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JEE Main 202522 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
22 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2025 · 22 January · Evening Shift · Mathematics · Binomial Theorem

Let α, β, γ and be the coefficients of x^7, x^5, x^3 and x...

Let α,β,γand\alpha, \beta, \gamma and\deltabethecoefficientsofbe the coefficients ofx^7, x^5, x^3 and xrespectivelyintheexpansionof(x+x3−1)5+(x−x3−1)5,x>1respectively in the expansion of \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x\gt1. If u and v satisfy the equations αu+βv=18γu+δv=20\begin{aligned} & \alpha u+\beta v=18 \\ & \gamma u+\delta v=20 \end{aligned} then u+vu+v equals :

Options

  1. A5
  2. B3
  3. C4
  4. D8

Correct answer

A. 5

Step-by-step solution

(x+x3−1)5+(x−x3−1)5−[5C0x5+5C1x4(x3−1)+……+5C5(x3−1)5]+[5C0x5−5C1x4(x3−1)+…….+−5C5(x3−1)5]=2[x5+5C2x3(x3−1)+5C4(x3−1)2]=10x7+20x6+2x5−20x4−20x3+10x\begin{aligned} & \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5 \\ & -\left[{ }^5 C_0 x^5+{ }^5 C_1 x^4\left(\sqrt{x^3-1}\right)+\ldots \ldots+{ }^5 C_5\left(\sqrt{x^3-1}\right)^5\right]+ \\ & {\left[{ }^5 C_0 x^5-{ }^5 C_1 x^4\left(\sqrt{x^3-1}\right)+\ldots \ldots .+{ }^{-5} C_5\left(\sqrt{x^3-1}\right)^5\right]} \\ & =2\left[x^5+{ }^5 C_2 x^3\left(x^3-1\right)+{ }^5 C_4\left(x^3-1\right)^2\right] \\ & =10 x^7+20 x^6+2 x^5-20 x^4-20 x^3+10 x \end{aligned} Now α=10,β=2,γ=−20,δ=10\alpha=10, \beta=2, \gamma=-20, \delta=10 Also, 10u+2v=18−20u+10v=20}u=1,v=4u+v=5\begin{aligned} & \left.\begin{array}{c} 10 u+2 v=18 \\ -20 u+10 v=20 \end{array}\right\} u=1, v=4 \\ & u+v=5 \end{aligned}
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