Let and be the coefficients of ands x respectively in the expansion of . If u and v satisfy the equations , then u + v equals : [2025]

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Solution

Q.
Let $α, β, γ$ and $δ$ be the coefficients of $x^{7}, x^{5}, x^{3}$ ands x respectively in the expansion of ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}, x > 1$ . If u and v satisfy the equations $α u + β v = 18, γ u + δ v = 20$ , then u + v equals : [2025]

1 5

2 3

3 4

4 8

Ans.
(1)

We have, ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}$

= $2 {{}^{5}C_{0} x^{5} + {}^{5}C_{2} x^{3} (x^{3} - 1) + {}^{5}C_{4} x {(x^{3} - 1)}^{2}}$

$∵ [{(x + a)}^{n} + {(x - a)}^{n} = 2 ({}^{n}C_{0} x^{n} + {}^{n}C_{2} x^{n - 2} a^{2} + . . .)]$

$= 2 {5 x^{7} + 10 x^{6} + x^{5} - 10 x^{4} - 10 x^{3} + 5 x}$

On comparing, we get

$α = 10, β = 2, γ = - 20, δ = 10$

Now, 10u + 2v = 18 and – 20u +10v = 20

$\Rightarrow$ u = 1, v = 4

Hence, u + v = 5.

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