Question

The value of ab if coefficient of $x^7$ in ${(a{x}^2+\frac{1}{bx})}^{11}$ and coefficient of $x^{-7}$ in ${(ax-\frac{1}{b{x}^2})}^{11}.$ is equal to:

Answer 1

Coefficient of $x^7$ in ${(a{x}^2+\frac{1}{bx})}^{11}$ is ${}^{11}{C}_5\frac{a^6}{b^5}$

And Coefficient of $x^{-7}$ in ${(ax-\frac{1}{b{x}^2})}^{11}$ is ${}^{11}{C}_6\frac{a^5}{b^6}$

coefficients are equal

${}^{11}\text{/}{C}_5\frac{a^6}{b^5}=$ ${}^{11}\text{/}{C}_6\frac{a^5}{b^6}$ $[{}^{11}{C}_5={}^{11}{C}_6{,}^n{C}_r={}^n{C}_{n-1}]$

$\Rightarrow a=\frac{1}{b}$

$\Rightarrow ab=1$.