If the co-efficient of x^9 in (ax^3 + 1/(β x) )^11 and the co-efficient of x^-9 in (ax - 1/(β x^3) )^11 are equal, then (αβ)^2 is equal to ________.

If the co-efficient of x^9 in (ax^3 + 1/(β x) )^11 and the… — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 29 Jan · Shift 1 · numerical answer

If the co-efficient of

x^9

\left(ax^3 + \frac{1}{\beta x}\right)^{11}

and the co-efficient of

x^{-9}

\left(ax - \frac{1}{\beta x^3}\right)^{11}

are equal, then

(\alpha\beta)^2

is equal to ________.

Show answer and solution

Answer: 01

Solution

In (ax³ + 1/(βx))¹¹, the general term is C(11,r)·(ax³)^(11-r)·(1/(βx))^r = C(11,r)·a^(11-r)/β^r · x^(33-3r-r) = C(11,r)·a^(11-r)/β^r · x^(33-4r).

For x⁹: 33-4r = 9 → r = 6. Coefficient = C(11,6)·a⁵/β⁶.

In (ax - 1/(βx³))¹¹, general term is C(11,r)·(ax)^(11-r)·(-1/(βx³))^r = C(11,r)·(-1)^r·a^(11-r)/β^r · x^(11-r-3r) = C(11,r)·(-1)^r·a^(11-r)/β^r · x^(11-4r).

For x^⁻⁹: 11-4r = -9 → r = 5. Coefficient = C(11,5)·(-1)⁵·a⁶/β⁵ = -C(11,5)·a⁶/β⁵.

Setting equal: C(11,6)·a⁵/β⁶ = -C(11,5)·a⁶/β⁵. Since C(11,6)=C(11,5), this gives 1/β = -a, so aβ = -1, and (αβ)² = 1.

[JEE Main 2023 · 29 Jan · Shift 1]

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