If alpha and beta are the coefficients of x^{ | vedclass

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(C) Let $f(x) = (x+\sqrt{x^{2}-1})^{6}+(x-\sqrt{x^{2}-1})^{6}$.Using the binomial expansion $(a+b)^n + (a-b)^n = 2[^{n}C_{0}a^n + ^{n}C_{2}a^{n-2}b^2

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$\alpha-\beta=-132$

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(C) Let $f(x) = (x+\sqrt{x^{2}-1})^{6}+(x-\sqrt{x^{2}-1})^{6}$.Using the binomial expansion $(a+b)^n + (a-b)^n = 2[^{n}C_{0}a^n + ^{n}C_{2}a^{n-2}b^2 + ^{n}C_{4}a^{n-4}b^4 + ^{n}C_{6}a^{n-6}b^6]$,we get:$f(x) = 2[^{6}C_{0}x^{6} + ^{6}C_{2}x^{4}(x^{2}-1) + ^{6}C_{4}x^{2}(x^{2}-1)^{2} + ^{6}C_{6}(x^{2}-1)^{3}]$.Expanding the terms:$f(x) = 2[1 \cdot x^{6} + 15(x^{6}-x^{4}) + 15x^{2}(x^{4}-2x^{2}+1) + 1(x^{6}-3x^{4}+3x^{2}-1)]$.$f(x) = 2[x^{6} + 15x^{6}-15x^{4} + 15x^{6}-30x^{4}+15x^{2} + x^{6}-3x^{4}+3x^{2}-1]$.$f(x) = 2[32x^{6} - 48x^{4} + 18x^{2} - 1]$.$f(x) = 64x^{6} - 96x^{4} + 36x^{2} - 2$.Comparing coefficients,$\alpha = -96$ and $\beta = 36$.Therefore,$\alpha - \beta = -96 - 36 = -132$.

If $\alpha$ and $\beta$ are 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:

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