Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1} ight)^5+\left(x-\sqrt{x^3-1} ight)^5, x\gt1$. If u and v satisfy the equations $\begin{aligned} & \alpha u+\beta v=18 \ & \gamma u+\delta v=20…

Question

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1}ight)^5+\left(x-\sqrt{x^3-1}ight)^5, x\gt1$. If u and v satisfy the equations $\begin{aligned} & \alpha u+\beta v=18 \ & \gamma u+\delta v=20 \end{aligned}$ then $u+v$ equals :

Aicharya · Accepted Answer

$\begin{aligned} & \left(x+\sqrt{x^3-1}ight)^5+\left(x-\sqrt{x^3-1}ight)^5 \ & -\left[{ }^5 C_0 x^5+{ }^5 C_1 x^4\left(\sqrt{x^3-1}ight)+\ldots \ldots+{ }^5 C_5\left(\sqrt{x^3-1}ight)^5ight]+ \ & {\left[{ }^5 C_0 x^5-{ }^5 C_1 x^4\left(\sqrt{x^3-1}ight)+\ldots \ldots .+{ }^{-5} C_5\left(\sqrt{x^3-1}ight)^5ight]} \ & =2\left[x^5+{ }^5 C_2 x^3\left(x^3-1ight)+{ }^5 C_4\left(x^3-1ight)^2ight] \ & =10 x^7+20 x^6+2 x^5-20 x^4-20 x^3+10 x \end{aligned}$ Now $\alpha=10, \beta=2, \gamma=-20, \delta=10$ Also, $\begin{aligned} & \left.\begin{array}{c} 10 u+2 v=18 \ -20 u+10 v=20 \end{array}ight\} u=1, v=4 \ & u+v=5 \end{aligned}$

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the…

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