JEE MAIN - Mathematics (2025 - 22nd January Evening Shift - No. 5)
Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of
$$\begin{aligned}
& \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1 \text {. If } u \text { and } v \text { satisfy the equations } \\\\
& \alpha u+\beta v=18, \\\\
& \gamma u+\delta v=20,
\end{aligned}$$
then $\mathrm{u+v}$ equals :
4
3
5
8
Explanation
$$\begin{aligned}
& \left(\mathrm{x}+\sqrt{\mathrm{x}^3-1}\right)^5+\left(\mathrm{x}-\sqrt{\mathrm{x}^3-1}\right)^5 \\
& =2\left\{{ }^5 \mathrm{C}_0 \cdot \mathrm{x}^5+{ }^5 \mathrm{C}_2 \cdot \mathrm{x}^3\left(\mathrm{x}^3-1\right)+{ }^5 \mathrm{C}_4 \cdot \mathrm{x}\left(\mathrm{x}^3-1\right)^2\right\} \\
& =2\left\{5 \mathrm{x}^7+10 \mathrm{x}^6+\mathrm{x}^5-10 \mathrm{x}^4-10 \mathrm{x}^3+5 \mathrm{x}\right\} \\
& \Rightarrow \alpha=10, \beta=2, \gamma=-20, \delta=10 \\
& \text { Now, } 10 \mathrm{u}+2 \mathrm{v}=18 \\
& -20 \mathrm{u}+10 \mathrm{v}=20 \\
& \Rightarrow \mathrm{u}=1, \mathrm{v}=4 \\
& \mathrm{u}+\mathrm{v}=5
\end{aligned}$$
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