Verified Solution • 2025–26 Session moderate

QJEEM23Ma4d6b: JEE Main 2023 Mathematics - Class 11 JEEM23Ma4d6b Solution: If the co-efficient of in and the co-efficient of in are equal, …

If the co-efficient of x9x^9 in (αx3+1βx)11{\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}} and the co-efficient of x−9x^{-9} in (αx−1βx3)11{\left( {\alpha x - {1 \over {\beta {x^3}}}} \right)^{11}} are equal, then (αβ)2(\alpha\beta)^2 is equal to ___________.

Step-by-Step Verified Solution

Coefficient of x9\mathrm{x}^{9} in (αx3+1βx)=11C6⋅α5β6\left(\alpha x^{3}+\frac{1}{\beta x}\right)={ }^{11} C_{6} \cdot \frac{\alpha^{5}}{\beta^{6}}

∵\because Both are equal

∴11C6⋅α5β6=−11C5⋅α6β5\therefore \frac{11}{C_{6}} \cdot \frac{\alpha^{5}}{\beta^{6}}=-\frac{11}{C_{5}} \cdot \frac{\alpha^{6}}{\beta^{5}}

⇒1β=−α\Rightarrow \frac{1}{\beta}=-\alpha

⇒αβ=−1\Rightarrow \alpha \beta=-1

⇒(αβ)2=1\Rightarrow(\alpha \beta)^{2}=1

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring