Verified Solution • 2025–26 Session moderate

QJEEM25Ma5fd7: JEE Main 2025 Mathematics - Class 11 JEEM25Ma5fd7 Solution: Let and be the coefficients of and respectively in the…

Let α,β,γ\alpha, \beta, \gamma and δ\delta be the coefficients of x7,x5,x3x^7, x^5, x^3 and xx respectively in the expansion of

amp;(x+x3−1)5+(x−x3−1)5,xgt;1. If u and v satisfy the equations amp;αu+βv=18,amp;γu+δv=20,\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 u+v\mathrm{u+v} equals :

Multiple Choice Options

  • A 4
  • B 3
  • C 5
  • D 8

Step-by-Step Verified Solution

 

amp;(x+x3−1)5+(x−x3−1)5amp;=2{5C0⋅x5+5C2⋅x3(x3−1)+5C4⋅x(x3−1)2}amp;=2{5x7+10x6+x5−10x4−10x3+5x}amp;⇒α=10,β=2,γ=−20,δ=10amp; Now, 10u+2v=18amp;−20u+10v=20amp;⇒u=1,v=4amp;u+v=5\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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