{x^{2} - 1})^{6} + (x - \sqrt{x^{2} - 1})^{6}$

$= 2 [^{6} C_{0} x^{6} +^{6} C_{2} x^{4} (x^{2} - 1) +^{6} C_{4} x^{2} (x^{2} - 1)^{2} +^{6} C_{6} (x^{2} - 1)^{3}]$

$= 2 [32 x^{6} - 48 x^{4} + 18 x^{2} - 1]$

$\Rightarrow α = - 96, β = 36$
$\Rightarrow α - β = - 132$

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If α and β be the coefficients of x4 and x2 respectively in the expansion of (x+√x2−1)6+(x−√x2−1)6, then :

The correct option is B α−β=−132(x+√x2−1)6+(x−√x2−1)6 =2[6C0x6+ 6C2x4(x2−1)+ 6C4x2(x2−1)2+ 6C6(x2−1)3] =2[32x6−48x4+18x2−1] ⇒α=−96,β=36 ⇒α−β=−132

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $(x + \sqrt{x^{2} - 1})^{6} + (x - \sqrt{x^{2} - 1})^{6}$ , then :