"If $\alpha$ and $\beta$ be the coefficients of x4 and x2 respectively in the expansion of $${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then"

Question

Accepted Answer

"(x+a)ⁿ+ (x – a)ⁿ = 2(T₁ + T₃ + T₅ +.....)

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

= 2[T₁ + T₃ + T₅ + T₇ ]

= 2[⁶C₀ x⁶ + ⁶C₂ x⁴(x² – 1) + ⁶C₄ x²(x² –1)² + ⁶C₆ x⁰(x²–1)³]

= 2[x⁶+ 15(x⁶ – x⁴) + 15x² (x⁴ + 1 –2x²) + (x⁶ – 3x⁴ +3x² –1)]

= 2[x⁶(2 + 15 + 15 + 1) + x⁴(–15 – 30 –3) + x²(15 + 3)]

Coefficient of x⁴ = $\alpha$ = -96

And coefficient of x² = $\beta$ = 36

$\therefore$ $\alpha - \beta = - 96 - 36 = -132$"

If $\alpha$ and $\beta$ be the coefficients of x⁴ and x² respectively in the expansion of
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then

Solution

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If $\alpha$ and $\beta$ be the coefficients of x4 and x2 respectively in the expansion of $${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then

Solution

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More Binomial Theorem questions

Drill 25 more like these. Every day. Free.

If $\alpha$ and $\beta$ be the coefficients of x⁴ and x² respectively in the expansion of
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then