If and be the coefficients of x 4 and x 2 respectively in the expansion of ( x + x 2 - 1 ) 6 + ( x - x 2 - 1 ) 6 , then

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

Smart Achievers · Accepted Answer

(2)

${(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}$

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

$= 2 [x^{6} + 15 (x^{6} - x^{4}) + 15 x^{2} (x^{4} - 2 x^{2} + 1) + (- 1 + 3 x^{2} - 3 x^{4} + x^{6})]$

$= 2 (32 x^{6} - 48 x^{4} + 18 x^{2} - 1)$

$∴$ $α = - 96$ $and$ $β = 36$

$∴$ $α - β = - 132$

Solution

Q.
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

1 $α + β = - 30$

2 $α - β = - 132$

3 $α + β = 60$

4 $α - β = 60$

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Solution

Q. 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

1 α+β=-30

2 α-β=-132

3 α+β=60

4 α-β=60