Question:
_____________If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of
$\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6},$ then :
_____________If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of
$\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6},$ then :
Updated On: Jul 15, 2024
- $\alpha+\beta=-30$
- $\alpha-\beta=-132$
- $\alpha+\beta=60$
- $\alpha-\beta=60$
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The Correct Option is B
Solution and Explanation
The correct answer is B:\(\alpha-\beta=-132\)
Given that;
\(\alpha,\beta\) are coefficient of \(x^4 and\space x^2\),and;
\(x+\sqrt{(x^2-1)^6}+x-\sqrt{(x^2-1)^2}\)
\(2\left[^{6}C_{0}.x^{6}+^{6}C_{2}x^{4}\left(x^{2}-1\right)+^{6}C_{4}x^{2}\left(x^{2}-1\right)^{2}+^{6}C_{6}\left(x^{2}-1\right)^{3}\right]\)
\(=2[x^6+15(x^6-x^4)+15x^2(x^4-2x^2+1)+(-1+3x^2-3x^2+x^6)]\)
\(=64x^6-96x^4+36x^2-2\)
as \(\alpha,\beta\) are the coefficient of \(x^4,x^2\)
\(\therefore \alpha = -96\,\&\,\beta = 36\)
\(\therefore \alpha - \beta = -132\)
Given that;
\(\alpha,\beta\) are coefficient of \(x^4 and\space x^2\),and;
\(x+\sqrt{(x^2-1)^6}+x-\sqrt{(x^2-1)^2}\)
\(2\left[^{6}C_{0}.x^{6}+^{6}C_{2}x^{4}\left(x^{2}-1\right)+^{6}C_{4}x^{2}\left(x^{2}-1\right)^{2}+^{6}C_{6}\left(x^{2}-1\right)^{3}\right]\)
\(=2[x^6+15(x^6-x^4)+15x^2(x^4-2x^2+1)+(-1+3x^2-3x^2+x^6)]\)
\(=64x^6-96x^4+36x^2-2\)
as \(\alpha,\beta\) are the coefficient of \(x^4,x^2\)
\(\therefore \alpha = -96\,\&\,\beta = 36\)
\(\therefore \alpha - \beta = -132\)
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.